diff --git a/.gitattributes b/.gitattributes index a60f5ac8356fb45cb3a14984db8a339aec631a1b..52a4778507f95f6f58e1849988f772935037946f 100644 --- a/.gitattributes +++ b/.gitattributes @@ -209,3 +209,4 @@ h9E0T4oBgHgl3EQfpgF0/content/2301.02540v1.pdf filter=lfs diff=lfs merge=lfs -tex c9E3T4oBgHgl3EQfeQoX/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text w9FJT4oBgHgl3EQfgiwX/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text X9FRT4oBgHgl3EQfODd9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +AtAyT4oBgHgl3EQfRvdA/content/2301.00071v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/0tAzT4oBgHgl3EQfRPvU/content/tmp_files/2301.01214v1.pdf.txt b/0tAzT4oBgHgl3EQfRPvU/content/tmp_files/2301.01214v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..26da472f833874482e545a61aa65f72812e8753f --- /dev/null +++ b/0tAzT4oBgHgl3EQfRPvU/content/tmp_files/2301.01214v1.pdf.txt @@ -0,0 +1,2052 @@ +Comparison of tree-based ensemble algorithms for merging satellite +and earth-observed precipitation data at the daily time scale +Georgia Papacharalampous1, Hristos Tyralis2, Anastasios Doulamis3, Nikolaos Doulamis4 +1 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, +National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece +(papacharalampous.georgia@gmail.com, https://orcid.org/0000-0001-5446-954X) +2 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, +National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece +(montchrister@gmail.com, +hristos@itia.ntua.gr, +https://orcid.org/0000-0002-8932- +4997) +3 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, +National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece +(adoulam@cs.ntua.gr, https://orcid.org/0000-0002-0612-5889) +4 Department of Topography, School of Rural, Surveying and Geoinformatics Engineering, +National Technical University of Athens, Iroon Polytechniou 5, 157 80 Zografou, Greece +(ndoulam@cs.ntua.gr, https://orcid.org/0000-0002-4064-8990) +Abstract: Merging satellite products and ground-based measurements is often required +for obtaining precipitation datasets that simultaneously cover large regions with high +density and are more accurate than pure satellite precipitation products. Machine and +statistical learning regression algorithms are regularly utilized in this endeavour. At the +same time, tree-based ensemble algorithms for regression are adopted in various fields +for solving algorithmic problems with high accuracy and low computational cost. The +latter can constitute a crucial factor for selecting algorithms for satellite precipitation +product correction at the daily and finer time scales, where the size of the datasets is +particularly large. Still, information on which tree-based ensemble algorithm to select in +such a case for the contiguous United States (US) is missing from the literature. In this +work, we conduct an extensive comparison between three tree-based ensemble +algorithms, specifically random forests, gradient boosting machines (gbm) and extreme +gradient boosting (XGBoost), in the context of interest. We use daily data from the +PERSIANN (Precipitation Estimation from Remotely Sensed Information using Artificial +Neural Networks) and the IMERG (Integrated Multi-satellitE Retrievals for GPM) gridded + +2 + +datasets. We also use earth-observed precipitation data from the Global Historical +Climatology Network daily (GHCNd) database. The experiments refer to the entire +contiguous US and additionally include the application of the linear regression algorithm +for benchmarking purposes. The results suggest that XGBoost is the best-performing tree- +based ensemble algorithm among those compared. They also suggest that IMERG is more +useful than PERSIANN in the context investigated. +Keywords: contiguous US; gradient boosting machines; IMERG; machine learning; +PERSIANN; random forests; remote sensing; satellite precipitation correction; spatial +interpolation; XGBoost +1. +Introduction +Machine and statistical learning algorithms (e.g., those documented in Hastie et al. 2009; +James et al. 2013; Efron and Hastie 2016) are increasingly adopted for solving a variety of +practical problems in hydrology (Dogulu et al. 2015; Xu et al. 2018; Quilty et al. 2019; +Curceac et al. 2020; Jehn et al. 2020; Quilty and Adamowski 2020; Rahman et al. 2020; +Althoff et al. 2021; Fischer and Schumann 2021; Papacharalampous and Tyralis 2022b) +and beyond (Ahmed et al. 2010; García-Gutiérrez et al. 2015; Goetz et al. 2015; Asri et al. +2016; Idowu et al. 2016; Bahl et al. 2018; Feng et al. 2020; Khanam and Foo 2021; Rustam +et al. 2021; Bamisile et al. 2022). Among the entire pool of such algorithms, the tree-based +ensemble ones (i.e., those combining decision trees under properly designed ensemble +learning strategies; Sagi and Rokach 2018) are of special interest for many practical +problems, as they can offer high predictive performance with low computational cost, +among their remaining benefits (Tyralis et al. 2019; Tyralis and Papacharalampous 2021). +Still, the known theoretical properties of the various tree-based ensemble algorithms +(including random forests, gradient boosting machines − gbm and extreme gradient +boosting – XGBoost; Breiman 2001, Friedman 2001, Chen and Guestrin 2016) cannot +support the selection of the most appropriate one among them for each practical problem. +Instead, such a selection could rely on attentively designed empirical comparisons. Thus, +such comparisons of tree-based ensemble algorithms are conducted with increasing +frequency in various scientific fields (Adler et al. 2011; Ahmad et al. 2018; Fan et al. 2018; +Besler et al. 2019; Ahmad and Zhang 2020; Ampomah et al. 2020; Liu et al. 2020; Rahaman +et al. 2021; Ziane et al. 2021; Khorrami et al. 2022; Mittendorf e al. 2022; Park and Kim +2022; Wei et al. 2022; Yao et al. 2022). + +3 + +Tree-based ensemble algorithms are regularly applied and compared to other machine +and statistical learning algorithms for the task of merging satellite products and ground- +based measurements. This task is the general focus of this work, together with the general +concept of tree-based ensemble algorithms, and is commonly executed in the literature in +the direction of obtaining precipitation datasets that cover large geographical regions +with high density and, simultaneously, are more accurate than pure satellite precipitation +products. The importance of this same task could be perceived through the inspection of +the major research topics appearing in the hydrological literature (see, e.g., those +discussed in Montanari et al. 2013, Blöschl et al. 2019). Relevant examples of applications +and comparisons are available in He et al. (2016), Meyer et al. (2016), Baez-Villanueva et +al. (2020), Chen et al. 2021, Nguyen et al. (2021), Shen and Yong (2021), Zhang et al. +(2021), Fernandez-Palomino et al. (2022), Lei et al. (2022), Lin et al. (2022), Zandi et al. +(2022) and Militino et al. (2023). +These examples refer to various temporal resolutions and many different geographical +regions around the globe (see also the reviews by Hu et al. 2019 and Abdollahipour et al. +2022), with the daily temporal resolution and the Unites States (US) being frequent cases. +Nonetheless, a relevant comparison of tree-based ensemble algorithms for the latter +temporal resolution and the latter geographical region is missing from the literature, with +the closest investigations at the moment being those available in the work by Lei et al. +(2022), which however focus on China. In this work, we fill this specific literature gap. +Notably, the selection of the most accurate regression algorithm from the tree-based +ensemble family could be particularly useful at the daily temporal scale, in which the size +of the datasets for large geographical areas might impose significant limitations on the +application of other accurate machine and statistical learning regression algorithms due +to their large computational cost. +The remainder of the paper is structured as follows: Section 2 describes the tree-based +ensemble algorithms compared in this work. It also describes a machine learning metric +that is utilized for ensuring some degree of explainability. Moreover, Section 3 presents +the data retrieved and utilized for the comparisons. The same section outlines the way in +which the tree-based ensemble algorithms are compared with each other. Furthermore, +Sections 4, 5 and 6 present the results, provide their discussion in light of the literature +and conclude the work, respectively. Lastly, Appendix A provides statistical software +information that assures the work’s reproducibility. + +4 + +2. +Methods +Random forests, gradient boosting machines (gbm) and extreme gradient boosting +(XGBoost) were applied in a cross-validation setting (see Section 3.2) for conducting an +extensive comparison in the context of merging gridded satellite products and gauge- +based measurements at the daily time scale. Additionally, the linear regression algorithm +was applied in the same setting for benchmarking purposes. In this section, we provide +brief descriptions of the four afore-mentioned algorithms. Extended descriptions are out +of the scope of this work, as they are widely available in the machine and statistical +learning literature (e.g., in Hastie et al. 2009; James et al. 2013; Efron and Hastie 2016). +2.1 Linear regression +The results of this work are reported in terms of relative scores (see Section 3.3). These +scores were computed with respect to the linear regression algorithm, which models the +dependent variable as a linear weighted sum of the predictor variables (Hastie et al. 2009, +pp 43–55). A squared error scoring function facilitates this algorithm’s fitting. +2.2 Random forests +Random forests (Breiman 2001) are the most commonly used algorithm in the context of +merging gridded satellite products and gauge-based measurements (see the examples in +Hengl et al. 2018). A detailed description of this algorithm can be found in Tyralis et al. +(2019b), along with a systematic review of its application in water resources. Notably, +random forests are an ensemble learning algorithm and, more precisely, an ensemble of +regression trees that is based on bagging (acronym for “bootstrap aggregation”) but with +an additional randomization procedure. The latter aims at reducing overfitting. In this +work, random forests were implemented with all their hyperparameters kept as default. +For instance, the number of trees was equal to 500. +2.3 Gradient boosting machines +Gradient boosting machines (Friedman 2001, Mayr et al. 2014) are also an ensemble +learning algorithm that is herein used with regression trees as base learners. The main +concept behind this ensemble algorithm and, more generally, behind all the boosting +algorithms (including the one described in Section 2.4) is the iterative training of new +base learners using the errors of previously trained base learners (Natekin and Knoll +2013, Tyralis and Papacharalampous 2021). For gradient boosting machines, a gradient + +5 + +descent algorithm adapts the loss function for achieving optimal fitting. This loss function +is the squared error scoring function herein. Consistency with the respect to the +implementation of random forests is ensured by setting the number of trees equal to 500. +The remaining hyperparameters were kept as default. +2.4 Extreme gradient boosting +Extreme gradient boosting (XGBoost; Chen and Guestrin 2016) consists the third tree- +based ensemble learning and the second boosting algorithm implemented in this work. In +the implementations of this work, all the hyperparameters were kept as default, except +for the maximum number of iterations that were set to 500. +Aside from applying XGBoost in a cross-validation setting for its comparison to the +remaining algorithms, we also utilized it with the same hyperparameter values for +assuring some degree of explainability in terms of variable importance under the more +general explainable machine learning culture (Linardatos et al. 2020, Roscher et al. 2020, +Belle and Papantonis 2021). Specifically, we computed the gain importance metric, which +is available in the XGBoost algorithm. This metric assesses the “fractional contribution of +each feature to the model based on the total gain of this feature’s splits”, with higher values +indicating more important features (Chen et al. 2022c). +3. +Data and application +3.1 Data +For our experiments, we retrieved and used daily earth-observed precipitation, gridded +satellite precipitation and elevation data for the gauged locations and grid points shown +in Figures 1 and 2. + +6 + + +Figure 1. Map of the geographical locations of the earth-located stations that offered data +for this work. + +-120 +-100 +-80 +Longitude (°)3itude97 + + + +Figure 2. Maps of the geographical locations of the points composing the (a) PERSIANN +and (b) IMERG grids utilized in this work. + +-120 +-100 +80 +Longitude (°)(b)8(a)&58 + +3.1.1 Earth-observed precipitation data +Daily precipitation totals from the Global Historical Climatology Network daily (GHCNd) +(Durre et al. 2008, 2010, Menne et al. 2012) were used for comparing the algorithms. +More precisely, data from 7 264 earth-located stations spanning across the contiguous +United States (see Figure 1) were extracted. This data cover the two-year time period +2014−2015. Data retrieval was made from the website of the NOAA National Climatic Data +Center (https://www1.ncdc.noaa.gov/pub/data/ghcn/daily; assessed on 2022-02-27). +3.1.2 Satellite precipitation data +For comparing the algorithms, we additionally used gridded satellite daily precipitation +data from the current operational PERSIANN (Precipitation Estimation from Remotely +Sensed Information using Artificial Neural Networks) system (see the geographical +locations of the extracted PERSIANN grid with a spatial resolution of 0.25 degree x 0.25 +degree in Figure 2a) and the GPM IMERG (Integrated Multi-satellitE Retrievals) Late +Precipitation L3 1 day 0.1 degree x 0.1 degree V06 (see the geographical locations of the +extracted IMERG grid in Figure 2b). These two gridded satellite precipitation databases +were developed by the Centre for Hydrometeorology and Remote Sensing (CHRS) at the +University of California, Irvine (UCI) and the NASA Goddard Earth Sciences (GES) Data +and Information Services Center (DISC), respectively. More precisely, the PERSIANN data +were retrieved from the website of the Center for Hydrometeorology and Remote Sensing +(CHRS) (https://chrsdata.eng.uci.edu; assessed on 2022-03-07) and the IMERG data were +retrieved from the website of NASA (National Aeronautics and Space Administration) +Earth Data (https://doi.org/10.5067/GPM/IMERGDL/DAY/06; assessed on 2022-12- +10). The extracted data cover the entire contiguous United States at the two-year time +period 2014−2015. +3.1.3 Elevation data +Elevation is a key predictor variable for many hydrological processes (Xiong et al. 2022). +Therefore, we estimated its value for all the geographical locations shown in Figures 1 +and 2. For this estimation, we relied on the Amazon Web Services (AWS) Terrain Tiles +(https://registry.opendata.aws/terrain-tiles; assessed on 2022-09-25). +3.2 Validation setting and predictor variables +To formulate the regression settings of this work, we first defined earth-observed daily + +9 + +total precipitation at a point of interest (which could be station 1 in Figure 3) as the +dependent variable. Then, we adopted procedures proposed in Papacharalampous et al. +(2023) to compute the observations of possible predictor variables. Separately for each +of the two satellite precipitation grids (see Figure 2), we determined the closest four grid +points to each ground-based station from those depicted in Figure 1. We also computed +the distances di, i = 1, 2, 3, 4 from these grid points and indexed the latter as Si, i = 1, 2, 3, +4 based on the following order: d1 < d2 Y +lax/oplax ∆1-bilimits +X +ш +F Y “ X +Ð> +F Y “ X +Ј +F Y “ X +Ñ> +F Y +11) +finite direct sums +‘k +s“1xs “ śk +s“1 xs “ šk +s“1 xs +general lax bilimits +Àlax +s:S Xs “ laxlims:S Xs “ laxcolims:S Xs +12) +matrices p m11 m12 +m21 m22 q +lax matrices +¨ +˚ +˝ +M11 +M12 +M21 +M22 +˛ +‹‚ +13) +matrix multiplication +pnmqus “ ř +t“t1pnut1 ˝ mtsq +lax matrix multiplication +pNMqus “ colimγ : tÑt1pNut1 ˝ γ ˝ Mtsq +14) +matrix multiplication, reparameterized +pnmqus “ ř +tp´1qtpnut ˝ mtsq +lax-oplax matrix multiplication +pNMqus “ tottpNut ˝ Mtsq +Accepting our basic premise that abelian groups are to be categorified by stable 8- +categories, Rule 9) requires no further comment. Rule 8) is a convenient intermediate step, +categorifying the situation where the addition on hom-sets does not necessarily have inverses; +just like the uncategorified case, many basic lemmas are most naturally expressed in this +generality leading to the notion of lax semi-additive p8, 2q-category. +The direct sum of abelian groups is both a categorical product and a categorical coproduct, +a universal property that is taken as the definition in general additive categories. Rule 11) +states that the same definition can be categorified, as long as we replace finite products and +coproducts, i.e., limits and colimits indexed by finite discrete categories S “ t1, . . . , ku, with +lax limits and colimits indexed by arbitrary 8-categories S. Apart from this change, the +theory is exactly analogous: if the hom-categories have colimits (categorifying addition) then +such lax limits and colimits always agree if they exits, yielding the notion of lax bilimits. +9 + +When the hom-categories are stable, we can say even more for certain special shapes, such +as S “ ∆1: in this case all four possible universal 2-categorical constructions (lax/oplax, +limit/colimit) associated to an S-diagram X FÝÑ Y agree with each other. +An extremely convenient feature of additive categories is that maps m: x1 ‘ x2 Ñ y1 ‘ y2 +between direct sums can be represented as matrices of the form +m “ +ˆm11 : x1 Ñ y1, +m12 : x2 Ñ y1 +m21 : x1 Ñ y2, +m22 : x2 Ñ y2 +˙ +. +Composing such maps then just amounts to the usual matrix multiplication. Rule 13) shows +how the usual matrix multiplication formula can be categorified, yielding an analogous theory +of matrices indexed in each coordinate not by a finite set but by arbitrary 8-categories. These +matrices are just a dependent version of bimodules, which by Morita theory encode functors +between module categories. +In §1.1 we have already seen how it is conceptually easier to categorify subtraction rather +than addition and more generally alternating rather than ordinary sums. In the lax additive +setting we see a similar feature, expressed in Rule 14), where a convenient “coordinate change” +yields a much nicer formula for the categorified matrix multiplication when we reparameterize +it to use alternating rather than ordinary sums. Categorically speaking this reparameteriza- +tion involves the identification of lax and oplax limits and is therefore only available for certain +special indexing categories such as ∆1. +1.7 +Relations to previous work +As already indicated in the very beginning, the idea to consider categorical complexes is, +in principle, by no means new: short exact categorical sequences in the form of localization +sequences already play a central role in algebraic K-theory. Also the directed pullback con- +struction has already been used in this context, cf. [LT19], where (implicitly) also the product +totalization of a square is being considered. +As the authors have told us, they have also +observed a type of Beck–Chevalley phenomenon, that will be relevant for applications to K- +theory. A version of categorical complexes for non-stable symmetric monoidal 8-categories +was introduced in [Lur09b], but the motivation in this context is quite different from ours. +The Calabi-Yau structures introduced in §6 can be interpreted in the framework of [ST16] +but our examples have not been considered there. The work in progress [AGHJ22] studies +Fukaya categories for Landau-Ginzburg models with multi-potentials. As indicated to us by +the authors, this theory will provide a means to introduce cubical categorical diagrams in +terms of partially wrapped Fukaya categories which one would expect to mirror the cubical +diagrams in §7.1. Variants of this homological mirror symmetry conjecture for categorical +cubes were already recorded in [Lee22]. The expected relation to our formulation will be +explained below. The idea to use matrices to describe coordinate changes for semiorthogonal +decompositions already appears in [DKS23], similar in spirit to the lax matrices in §8. +1.8 +Acknowledgements +We are very grateful to Jeffrey Hicks, Sukjoo Lee, Nick Sheridan, and the participants of +the Edinburgh Hodge Seminar in general for interesting conversations about the topic of this +work. In particular, we thank them for making us aware of the work in progress on Fukaya +10 + +categories of multi-potentials. Many thanks to Denis Auroux, Sheel Ganatra, Andrew Han- +lon, and Maxim Jeffs for taking the time and effort to provide us with a report on their work +in progress [AGHJ22] (and to Nick Sheridan for organizing the lecture series). We further +thank Jacob Lurie for interesting remarks on categorical complexes. We also thank Federico +Barbacovi for discussions about spherical functors. T.D. thanks M. Kapranov and V. Schecht- +man for many inspiring discussions on perverse sheaves and schobers, in particular, this work +draws substantial inspiration from their perspectives proposed in [KS14] and related subse- +quent work. T.W. thanks Claudia Scheimbauer for conversations in the context of their joint +work that helped shape many of the key ideas regarding higher categorical additivity. M.C. +thanks the Hausdorff Research Institute for the hospitality during his stay, during which +part of this work was written. M.C. was funded by the Deutsche Forschungsgemeinschaft +(DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 +– 390685813. M.C. and T.D. acknowledge support by the Deutsche Forschungsgemeinschaft +under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. T.D. +acknowledges support of the VolkswagenStiftung through the Lichtenberg Professorship Pro- +gramme. T.W acknowledges support by the SFB 1085 – Higher Invariants, funded by the +DFG. +2 +Basic concepts +2.1 +Stable 8-categories +As a model for “enhanced triangulated categories”, we will use stable 8-categories as intro- +duced in [Lur17] which, along with [Lur09a] and [GR17], are our standard references. It will +be most convenient to work with presentable stable 8-categories – the 8-category formed by +these with colimit–preserving functors will be denoted by St. +For applications, it is also important to consider presentable stable 8-categories linear +over a given field, or more generally over a commutative ring spectrum, denoted k. Following +[Lur17], we define these as modules in St over the symmetric monoidal stable 8-category Modk +of k-module spectra. The resulting 8-category of k-linear stable 8-categories will be denoted +by Stk. Given objects C, D P Stk, we denote by StkpC, Dq the 8-category of morphisms in +Stk, i.e., colimit–preserving k-linear functors. The 8-category StkpC, Dq can be regarded as +an object of Stk and thus defines an internal hom. Further, Stk admits a symmetric monoidal +structure (adjoint to the internal hom) and we denote the corresponding binary tensor product +by b. +Any compactly generated 8-category C P Stk is dualizable with respect to this monoidal +structure and we denote its dual by C_. For the subcategory C0 Ă C of compact objects, the +8-category C is equivalent to the 8-category IndpC0q of Ind-objects of C0, and we have +C_ » StkpC, Modkq » IndpCop +0 q. +For a morphism F : C Ñ D in Stk between compactly generated 8-categories, we denote its +dual by +F _ : D_ ÝÑ C_. +By the adjoint functor theorem, any functor F in Stk admits a right adjoint, denoted by +F R. The adjoint F R is not necessarily a morphism in Stk, since it may not preserve colimits. +11 + +The condition that F R preserves colimits and admits a k-linear structure is equivalent to the +condition that F preserves compact objects. When we speak of a k-linear adjoint, we mean +that the adjoint also lies in Stk, and the adjunction is in this case given in a 2-categorical +sense in Stk. +2.2 +Complexes of stable 8-categories +We define a chain complex of stable 8-categories to consist of a sequence +. . . +A2 +A1 +A0 +. . . +d +d +d +d +of morphisms d: Ai Ñ Ai´1 in Stk satisfying d2 » 0. Note, that the existence of an equiva- +lence d2 » 0 simply amounts to the statement that d2 is a zero object in the stable 8-category +StkpAi, Ai´2q and the space of zero objects is contractible. Thus, in order to specify a com- +plex of stable 8-categories, one only needs to provide the list of differentials, and verify the +condition d2 » 0. In particular, in contrast to the notion of a complex of objects in a fixed +stable 8-category, no higher coherence data is involved. We justify this a bit more formally: +Definition 2.2.1. Following [Wal22], we define the category +Ch :“ Z˚{ „ +where Z˚ denotes the category obtained from Z by adjoining a zero object ˚ and Ch is the +quotient obtained by identifying each composite d2 : i Ñ i ` 2 with the 0 morphism (i.e. the +unique composite i Ñ ˚ Ñ i ` 2). We then define a coherent chain complex in a pointed +8-category C to be a functor Chop Ñ C that preserve zero objects. +Furthermore, we denote by +G :“ ¨ ¨ ¨ >t´1u t´1 Ñ 0u >t0u t0 Ñ 1u >t1u t1 Ñ 2u >t2u ¨ ¨ ¨ Ă Z +(2.2.2) +the 1-skeleton of (the nerve of) Z. By definition, a map of simplicial sets G Ñ K just consists +of a sequence of composable edges in the simplicial set K. +Lemma 2.2.3. We regard Stk as an 8-category (by discarding noninvertible 2-morphisms) +and consider further the 8-categories +• Fun˚pChop, Stkq of functors preserving zero objects, +• Funpd2qpZop, Stkq of functors such that each d2 is equivalent to the 0 functor, +• Funpd2qpGop, Stkq of functors such that each d2 is equivalent to the 0 functor. +The restriction functors +Fun˚pChop, Stkq ÝÑ Funpd2qpZop, Stkq ÝÑ Funpd2qpGop, Stkq +are equivalences of 8-categories. +12 + +Proof sketch. The rightmost restriction functor is an equivalence because the inclusion G Ă Z +is an inner anodyne extension of simplicial sets. +Extending a functor F : Zop Ñ Stk to Chop involves various lifting problems in the hom- +spaces StkpFpiq, Fpjqq. If we assume that for each k ě 2 the edge Fpdkq is equivalent to +zero, then all of these lifting problems take place in the zero component of the mapping space +StkpFpiq, Fpjqq. Hence the statement follows from the key fact that for each A, B in Stk, the +zero connected component of StkpA, Bq is just the groupoid of zero functors A Ñ B, which is +contractible. We omit the remaining details. +In virtue of Lemma 2.2.3, we may always extend a given complex in Stk presented as a +sequence of differentials d satisfying d2 » 0, to a fully coherent complex in an essentially +unique way. Further, we will sometimes want to turn a coherent complex concentrated in +degrees n ě i ě 0 into a cubical diagram: To this end, we may define the functor +q: Ppt1, ..., nuq ÝÑ Ch, M ÞÑ +$ +’ +& +’ +% +0 +if M “ H, +i +if M “ t1, ..., iu, +˚ +else +and consider the pullback +q˚ : Fun˚pChop, Stkq ÝÑ Fun˚pPpt1, ..., nuqop, Stkq. +(2.2.4) +For the sake of brevity, we will refer to a complex in Stk as a categorical complex. +Definition 2.2.5. We denote by ChpStkq the full subcategory of FunpZop, Stkq spanned by +categorical complexes. +An categorical n-complex is a functor A˚ : Zop,n Ñ Stk, meaning collections of commuting +n differentials +di : Aa1,...,ai,...,an ÝÑ Aa1,...,ai´1,...,an , +satisfying that d2 +i » 0 for all 1 ď i ď n. +Definition 2.2.6. Let n ě 1. +(1) We denote by ChnpStkq the full subcategory of FunppZopqn, Stkq spanned by categorical +n-complexes. +(2) A categorical n-complex A˚ is called a categorical n-cube if Aj » 0 for all j P pZopqn +and j R In, where I “ r1sop “ t1 Ñ 0u Ă Zop. We denote by CubenpStkq Ă ChnpStkq the +full subcategory spanned by categorical n-cubes. +2.3 +Partially lax limits +There are different types of limits of a given diagram tAiui:I in an p8, 2q-category. Namely, +for each triangle +lim +Ai +Aj +in the limit cone, one can require either of the following: +13 + +(i) strict: it commutes up to natural equivalence: +lim +Ai +Aj +(ii) lax: it commutes up to a possibly noninvertible 2-morphism as in +lim +Ai +Aj +(iii) oplax: it commutes up to a possibly noninvertible 2-morphism as in +lim +Ai +Aj +The oplax triangle may be modelled as an amalgamate of a strict and a lax triangle as in +lim +Ai +Aj +Aj +id +so that it suffices to distinguish between strict and lax triangles. One way to encode this +“partial laxness” is to simply mark those edges in the diagram tAiui:I over which we require +the cone to strictly commute. The resulting notion of marked limit has been developed and +studied by several groups of authors, see, e.g., [AGS22] for a formal definition. +Throughout this work, we will use partially lax universal constructions to describe stable 8 +categories and functors among them (see §2.3). In §3 through §7, the focus lies on concreteness +and explicit examples, so that we take a hands-on approach and use explicit models for the +relevant constructions. The framework of 8-categories is sufficient to describe the needed +universal properties, even though we often formulate them in p8, 2q-categorical terms. +In §8 we will discuss (some of) these universal constructions from a more foundational +point of view within a framework of lax additive p8, 2q-categories (also see §2.4). Here, we do +not make use of a concrete model for p8, 2q-categories, but rather provide a list of features +that will be used (see §8.2). +2.4 +Lax additivity +For given abelian groups A and B, their product and coproduct are characterized by the +universal cones +A ˆ B +B +A > B +B +A +and +A, +(2.4.1) +14 + +respectively. The fact that these constructions agree up to canonical isomorphism is typically +indicated by using the terminology direct sum, and the notation A ‘ B. This phenomenon is +referred to as the semiadditivity of the category of abelian groups. Additivity then amounts +to the extra condition of the resulting monoid structure on HompA, Bq being a group. +As already indicated in §1.1, the analogous categorified universal constructions typically +involve more data, invisible upon passing to K0. Namely, beyond a pair of stable 8-categories +A, B, we are also given an exact functor F : A Ñ B. The relevant universal constructions +that we may build from this data are characterized by the universal cones depicted in Figure +2.4.1. +lax limit: +A +ш +F B +A +B +F +oplax limit: +A +Ј +F B +A +B +F +lax colimit: +A +Ð> +F B +A +B +F +oplax colimit: +A +Ñ> +F B +A +B +F +Figure 2.4.1: The four lax cones with base given by a functor F : A Ñ B. +Theorem 2.4.2 (Lax Additivity). In the p8, 2q-category Stk of presentable stable 8-categories, +the four lax universal constructions from Figure 2.4.1 exist and are canonically equivalent. +As explained in detail in §8, the analogous statement holds in any lax additive p8, 2q- +category and Stk is an example of such. +In fact, §8 provides a more refined systematic +analysis also introducing the notion of a lax semiadditive p8, 2q-category as an intermediate +step. +From an axiomatic perspective, the canonical equivalence of these four universal construc- +tions captures the essence of what seems to make lax additive p8, 2q-categories a suitable +context for categorified homological algebra (just like additive categories are used for classical +homological algebra). +While complete proofs will be deferred to §8, we explain here, how to construct a particular +model for the lax limit in St and describe the four universal lax cones from Figure 2.4.1 in +terms of this model (the analogous statement for Stk is somewhat more involved as we need +to keep track of the Modk-module structures – we don’t discuss this here). Let ΣpFq denote +the 8-category of sections of the covariant Grothendieck construction of F, considered as a +diagram ∆1 Ñ St. Due to the simplicity of the indexing category, this can be described even +more concretely as the pullback of simplicial sets +ΣpFq +Funp∆1, Bq +A +B +ev0 +F +15 + +Notationally, we simply write +ΣpFq “ tpa, b, ηq | a P A, b P B, η: Fpaq Ñ bu. +We now describe the universal lax cones that characterize ΣpFq as each of the four universal +constructions: +1. The lax limit cone is evident: The functors to A and B, respectively, are simply given by +projecting to the components a P A and b P B while the natural transformation which is +part of the cone is given by Fpaq Ñ b. Note that, while we specify the data of the cone +on the level of objects, it is evident how to extend these formulas to actual functors and +natural transformations of 8-categories. +2. The lax colimit cone of ΣpFq corresponds to the functors +A Ñ ΣpFq, +a ÞÑ pa, Fpaq, Fpaq id +Ñ Fpaqq +B Ñ ΣpFq, +b ÞÑ p0, b, 0 Ñ bq +where the natural transformation assigns to a P A, the morphism in ΣpFq given by +p +0, +Fpaq, +0 +Fpaq +q +p +a, +Fpaq, +Fpaq +Fpaq +q +id +id +id +3. The oplax limit cone is given by the functors +ΣpFq Ñ A, +pa, b, ηq ÞÑ a, +ΣpFq Ñ B, +pa, b, ηq ÞÑ fibpηq, +and the natural transformation which assigns to pa, b, ηq P Σ, the morphism in B given +by +fibpηq Ñ Fpaq +which is part of the fiber square +fibpηq +Fpaq +0 +b. +4. Finally, the oplax colimit cone is determined by the functors +A Ñ ΣpFq, +a ÞÑ pa, 0, Fpaq Ñ 0q +B Ñ ΣpFq, +b ÞÑ p0, br1s, 0 Ñ br1sq +while the natural transformation assigns to a P A, the morphism in ΣpFq given by +p +a, +0, +Fpaq +0 +q +p +0, +Fpaqr1s, +0 +Fpaqr1s +q +where the square is the biCartesian square exhibiting Fpaqr1s as the suspension of Fpaq. +16 + +Remark 2.4.3. Note that, for F “ id: A Ñ A, we have +ΣpFq “ Funp∆1, Aq +so that ΣpFq is simply a presheaf category, or put differently, the category of A-valued repre- +sentations of the quiver of Dynkin type A2. In this situation, the fact that ΣpFq, which most +evidently is a lax limit of F, is in fact also a lax colimit of F amounts to the statement that +any presheaf category is generated by the subcategory of representable presheaves (tensored +with objects of A). As explained in more detail in §8, the equivalence (for general F) +A +ш +F B » A +Ð> +F B +generalizes to any so-called lax semiadditive p8, 2q-category. In particular, it does not require +stability and also holds, for example, in the p8, 2q-category PrL of presentable 8-categories. +Remark 2.4.4. In contrast to the equivalence between lax limit and colimit discussed in +Remark 2.4.3, the phenomenon that the lax and oplax limit are equivalent does not hold +in a general lax semiadditive p8, 2q-category but requires lax additivity. For example, the +p8, 2q-category St of stable presentable 8-categories has this property. +The resulting equivalence +A +ш +F B » A +Ј +F B +(2.4.5) +is exploited in [DJW21] where it is described explicitly, in terms of models for lax and oplax +limits, and used to provide a framework for generalized Bernstein-Gelfand-Ponomarev reflec- +tion functors. +This equivalence also has a natural interpretation in terms of semiorthogonal decomposi- +tions, as we now explain. Recall (cf. [DKSS21]) that a semiorthogonal decomposition of a +stable 8-category C consists of a pair pX, Yq of full stable subcategories of C, such that the +functor of 8-categories +tx Ñ y | x : X, y : Yu ÝÑ C, +given by associating to a morphism x Ñ y its cofiber, is an equivalence. The lax and oplax +limit cones for ΣpFq naturally induce semiorthogonal decompositions with components given +by the kernels of the projection maps. This yields concretely: +1. lax limit: ptpa, 0, Fpaq Ñ 0qu, tp0, b, 0 Ñ bquq +2. oplax limit: ptp0, b, 0 Ñ bqu, tpa, b, Fpaq » +Ñ bquq +In terms of the terminology developed in [DKSS21], these two semiorthogonal decompositions +of ΣpFq are related by mutation where the first decomposition is coCartesian and the second +decomposition Cartesian. +Within this context, the equivalence (2.4.5) corresponds to the +statement that the underlying category of a coCartesian (resp. Cartesian) semiorthogonal +decomposition can be recovered from its gluing functor as a lax (resp. oplax) limit. +When we wish to make the point that the above universal constructions are equivalent +(where the equivalences are determined by the choices of universal cones in Figure 2.4.1), then +we will use the notation +A +Ø‘ +F B +17 + +to denote any of them, and refer to it as the lax sum of A and B along F. Whenever we +would like to refer to one of the above universal lax cones, then we will use the notation for +the respective universal construction from Figure 2.4.1. +We conclude this section with a further basic equivalence which exists in the presence of +adjoints. +Proposition 2.4.6. Let F : A Ñ B be a functor in Stk and suppose that F has a right adjoint +F R. Then there is a canonical equivalence +A +ш +F B +» +ÐÑ B +Ј +F R A, +(2.4.7) +in Stk described by the formula +tpa, b, Fa Ñ bqu Ø tpa, b, a Ñ F Rbqu. +(2.4.8) +Proof. This is a special case of Corollary A.1.15. +3 +Simplicial totalization and the Dold-Kan correspondence +3.1 +The categorical cochain complex of a simplex +To convey a first impression of the workings of categorical complexes, we present a class of +examples which is easy to describe directly yet illustrates some of the key phenomena. +Let A be an abelian group. We will describe categorifications of the simplicial cochain +complex C‚p∆n, Aq of an n-simplex. We have +Ckp∆n, Aq “ Mapptσ: rks ãÑ rnsu, Aq +with differential given by the formula +pdaqpσq “ +nÿ +i“0 +p´1qiapσ ˝ Biq. +Let us further focus on the 2-simplex, where we have +C‚p∆2, Aq : +C‚p∆2, Aq : +A – +tpx012qu +tX012u +» A +A3 – +tpx01, x02, x12qu +tX01 Ñ X02 Ñ X12u +» xA, A, Ay +A3 – +tpx0, x1, x2qu +tX0 Ñ X1 Ñ X2u +» xA, A, Ay +x012“x12´x02`x01 +X012“totpX01ÑX02ÑX12q +xij“xj´xi +Xij“conepXiÑXjq +We note the following features of the categorical complex C‚p∆2, Aq: +18 + +(1) To be able to define the first differential, we have adjoined morphisms Xi Ñ Xj so that +we can take their “difference” according to rule 3). +(2) To define the second differential, the objects Xij need to form a complex so that we +may totalize it. +(3) The octahedral “axiom” (or rather the third isomorphism theorem) guarantees that the +first differential is well defined (the cones conepXi Ñ Xjq form a complex) and further, +that d2 » 0 (since the just-mentioned complex is exact). +The additional lax data we needed to implement the rules of categorification from §1.1 +actually has a natural interpretation: The simplex category ∆ can be defined as the full +subcategory of the 1-category Cat of small categories spanned by the standard ordinals. As +a result, given ordinals rks and rns we obtain a set ∆prks, rnsq of k-simplices in ∆n described +as the corresponding set of maps in ∆. However, the collection of categories has a natural +2-categorical structure taking in to account natural transformations. Defining ∆ to be the +full 2-subcategory of the 2-category Cat instead yields morphism categories, in fact posets, +∆prks, rnsq. +Definition 3.1.1. Within this context, we can now give the general definition of the complex +of the categorical simplicial cochain complex C‚ :“ C‚p∆n, Aq: +(1) For 0 ď k ď n, the 8-category Ck is defined to be the full subcategory +Ck Ă Funp∆prks, rnsq, Aq +consisting of those diagrams X satisfying: +• for every non-injective map τ : rks Ñ rns, we have Xpτq » 0. +(2) The differential d: Ck Ñ Ck`1 is given by associating to a diagram X P Ck, the diagram +dX : ∆prk ` 1s, rnsq Ñ A, σ ÞÑ totpXpdnσq Ñ ... Ñ Xpd0σqq. +Remark 3.1.2. We comment on the defining formula for the diagram dX in (2) which needs +to be made precise in several ways. +1. First, the formula +totpXpdnσq Ñ ... Ñ Xpd0σqq +needs to be explained. The expression +Xpdnσq Ñ ... Ñ Xpd0σq +refers to an n ` 1-term complex in the stable 8-category A. Formally, this is real- +ized as a cubical diagram In Ñ A, with I “ r1sop “ t1 Ñ 0u, with zero relations +encoded by certain vertices of this cube being mapped to zero objects. The (cofiber) +totalization of this complex is then obtained by extending with zeros to a punctured +pn ` 1q-cube In`1 ztp0, . . . , 0qu, i.e. a right Kan extension, and taking the colimit over +In`1 ztp0, . . . , 0qu, i.e. a left Kan extension to In`1. +19 + +2. Second, we need to explain how the various values dXpσq organize into an actual diagram +∆prk ` 1s, rnsq Ñ A and, further, how the association X ÞÑ dX defines a functor in Stk. +This is achieved by means of the formalism of Kan extensions as developed in [Lur09a] +For most parts of this paper, we will not delve into the technical details of constructions +such as 1. and 2. as this would have a tendency of hiding the main ideas behind the (routine) +technical details. Rather, we assume that the reader is familiar with the techniques alluded +to and keep the treatment somewhat informal as in Definition 3.1.1. Typically, it is rather +straightforward how to make things formally precise so that we hope that not much is lost by +this approach. +3.2 +The categorified Dold–Kan correspondence +The discussion in § 3.1 shows that interesting examples of categorical complexes arise when +studying simplicial (or rather 2-simplicial) objects. These ideas can be put in a more system- +atic context, the categorified Dold–Kan correspondence as established in [Dyc21], which will +be briefly recalled here. +Denote by Cat the 2-category of small categories, and further, by ∆ Ă Cat the full sub +2-category spanned by the standard ordinals trns|n ě 0u. Treating ∆ as an p8, 2q-category, +we define a 2-stable 8-category to be a functor +X: ∆op Ñ St +of p8, 2q-categories. For example, this can be modelled concretely as a Set∆-enriched functor +from ∆, considered as Set∆-enriched by taking the nerve of the categories of morphisms, to +Set∆ taking values in stable 8-categories with exact functors. +We explain how to associate to X a complex of stable 8-categories by a construction which +is, in a sense, dual to the one discussed in §3.1. To this end, for n ą 0, consider the cubical +diagram +q: t0, 1un ÝÑ Funprn ´ 1s, rnsq, pi0, ..., in´1q ÞÑ 0 ` i0 ď 1 ` i1 ď ... ď pn ´ 1q ` in´1. +Note that t0, 1u “ r1s as posets, we use the former notation in this section to avoid confusions. +The 2-functoriality of X induces a cubical diagram +qX : t0, 1un ÝÑ FunpXn, Xn´1q . +We now denote by +Xn :“ Xn{tdegenerate simplicesu +the “Verdier quotient” by the subcategory of degenerate simplices, i.e., the full subcategory +spanned by the images of all degeneracy functors. The above cube qX descends to define a +cubical diagram +qX : t0, 1un ÝÑ FunpXn, Xn´1q +which represents a homotopy-coherent pn ` 1q-term complex formed by the face maps +dn Ñ dn´1 Ñ ... Ñ d0. +20 + +Passing to the totalization of this complex, we obtain a functor +totpqXq: Xn Ñ Xn´1 +and further, noting that this totalization maps degenerate simplices to zero objects, an induced +functor +d: Xn Ñ Xn´1 +(3.2.1) +We leave the proof of the following lemma as an entertaining exercise to the reader: +Lemma 3.2.2. The differential constructed in (3.2.1) satisfies d ˝ d » 0. +Definition 3.2.3. The resulting complex pX‚, dq is called the simplicial totalization of X. +As shown in [Dyc21], the association X ÞÑ pX‚, dq defines an equivalence which can be +interpreted as a categorified variant of the classical Dold-Kan correspondence: +Theorem 3.2.4. The simplicial totalization functor defines an equivalence of 8-categories +C: St∆ ÐÑ Chě0pStq :N +Note that in [Dyc21] a different description of the functor C is given, but this can be shown +to be equivalent to the above, by investigating the fully faithful adjoints of the localization +functor Xn Ñ Xn. +One interesting aspect of the categorified Dold-Kan correspondence is the reverse proce- +dure of constructing 2-simplicial objects from categorical complexes. In the classical context, +this provides a bridge between homological and homotopical data, and we expect the cate- +gorified Dold-Kan correspondence to play a similar role for categorical complexes. +4 +Totalizations of categorical multicomplexes +Besides the totalization of simplicial objects, another natural class of examples of categorical +complexes arises by totalizing bicomplexes, or more generally multi-dimensional complexes. +In this section, we discuss these totalization constructions and establish some basic proper- +ties. This will allow us to introduce and investigate several interesting classes of categorical +complexes. +4.1 +Directed pushouts and pullbacks +We introduce two universal p8, 2q-categorical constructions which will be crucial in this sec- +tion. +(1) The directed pullback of a diagram +B +C +D +S +G +(4.1.1) +21 + +in Stk is an object B +ñˆ +D C together with a universal cone of the form +B +ñˆ +D C +B +C +D +S +G +(4.1.2) +It may be expressed in terms of the lax limit from §2.4 as +B +ñˆ +D C » pB +ш +S Dq ˆD C +(4.1.3) +or explicitly as the fiber product of simplicial sets +B +ñˆ +D C “ B ˆD Funp∆1, Dq ˆD C “ tpb, c, ηq | b P B, c P C, η: Spbq Ñ Gpcqu. +(2) Dually, the directed pushout of a diagram +A +B +C +R +F +(4.1.4) +in Stk is an object B +ñ> +A C together with a universal cone of the form +A +B +C +B +ñ> +A C +F +R +(4.1.5) +We may construct it in terms of the oplax colimit via the formula +B +ñ> +A C “ B >A pA +Ñ> +R Cq +or as a pushout +B >A Funp∆1, Aq >A C +in Stk, where the functors A Ñ Funp∆1, Aq are given by the formulas +a ÞÑ p0 Ñ aq +and +a ÞÑ pa id +Ñ aq, +respectively. +In the presence of suitable adjoints, we may express the directed pullbacks (resp. pushouts) +introduced in this section in terms of the (op)lax (co)limits of §2.4. This will be most relevant +for §4.5 where we investigate Beck–Chevalley conditions and is documented in the following +proposition. +22 + +Proposition 4.1.6. +(1) Suppose we are given a diagram as in (4.1.1). +(a) There is a canonical equivalence +B +ñˆ +D C » C +Ј +SR˝G +B. +(b) Suppose further that G has a left adjoint GL. Then there is a canonical equivalence +B +ñˆ +D C » B +ш +GL˝S +C. +(2) Suppose we are given a diagram as in (4.1.4). +(a) Then there is a canonical equivalence +B +ñ> +A C » C +Ð> +F˝RR B. +(b) Suppose further that F has a left adjoint F L. Then there is a canonical equivalence +B +ñ> +A C » B +Ñ> +R˝F L C. +Proof. Informally, the equivalences are simply described by transposition using the provided +adjunctions, for example +pb, c, Sb Ñ Gcq Ø pb, c, b Ñ SRGcq +(4.1.7) +for the first statement. See Corollary A.1.15 for a general p8, 2q-categorical proof. +4.2 +Commutative squares +Let +A +B +C +D +F +R +S +G +(4.2.1) +be a commutative square in Stk. We introduce two means of totalizing (4.2.1) to obtain a +categorical complex: +(1) We define the product totalization to be the sequence of functors +A +B +ñˆ +D C +D +d2 +d1 +(4.2.2) +where d2 is the canonical functor arising from the square (4.2.1), interpreted as a directed +cone over (4.1.1), i.e. +a ÞÑ pFpaq, Rpaq, SpFpaqq » GpRpaqqq +while the functor d1 is given by +pb, c, η: Spbq Ñ Gpcqq ÞÑ fibpηq. +It is immediate that we have d1 ˝d2 » 0 so that we may interpret (4.2.2) as a categorical +complex with D in degree 0. +23 + +(2) We define the coproduct totalization to be the sequence of functors +A +B +ñ> +A C +D +d2 +d1 +(4.2.3) +where the functor d2 is the cofiber of the natural transformation in the universal cone +(4.1.5) while d1 is the canonical functor arising from the square (4.2.1), interpreted as a +directed cone under (4.1.4). Again, it is evident, that d1 ˝ d2 » 0: The postcomposition +of the said natural transformation from (4.1.5) with d1 is a natural equivalence, since +the square (4.2.1) commutes. Thus, we may interpret (4.2.3) as a categorical complex +as well, concentrated in degrees 2 to 0. +While the two categorical totalizations of the square (4.2.1) yield isomorphic complexes of +abelian groups when passing to K0 (as can be deduced from the existence of semiorthogonal +decompositions pB, Cq on both directed pushout and directed pullback), they are not in gen- +eral equivalent as categorical complexes. Nevertheless, there exists a canonical comparison +morphism which will be investigated next. +We construct a functor +χ: B +ñ> +A C ÝÑ B +ñˆ +D C +(4.2.4) +in terms of the universal properties of both sides: +1. the functor B Ñ B +ñˆ +D C is given by +b ÞÑ pb, 0, Spbq Ñ 0q +2. the functor C Ñ B +ñˆ +D C is given by +c ÞÑ p0, cr1s, 0 Ñ Gpcqr1sq +3. the functor A Ñ Funp∆1, B +ñˆ +D Cq is given by +a ÞÑ +p +Fpaq, +0, +S ˝ Fpaq +0 +q +p +0, +Rpaqr1s, +0 +G ˝ Rpaqr1s +q +where the square is the biCartesian square exhibiting the equivalence S ˝ F » G ˝ R. +Proposition 4.2.5. The functor χ extends to a morphism of categorical complexes +A +B +ñ> +A C +D +A +B +ñˆ +D C +D +d2 +r1s +d1 +χ +id +d2 +d1 +between the coproduct and product totalizations of (4.2.1). +24 + +Proof. Direct computation. +We now provide a criterion for when the functor χ is an equivalence so that in particular, +by Proposition 4.2.5, the coproduct and product totalization of (4.2.1) will be equivalent as +categorical complexes. The square (4.2.1) is called vertically right adjointable, if the functors +R and S have right adjoints and the resulting mate +C +D +A +B +G +RR +SR +F +(4.2.6) +commutes (i.e. the natural transformation is an equivalence). +Proposition 4.2.7. Suppose that the square (4.2.1) is vertically right adjointable. +Then +the functor χ is an equivalence. In particular, the product and coproduct totalizations are +canonically equivalent. +Proof. One may explicitly verify that we have a commutative square +B +ñ> +A C +B +ñˆ +D C +C +Ð> +F˝RR B +C +Ð> +SR˝G B +C +Ј +SR˝G +B +χr´1s +» +» +» +» +where all equivalences are given by the canonical identifications that arise from the various +universal cones described in §2.4. +The goal of the subsequent parts of §4 will be to generalize the totalization of squares +to bicomplexes, and finally multicomplexes – with a special focus on cubes, as these are our +main examples. +4.3 +Categorical bicomplexes +A categorical bicomplex A‚,‚ consists of the datum of +• a family tApi,jqupi,jqPZop,2 of objects in Stk, +• functors d: Api,jq Ñ Api´1,jq and δ: Api,jq Ñ Api,j´1q, +• equivalences dδ » δd. +such that +• d2 » 0 and δ2 » 0. +Up to contractible choices, we may identify a categorical bicomplex with a functor Zop,2 Ñ St +(with the property d2 “ 0 and δ2 “ 0. Given a categorical bicomplex A‚,‚, we define the +product totalization to be the categorical complex +C‚ “ totˆpA‚,‚q +given as follows: +25 + +(1) for n P Zop, the category Cn is the iterated directed pullback +¨ ¨ ¨ +ñˆ +A1,n´2 +A1,n´1 +ñˆ +A0,n´1 +A0,n +ñˆ +A´1,n +A´1,n`1 +ñˆ +A´2,n`1 +¨ ¨ ¨ +Thus, an object of this category consists of a sequence tai,jui`j“n together with, for +every i, j, a specified morphisms dai,j Ñ δai´1,j`1 in Ai´1,j. +(2) the differential dC : Cn Ñ Cn´1 is defined as follows: for an object tai,ju of Cn, the +component in Ak,l of its image under dC is given by +fibpdak`1,l Ñ δak,l`1qrks +The images of adjacent components of dCptai,juq under d and δ are canonically equivalent +via the chosen equivalence dδ » δd. The components, equipped with these equivalences +then define an object of Cn´1. +Note that, in contrast to its classical counterpart, the definition of the categorical product +totalization is not symmetric in each term (ignoring the differential) with respect to swapping +the coordinates of the bicomplex, due to the appearance of the iterated directed pullback in +(1). Our convention is to use the linear order of the coordinates of the bicomplex to determine +the chosen direction. +The coproduct totalization C1 +‚ “ tot>pA‚,‚q is defined in analogy to (4.2.3): +(1) for n P Zop, the category C1 +n is the iterated directed pushout +¨ ¨ ¨ +ñ> +A2,n´1 A1,n´1 +ñ> +A1,n A0,n +ñ> +A0,n`1 A´1,n`1 +ñ> +A´1,n`2 ¨ ¨ ¨ +Thus, via the universal property, a functor f : C1 +n Ñ D corresponds to a collection of +functors αi,j : Ai,j Ñ D with i ` j “ n and natural transformations ηi,j fitting into +diagrams of the following form: +Ai`1,j +Ai,j +Ai`1,j´1 +D +d +δ +αi,j +ηi,j +αi`1,j´1 +(2) We specify the composite of the differential dC1 : C1 +n`1 Ñ C1 +n with any functor f : C1 +n Ñ D +as above. Specializing to f “ idC1n yields the differential dC1. We describe f ˝ dC1 via the +universal property of the iterated directed pushout in terms of functors +βi`1,j “ fibpηi,jqri ` 1s: Ai`1,j ÝÑ D +and natural transformations +βi`1,j ˝ δ » βi,j`1 ˝ d +(4.3.1) +arising from the identities d2 » 0, δ2 » 0 and δ ˝ d » d ˝ δ. It follows from the fact that +the natural transformation (4.3.1) is a natural equivalence, that d2 +C1 » 0. +26 + +Remark 4.3.2. With some more effort, one can show that product and coproduct totaliza- +tions form functors +totˆ, tot> : Ch2pStkq Ñ ChpStkq , +but we omit the details in this work. +The most important example of a totalization of a bicomplex for us will be the following +special case. +Construction 4.3.3. Let F : A‚ Ñ B‚ be a morphism of categorical complexes, depicted as +follows. +. . . +A2 +A1 +A0 +. . . +. . . +B2 +B1 +B0 +. . . +F2 +dA +dA +F1 +F0 +dB +dB +(4.3.4) +We may interpret F as a bicomplex C‚,‚ with C0,‚ “ A‚ and C´1,‚ “ B‚. We define the fiber +of F as the product totalization FibpFq :“ totˆpC‚,‚q. Explicitly, we have +FibpFqi “ Ai +ñˆ +Bi +Bi`1, +The differential d: FibpFqi Ñ FibpFqi´1 is given by +pa, b, η: Fipaq Ñ dpbqq ÞÑ pdpaq, fibpηq, Fi´1dpaq » dFipaqq, +(4.3.5) +where we note that d fibpηq » dFipaq. +Dually, we define the cofiber CofpFq of F as the coproduct totalization of the bicomplex +(4.3.4), with a degree shift of ´1, so that we have +CofpFqi » Ai´1 +ñ> +Ai Bi. +Definition 4.3.6. For a chain complex pA‚, dAq we define its shift Ar1s :“ CofpA Ñ 0q, +which explicitly is given by +Ar1si :“ Ai´1 +and +dAr1s +i`1 :“ dA +i r1s: Ai Ñ Ai´1. +(4.3.7) +Dually we define Ar´1s :“ Fibp0 Ñ Aq and observe +Ar1sr´1s » A » Ar´1sr1s. +(4.3.8) +Remark 4.3.9. The cofiber (resp. fiber) satisfy universal properties which can be formulated +in terms of the notion of categorical homotopy introduced in §4.6. For example, the cofiber +is the universal example of a categorical complex C‚ equipped with a morphism G: B‚ Ñ C‚ +together with a categorical zero homotopy of the composite G ˝ F. See also §8.10 for more +details. We expect that this phenomenon will become relevant when trying to introduce a +notion of derived category of categorical complexes. +In light of Remark 4.3.9 one may anticipate that fiber and cofiber of a given morphism +are equivalent up to a shift. While this is not the case for a general morphism, remarkably, +there is a natural class of morphisms for which the statement holds and which we introduce +next. +27 + +Definition 4.3.10. A chain map F : A‚ Ñ B‚ between categorical complexes is called right +adjointable if each square +Ai +Ai´1 +Bi +Bi´1 +dA +Fi +Fi´1 +dB +(4.3.11) +is vertically right adjointable. +Proposition 4.3.12 (Beck–Chevalley). Let F : A‚ Ñ B‚ be a right adjointable morphism. +Then there is a canonical equivalence of categorical complexes +χ‚ : CofpFq +» +ÝÑ FibpFqr1s +where +χi : Ai´1 +ñ> +Ai Bi ÝÑ Ai´1 +ñˆ +Bi´1 +Bi, +is the functor defined in (4.2.4) corresponding to the commutative square (4.3.11). +Proof. Each functor χi is an equivalence by Proposition 4.2.7. The fact that χ‚ defines a +morphism of complexes is verified by direct computation using the universal properties of the +terms involved. An alternative argument is also provided in Corollary 8.9.62. +4.4 +Categorical multicomplexes +Consider a categorical n-complex as in Definition 2.2.6. To iteratively define its totalization, +we begin by introducing the partial totalization at adjacent coordinates. +Construction 4.4.1. Let n ě 2 and A˚ : Zop,n Ñ Stk be a categorical n-complex. We +consider A˚ as an object of FunpZop,n´2, Ch2pStkqq, valued in the bicomplexes describing +chosen coordinates 1 ď i, i`1 ď n. Using the functoriality of the totalization constructions of +Section 4.3, we may partially product totalize the chosen coordinates to obtain a categorical +pn ´ 1q-complex totˆ +i,i`1pA˚q, defined as the image under the functor +totˆ +i,i`1 : FunpZop,n´2, Ch2pStkqq +FunpZop,n´2,totˆq +ÝÝÝÝÝÝÝÝÝÝÝÝÑ FunpZop,n´2, ChpStkqq . +The partial coproduct totalization tot> +i,i`1pA˚q P Chn´1pStkq is defined similarly. +Given a categorical n-complex A˚, we obtain its product and coproduct totalizations by +iterating the above partial totalizations. For the moment, we fix one particular order for these +partial totalizations. +Definition 4.4.2. Let A˚ be a categorical n-complex. We define its product totalization +totˆpA˚q as the iterated partial totalization +totˆ +1,2 ˝ totˆ +2,3 ˝ ¨ ¨ ¨ ˝ totˆ +n´1,npA˚q . +Similarly, the coproduct totalization tot>pA˚q of A˚ is defined as +tot> +1,2 ˝ tot> +2,3 ˝ ¨ ¨ ¨ ˝ tot> +n´1,npA˚q . +28 + +With our conventions, both the product and coproduct totalizations of a categorical n- +cube lie in degrees n to 0. +Example 4.4.3. Suppose that q : In Ñ Stk is a constant cube with value A. By an inductive +argument, we obtain the following description of the product totalization of q: +• Cn`1 “ A, +• the category Cn´k consists of diagrams +X : Funprks, rnsq Ñ A +mapping non-injective maps rks Ñ rns to zero objects, +• the differential d: Cn´k Ñ Cn´k´1 is given by associating to a diagram +X : Funprks, rnsq Ñ A +the diagram +X1 : Funprk ` 1s, rnsq Ñ A +where X1pσq is the total fiber of the complex +Xpσ ˝ Bnq Ñ Xpσ ˝ Bn´1q Ñ ... Ñ Xpσ ˝ B0q +in A, where X1 is formally constructed from X by means of Kan extensions. +In particular, we observe that the n-cells of this complex are representations in A of higher +Dynkin type A. Note that this complex is an augmented variant of categorical cochain complex +C‚p∆, Aq from Definition 3.1.1 (normalized slightly differently there, since the differentials are +given by taking the total cofiber instead of the fiber). The example illustrates the general +phenomenon that, even for simple cubical diagrams, the terms of the totalization turn out to be +rather interesting stable 8-categories. This is somewhat in contrast to the usual totalization of +cubical diagrams of abelian groups, where the terms of the totalization are simply direct sums +of the terms of the cube. The coproduct totalization of q admits a dual description, we leave +the analogous detailed description to the reader. It follows from the Beck–Chevalley property +of the cube q, that the product and coproduct totalizations are in fact equivalent as categorical +complexes (repeatedly apply Proposition 4.3.12). The totalization of q is also an example of +a categorical Koszul complex, discussed in §4.7. See Theorem 4.7.2 and Corollary 4.7.8 for +categorical Koszul duality and its implications when applied to q. +Remark 4.4.4. We expect a fully coherent associativity statement for the totalization of +multicomplexes, meaning that the totalization does not depend on the order of the totaliza- +tions of the adjacent coordinates. A systematic analysis of the associativity of the totalizaton +would however go beyond the scope of this work. Instead, we sketch how to prove a partial +result in Proposition 4.4.5 below, needed for concrete applications (e.g. in §4.7). +The totalization of a cube depends on a further choice, namely the given total order of +the coordinates. We leave it as an interesting problem to determine under which conditions +the totalization does not depend on this order, up to equivalence. +29 + +Proposition 4.4.5. Let A˚ be a categorical n-cube. Then there exist equivalences of categor- +ical complexes +totˆpA˚q » totˆ +1,2 ˝ ¨ ¨ ¨ ˝ totˆ +1,2pA˚q +and +tot>pA˚q » tot> +1,2 ˝ ¨ ¨ ¨ ˝ tot> +1,2pA˚q . +Informally, Proposition 4.4.5 states that iteratively totalizing the last two coordinates is +equivalent to iteratively totalizing the first two coordinates. +To prove Proposition 4.4.5, we begin with the case of categorical 3-complexes of the form +I ˆZop ˆ I Ñ Stk, where I “ r1sop Ă Zop. +Lemma 4.4.6. Let FunChpI ˆZop ˆ I, Stkq Ă FunpI ˆZop ˆ I, Stkq be the full subcategory +spanned by categorical 3-complexes. There exists a natural equivalence between the functors +totˆ +1,2 ˝ totˆ +1,2 : FunChpI ˆZop ˆ I, Stkq Ñ ChpStkq +and +totˆ +1,2 ˝ totˆ +2,3 : FunChpI ˆZop ˆ I, Stkq Ñ ChpStkq . +Proof. We can depict a part of A˚ as follows, +A1,i`1,1 +A1,i`1,0 +A0,i`1,1 +A0,i`1,0 +A1,i,1 +A1,i,0 +A0,i,1 +A0,i,0 +A1,i´1,1 +A1,i´1,0 +A0,i´1,1 +A0,i´1,0 +and relabel this part for better readability as follows: +A2 +B1 +B4 +C3 +B2 +C1 +C4 +D3 +C2 +D1 +D4 +E3 +We have by definition +totˆ +1,2 ˝ totˆ +1,2pA˚qi`1 “ pC1 +ñˆ +D3 +C3q +ñˆ +pD1 +ñ +ˆ +E3 +D3q +pC2 +ñˆ +D4 +C4q +30 + +and +totˆ +1,2 ˝ totˆ +2,3pA˚qi`1 “ pC1 +ñˆ +D1 +C2q +ñˆ +pD3 +ñ +ˆ +E3 +D4q +pC3 +ñˆ +D3 +C4q . +Unraveling the definition, we find that both of these 8-categories satisfy the following uni- +versal property: a functor from X P Stk corresponds to +• functors Fi : X Ñ Cj in Stk for all 1 ď j ď 4, +• natural transformations +C1 +X +D1 +C2 +α +F1 +F2 +C1 +X +D3 +C3 +γ +F1r´1s +F3 +C2 +X +D4 +C4 +β +F2 +F4 +C3 +X +D3 +C4 +δ +F3 +F4 +and a null-homotopy of δ ˝ γ. This null-homotopy induces a natural transformation ν +from the functor X F1 +ÝÑ C1 Ñ D3 to cofpδq. +• A natural equivalence between the two composite natural transformations +C1 +X +D3 +E3 +C4 +D3 +ν +F1 +cofpδq +F4 +» +C1 +X +C2 +D1 +E3 +C4 +D4 +α +F1 +F4 +F2 +β +» +between the functors X F1 +ÝÑ C1 Ñ E3 and X F4 +ÝÑ C4 Ñ E3. The lower natural equivalence +in the left diagram arises from the fact that the functor C3 Ñ D3 Ñ E3 is zero, since d2 +is zero. +One checks that the differentials of both complexes are identified in terms of degreewise +equivalences arising from the matching universal properties. All arising suspensions can be +dealt with by including suitable equivalences. +Using (4.1.3), we can express all directed +pullbacks appearing above in terms of 8-categorical limits and the above equivalences in +terms of functorial equivalences between these limits, yielding the desired equivalence between +totˆ +1,2 ˝ totˆ +1,2 and totˆ +1,2 ˝ totˆ +2,3. +31 + +Proof of Proposition 4.4.5. We only prove the statement about the product totalization, the +case of the coproduct totalization is analogous. Repeatedly applying Lemma 4.4.6, we find +equivalences for all 1 ă j ă n +totˆ +j´1,j ˝ totˆ +j,j`1 ¨ ¨ ¨ ˝ totˆ +j,j`1pA˚q » totˆ +j´1,j ˝ totˆ +j´1,j ˝ totˆ +j`1,j`2 ˝ ¨ ¨ ¨ ˝ totˆ +j`1,j`2pA˚q +» . . . +» totˆ +j´1,j ˝ ¨ ¨ ¨ ˝ totˆ +j´1,j ˝ totˆ +j´1,jpA˚q . +Combining these equivalences, we have +totˆ +1,2 ˝ ¨ ¨ ¨ ˝ totˆ +n´1,npA˚q » totˆ +1,2 ˝ ¨ ¨ ¨ ˝ totˆ +n´3,n´2 ˝ totˆ +n´2,n´1 ˝ totˆ +n´2,n´1pA˚q +» totˆ +1,2 ˝ ¨ ¨ ¨ ˝ totˆ +n´3,n´2 ˝ totˆ +n´3,n´2 ˝ totˆ +n´3,n´2pA˚q +» . . . +» totˆ +1,2 ˝ ¨ ¨ ¨ ˝ totˆ +1,2pA˚q , +as desired. +4.5 +Beck-Chevalley conditions and spherical functors +Definition 4.5.1. Consider a commutate square in Stk: +A +B +C +D +G +F +F 1 +G1 +(4.5.2) +We say that the square is +(1) horizontally right adjointable, or simply right adjointable, if G and G1 admit k-linear +right adjoints GR and pG1qR and the natural transformation +F ˝ GR u ˝F˝GR +ùùùùùùñ pG1qR ˝ G1 ˝ F ˝ GR » +ùñ pG1qR ˝ F 1 ˝ G ˝ GR pG1qR˝F 1˝cu +ùùùùùùùùñ pG1qR ˝ F 1 +is a natural equivalence. +(2) horizontally left adjointable, if G and G1 admit k-linear left adjoints GL and pG1qL and +the natural transformation +pG1qL ˝F 1 pG1qL˝F 1˝u +ùùùùùùùñ pG1qL ˝F 1 ˝G˝GL » +ùñ pG1qL ˝G1 ˝F ˝GL cu˝F˝GL +ùùùùùùñ F ˝GL (4.5.3) +is a natural equivalence. +(3) vertically left adjointable, if F and F 1 admit k-linear right adjoints F R and pF 1qR and +the natural transformation +G˝F R u ˝G˝F R +ùùùùùùñ pF 1qR˝F 1˝G˝F R » +ùñ pF 1qR˝G1˝F ˝F R pF 1qR˝G1˝cu +ùùùùùùùùñ pF 1qR˝G1 (4.5.4) +is a natural equivalence. +32 + +(4) vertically right adjointable, if F and F 1 admit k-linear left adjoints F L and pF 1qL and +the natural transformation +pF 1qL ˝ G1 pF 1qL˝G1˝u +ùùùùùùùñ pF 1qL ˝ G1 ˝ F ˝ F L » +ùñ pF 1qL ˝ F 1 ˝ G ˝ F L cu˝G˝F L +ùùùùùùñ G ˝ F L +is a natural equivalence. +Remark 4.5.5. If F and F 1 admit right adjoints and G and G1 admit left adjoints, then the +natural transformations (4.5.3) and (4.5.4) are adjoint to another and conditions (2) and (3) +from Definition 4.5.1 hence equivalent. An analogous statement holds for conditions (1) and +(4). +Remarkably, all adjointability conditions are equivalent if the square is spherical. +Proposition 4.5.6. Consider a commutative square in Stk as in (4.5.2) and suppose that all +functors F, F 1, G, G1 are spherical functors, see Definition 5.0.1. Then all four conditions of +Definition 4.5.1 are equivalent. +Proof. To begin with, we note that spherical functors admit all repeated left and right adjoints, +see [DKSS21, Cor. 2.5.16]. It thus follows that conditions (2) and (3) of Definition 4.5.1 are +equivalent, as are conditions (1) and (4). It remains to show that conditions (1) and (2) are +equivalent. Let T be the cotwist functor of G % GR and T 1 the twist functor of G1 % pG1qR. +Then we have GL ˝ T » GR and T 1 ˝ pG1qL » pG1qR. We find the following commutative +diagram, +0 +T 1 ˝ pG1qL ˝ F 1 ˝ T +0 +F ˝ GR +pG1qR ˝ G1 ˝ F ˝ GR +pG1qR ˝ F 1 +0 +T 1 ˝ F ˝ GL ˝ T +0 +T 1˝pG1qL˝F 1˝u1˝T +˝ +u ˝F˝GR +˝ +pG1qR˝F 1˝cu +T 1˝cu1˝F˝GL˝T +omitting some identifications in the center. Here, u is the unit of G1 % pG1qR and cu the +counit of G % GR, u1 the unit of GL % G and cu1 the counit of pG1qL % G1. +The fact +that the upper right square and the lower left square are biCartesian, i.e. both pullback and +pushout, follows from the general properties of units and counit of spherical adjunctions, see +for instance Remark 2.9 and Lemma 2.10 in [Chr22c]. By the pasting laws for biCartesian +squares, we find that the horizontal middle composite is a natural equivalence if and only if +the vertical middle composite is a natural equivalence. The equivalence of conditions (1) and +(2) now follows from the fact that T and T 1 are invertible. +Definition 4.5.7. A categorical n-cube is called Beck-Chevalley if each rectilinear face is +both horizontally and vertically right adjointable. +33 + +Lemma 4.5.8. Suppose that A˚ is a Beck-Chevalley categorical n-cube. Then each partial +product totalization +totˆ +i,i`1 ˝ ¨ ¨ ¨ ˝ totˆ +n´1,npA˚q +and partial coproduct totalization +tot> +i,i`1 ˝ ¨ ¨ ¨ ˝ tot> +n´1,npA˚q +with 1 ď i ď n ´ 1 is a Beck-Chevalley pn ´ iq-cube. +Proof. We only show that totˆ +n´1,npA˚q˚ : Iˆn´2 ˆr2sop Ñ Stk is Beck-Chevalley. A similar +argument applies to tot> +n´1,npA˚q. The general case is proven analogously, by using that the +a-times repeated partial totalization consists of a stacked cubes, to each of which an analogous +argument as below applies. +Let J P In´2. For l “ 0, 2, we have +totˆ +n´1,npA˚qpJ,lq “ ApJ,l,lq . +If instead l “ 1, let J1 “ pJ, 1, 0q, J2 “ pJ, 0, 1q , J0 “ pJ, 0, 0q P In . Then we have +totˆ +n´1,npA˚qpJ,1q » AJ1 +ñˆ +AJ0 +AJ2 . +Consider a rectilinear face f : I2 Ñ In´2 ˆr2sop with fp1, 1q differing from fp0, 0q by sub- +tracting 1 in the entries i and j with 1 ď i, j ď n ´ 1 and i ‰ j. There are a few cases to +distinguish, in which similar arguments apply. We highlight two such cases. If i, j ‰ n ´ 1 +and fp0, 0qn´1 “ 0, 2 P r2sop, then the face of totˆ +n´1,npA˚q is equivalent to a corresponding +face of A˚ and hence both vertically and horizontally right adjointable. If i, j ‰ n ´ 1 and +fp0, 0qn´1 “ 1 P r2sop, then the face of totˆ +n´1,npA˚q is of the following form. +Afp0,0q1 +ñˆ +Afp0,0q0 +Afp0,0q2 +Afp1,0q1 +ñˆ +Afp1,0q0 +Afp1,0q2 +Afp0,1q1 +ñˆ +Afp0,1q0 +Afp0,1q2 +Afp1,1q1 +ñˆ +Afp1,1q0 +Afp1,1q2 +This diagram arises from the functoriality of directed pullbacks applied to a diagram con- +taining two Beck-Chevalley faces of A˚. The adjointability properties now follow from the +adjointability properties of these two squares and the observation that the right adjoints of +the morphisms in the above diagrams are computed componentwise, see Remark A.2.5 +4.6 +Mapping complexes +Given complexes A‚, B‚ of abelian groups, there is an associated mapping complex MappA‚, B‚q‚ +with +MappA‚, B‚qn “ +ź +iPZ +HompAi, Bn`iq +34 + +and differential given by the formula +dptfiuqk “ dB ˝ fk ´ p´1qk´1fk´1 ˝ dA. +We now introduce a categorical variant of this construction. Let A‚, B‚ P ChpStkq be +categorical complexes. The oplax categorical mapping complex between A‚ and B‚ is defined +as the product totalization +Mapoplax +‚ +pA‚, B‚q :“ totˆ StkpA´‚, B‚q +(4.6.1) +of the bicomplex obtained by applying Stkp´, ´q. Explicity: +• an object of Mapoplax +n +pA, Bq consists of a sequence tFiuiPZ of k-linear functors Fi : Ai Ñ +Bn`i equipped with natural transformations +φi : Fi´1 ˝ dA ñ dB ˝ Fi; +• the differential Mapoplax +n +pA‚, B‚q Ñ Mapoplax +n´1 pA‚, B‚q associates to this datum the se- +quence +tGi “ fibpφiqr´isu +(4.6.2) +of functors Gi : Ai Ñ Bi`n´1 together with the natural equivalences +Gi´1dA “ fibpFi´2dA Ñ dBFi´1qr´i ` 1sdA +(4.6.3) +» dBFi´1dAr´is » dB fibpFi´1dA Ñ dBFiqr´is +(4.6.4) +“ dBGi +(4.6.5) +Note that the “categorical signs”, i.e., the powers of the suspension r1s, do not agree with +the usual Koszul signs upon passing to K0. Since we will not use categorical mapping com- +plex systematically in this work, we will not attempt to address this (purely conventional) +discrepancy. +More pictorially, we may interpret Mapoplax +n +pA‚, B‚q as the stable 8-category of diagrams +in Stk of the form +. . . +A2 +A1 +A0 +A´1 +. . . +. . . +Bn`2 +Bn`1 +Bn +Bn´1 +. . . +d +d +F2 +d +F1 +d +F0 +d +F´1 +d +d +φ1 +d +φ1 +d +φ´1 +d +(4.6.6) +which may be called an oplax morphisms from A‚ to Bn`‚. The differential then associates +to this oplax morphism the (strict) morphism +. . . +A2 +A1 +A0 +A´1 +. . . +. . . +Bn`1 +Bn +Bn´1 +Bn´2 +. . . +d +d +G2 +d +G1 +d +G0 +d +G´1 +d +d +d +d +d +(4.6.7) +with components Gi “ fibpφiqr´is. +35 + +Remark 4.6.8. Implicit in the product totalization is a choice of ordering of the two directions +in the bicomplex StkpA´‚, B‚q which corresponds to the choice of direction of the 2-cells +appearing in the chain maps (4.6.6). +Flipping this choice yields the notion of lax chain +morphisms and the corresponding lax mapping complex. +See §8.8 for a more systematic discussion of lax and oplax chain maps and an alternative +construction of the lax/oplax mapping complex. +The classical Dold-Kan correspondence offers a means of turning the homological data +captured by a connective chain complex C‚ into homotopical data described by the Kan +complex underlying the associated Dold-Kan nerve of C‚. This transformation is of particular +interest when applied to mapping complexes where it leads to a model for the derived category +of complexes as a topological category (and further an 8-category by passing to coherent +nerves, cf. [Lur17]). +We may therefore hope to gain insights as to how the derived category of categorical com- +plexes should be defined by investigating the “homotopical data” captured by the categorified +Dold–Kan nerve of a categorical complex, and in particular, of the categorical mapping com- +plex. Let A‚ and B‚ be categorical complexes and let M‚ “ τě0 Mapoplax +‚ +pA‚, B‚q P ChpStq be +the categorical mapping complex, truncated below and with M0 “ kerpd0q. Recall (cf. §3.2), +that the categorified Dold–Kan nerve X‚ “ NDKpM‚q is a 2-simplicial stable 8-category, i.e. +a functor +X‚ : ∆ ÝÑ St +of p8, 2q-categories. We describe its low-dimensional simplices: +• A vertex corresponds to a morphism F : A‚ Ñ B‚ of categorical complexes. +• An edge corresponds to the datum of +1. a natural transformation +ηp01q : F p0q ñ F p1q, +of (strict) morphisms F p0q, F p1q : A‚ Ñ B‚ of categorical complexes, +2. an oplax morphism +Hp01q : A‚ Ñ B‚`1 +which we refer to as a categorical homotopy, +3. an exact triangle +dHp01q +F p0q +0 +F p1q +ηp01q +• A 2-simplex corresponds to the datum of +1. a diagram +F p0q +F p1q +F p2q +(4.6.9) +of natural transformations of morphisms A‚ Ñ B‚, +36 + +2. a diagram (not necessarily bicartesian) +Hp01q +Hp02q +0 +Hp12q +(4.6.10) +of oplax morphisms A‚ Ñ B‚`1, +3. an oplax morphism +Hp012q : A‚ Ñ B‚`2, +4. an extension of (4.6.9) and d(4.6.10) to a diagram +dHp01q +dHp02q +F p0q +0 +dHp12q +F p1q +0 +F p2q +of morphisms A‚ Ñ B‚ with all squares biCartesian, +5. an extension of (4.6.10) to a biCartesian cube +dHp012q +Hp01q +0 +Hp02q +0 +0 +0 +Hp12q +of oplax morphisms A‚ Ñ B‚`1. +• ... +As we observe from this description, the 2-simplicial object M‚ carries meaningful data, +such as a reasonable notion of “categorical homotopy” between morphisms of categorical +complexes. However, when analyzing the structural properties of M‚ there are many open +questions that remain to be investigated in order to understand the “higher homotopical con- +tent” of M‚ in analogy to its decategorified counterpart. For example, the ordinary Dold-Kan +nerve is a Kan complex (as every simplicial abelian group) so that it carries intrinsic homotopi- +cal meaning. The categorified Dold–Kan nerve does indeed have certain categorified lifting +properties, but these are much weaker so that, in particular, it is not possible to even “com- +pose homotopies” in an obvious way. We hope to analyze this context more systematically in +future work. +37 + +4.7 +Koszul complexes and categorical Koszul duality +Let R be a commutative ring and let λ1, . . . , λn be elements of R. We may then form the +Koszul complex as the tensor product +Kpλ1, ..., λnq “ +n +â +i“1 +pR +λi +ÝÑ Rq. +Recall the classical “self-duality” of the Koszul complex: +Theorem 4.7.1. There is a (canonical) isomorphism of complexes +Kpλ1, ..., λnq_ +n´‚ – Kpλ1, ..., λnq‚ +where p´q_ “ HomRp´, Rq. +In this section, we explain how the notion of a Koszul complex, as well as the self-duality +statement admit a variant for categorical complexes. +Let k be a commutative ring spectrum and Modk the category of k-modules. Given an +object L P Modk and A P Stk, we obtain a functor +A +´bkL +ÝÑ A +which is functorial in A, i.e., it defines a natural transformation on the identity functor of +Stk. We define the categorical two-term complex +KpLq :“ Modk +´bkL +ÝÑ Modk, +concentrated in degrees 0 and 1, and further, for k-module spectra L1, ..., Ln, the categorical +complex +KpL1, ..., Lnq :“ tot>pKpL1q bk KpL2q bk ¨ ¨ ¨ bk KpLnqq. +The main result of this section is the following theorem, which can be regarded as a +categorical variant of the classical “self-duality” of the Koszul complex. +Theorem 4.7.2 (Categorical Koszul duality). Let L1, ..., Ln be dualizable k-module spectra. +Then there is a canonical equivalence of categorical complexes +KpL1, ..., Lnq_ +n´‚ » KpLn, ..., L1q‚ +where p´q_ “ Stkp´, Modkq denotes the dual with respect to the symmetric monoidal structure +on Stk. +Proof. We prove the statement by induction on n. For n “ 1, the identification +StkpModk, Modkq » Modk +via the evaluation functor F ÞÑ Fpkq, extends to an equivalence between the complex KpL1q_ +1´‚ +and KpL1q. +We now assume given the equivalence +KpL1, ..., Ln´1q_ +n´1´‚ » KpLn´1, ..., L1q +38 + +and write A‚ :“ KpL1, ..., Ln´1q and A1 +‚ :“ KpLn´1, ..., L1q. We have +KpL1, L2, ..., Lnq “ tot>pA‚ b KpLnqq +We compute +tot>pA‚ b KpLnqq_ +n´‚ » pA_ +n´i +ñˆ +A_ +n´i +A_ +n´i´1qiPZ +(4.7.3) +» pA1 +i´1 +ñˆ +A1 +i´1 +A1 +iqiPZ +(4.7.4) +» pA1 +i´1 +ñ> +A1 +i +A1 +iqiPZ +(4.7.5) +» tot>pKpLnq b A1 +‚q +(4.7.6) +» KpLn, ..., L1q +(4.7.7) +where +• (4.7.3) holds since the dual of tot> is equivalent to totˆ of the dual. +• (4.7.4) holds by the induction hypothesis. +• (4.7.5) follows from Proposition 4.3.12 applied to the morphism given by tensoring with +Ln. This has an adjoint given by tensoring with L_ +n (since we assumed the Li to be +dualizable) which is again central. +• (4.7.6) holds by definition. +• (4.7.7) follows from Proposition 4.4.5. +We give an application of Theorem 4.7.2: +Corollary 4.7.8. Set Li “ k. Then the categorical Koszul complex can be identified with the +totalization of a constant cube. By Example 4.4.3, the categorical Koszul duality amounts to +an equivalence between +• the category of diagrams +X : Funprks, rnsq Ñ Modk +mapping every non-injective map τ : rks Ñ rns to a zero object in Modk, and +• the category of diagrams +Y : Funprn ´ ks, rnsq Ñ Modk +mapping every non-injective map τ : rn ´ ks Ñ rns to a zero object in Modk. +The duality described in Corollary 4.7.8 was first obtained by Beckert [Bec18] using the +theory of derivators. A geometric proof based on Fukaya categories was given in [DJL21]. +The novelty in our proof based on categorical Koszul complexes is the inductive nature which +does not feature in the previous proofs (and seems to make the argument both simpler and +more conceptual). +39 + +5 +Spherical complexes and perverse schobers +Given an adjunction F : A Ø B :G of stable 8-categories, we define +• the twist functor TA : A Ñ A as the cofiber of the unit u : idA Ñ FG in the stable +8-category FunpA, Aq. +• the cotwist functor TB : B Ñ B as the fiber of the counit c : FG Ñ idB in the stable +8-category FunpB, Bq. +Definition 5.0.1 ([AL17, DKSS21]). An adjunction F % G is called spherical if both TA and +TB are invertible. A functor F : A Ñ B of stable 8-categories is called spherical if it admits +a right adjoint G and the adjunction F % G is spherical. +Spherical adjunctions were originally conceived by R. Anno, to describe “family versions” +of the spherical objects introduced by P. Seidel and R. Thomas in [ST01]. Natural examples +can be found among the various categorical structures that arise within the context of Kont- +sevich’s homological mirror symmetry program. +More recently, it has been proposed in [KS14] to interpret spherical functors (and, more +generally, suitable diagram categories built from spherical functors) as categorified analogues +of perverse sheaves (referred to as perverse schobers). This is motivated by the observation +that the abelian category of perverse sheaves on the complex plane C, with stratification given +by the origin t0u and its complement, is classically known to be equivalent to the category of +diagrams +Φ +Ψ, +f +g +(5.0.2) +of vector spaces Φ and Ψ with id ´fg and id ´gf invertible. +While a fully satisfying intrinsic definition of perverse schobers still remains open in gen- +eral and work in progress in two dimensions (see [DKS20, DKSS21] for partial results), in +many situations one can guess ad-hoc definitions, based on diagrammatic descriptions of the +respective categories of perverse sheaves such as (5.0.2). The resulting notions of perverse +schobers are not intrinsic and depend on auxiliary choices – however, it is still worthwhile to +construct interesting examples and study applications, since these descriptions will hopefully +become part of a more intrinsic theory in the future. +In our current context of categorical complexes, another instance of such an ad-hoc notion +of perverse schober arises from a diagrammatic description of perverse sheaves on Cn with +respect to the stratification given by the hyperplane inclusions +t0u Ă C Ă C2 Ă C3 Ă ¨ ¨ ¨ Ă Cn +and their complements. The resulting notion of perverse schober in this context is a spherical +complex (see §5.3): a categorical complex of length n all of whose differentials are spherical +functors. In §5.4 we also consider the further stratification of Cn given by the coordinate +hyperplanes, their intersections and complements. In this case, the resulting notion of per- +verse schober is a Beck-Chevalley categorical n-cube, whose edges are described by spherical +functors. We call such a categorical cube a spherical cube. +40 + +5.1 +Spherical adjunctions and perverse schobers on C +As mentioned above, the abelian category of perverse sheaves on C with respect to the strat- +ification given by t0u and its complement is equivalent to the category of diagrams +Φ +Ψ, +f +g +of vector spaces Φ and Ψ with id ´fg and id ´gf invertible. Geometrically, the vector spaces +Φ (resp. Ψ) correspond to the spaces of vanishing cycles (resp. nearby cycles) associated to +a given perverse sheaf. +Remark 5.1.1. It is instructive to investigate how this linear algebraic data describes the +perverse sheaf, when considering it as an object of the derived category of constructible +sheaves. To this end, we may also interpret this latter category as the category of constructible +sheaves on C valued in the 8-category of cochain complexes1 of vector spaces. Such a sheaf +F of cochain complexes may then be described by assembling the linear algebraic data into +the diagram +Ψ +Φ +Ψ +0 +g +id +f +öT +(5.1.2) +where +• the stalk F0 of F at 0 P C is given by the cochain complex +Ψ +Φ +g +concentrated in degrees ´1 and 0, +• the stalk F1 of F at 1 P C is given by the cochain complex +Ψ +0 +concentrated in degree ´1, +• the restriction map res: F0 Ñ F1 corresponds to the morphism represented by the +commutative square in (5.1.2), +• the monodromy of the stalk F1 is induced by the automorphism T “ id ´fg of Ψ, and +finally, +• we may interpret the map f : Φ Ñ Ψ as a homotopy, as depicted in (5.1.2), between res +and the composite T ˝ res. This homotopy expresses the compatibility relations arising +from the C˚-family of restriction maps F0 Ñ Fp, with p P C˚, of which we only need to +remember res: F0 Ñ F1. +1When discussing perverse sheaves, we use cochain complexes as this is the standard convention in the +literature. In most other places in this paper, we however use chain complexes. +41 + +It is further interesting to note that we may apply Verdier duality to describe F equivalently +as a constructible cosheaf F_ valued in cochain complexes. This cosheaf then admits the +following analogous description: +Φ +Ψ +0 +Ψ +f +öT +id +g +(5.1.3) +• the costalk F_ +0 of F_ at 0 P C is given by the cochain complex +Φ +Ψ +f +concentrated in degrees 0 and 1, +• the costalk F_ +1 of F_ at 1 P C is given by the cochain complex +0 +Ψ +concentrated in degree 1, +• the corestriction map cores: F_ +1 Ñ F_ +0 corresponds to the commutative square in (5.1.3), +• the monodromy of the stalk F_ +1 is induced by the automorphism T “ id ´fg of Ψ, and, +• we may interpret the map g: Ψ Ñ Φ as a homotopy, as depicted in (5.1.3) between +cores and the composite cores ˝T, completing the data needed to define a constructible +cosheaf valued in cochain complexes. +Note that the invertibility of id ´fg is in fact equivalent to the invertibility of id ´gf, ex- +plaining why this condition does not appear in an apparent way in the description of F or +F_. +Remark 5.1.4. The datum of a spherical adjunction +F : A ÐÑ B :G +can be interpreted in a fashion analogous to Remark 5.1.1. In other words, we may repackage +the data into the diagram +B +A +B +0 +G +id +F +öT +where the exact triangles +TB Ñ id Ñ FG +exhibit F as a categorical homotopy, as defined in Section 4.6. In the categorical context, +however, it is not the case, that the invertibility of TB implies the invertibility of TA so that this +perspective only partially characterizes spherical adjunctions. Nevertheless, it motivates our +proposal to interpret spherical functors as 2-term categorical complexes and use this intuition +to generalize to spherical complexes as explained below in §5.3. +42 + +5.2 +Barbacovi’s Theorem and its geometric interpretation +A remarkable result by Ed Segal ([Seg18]) says that every autoequivalence on a category +arises as a spherical twist. In particular, the composition of the spherical twists associated to +a pair of spherical adjunctions with the same target, must again be a spherical twist (also cf. +[Chr22c] for an 8-categorical generalization). The main result of [Bar20] provides an explicit +description of a spherical adjunction that describes this composite twist in terms of the given +spherical adjunctions. +In this section, we explain how Barbacovi’s construction fits into our context, provide a +geometric interpretation in terms of perverse schobers, and finally prove a generalization to +stable 8-categories which we will need below. +To begin with, consider the complex plane C equipped the stratification given by two +points tx, x1u, say x “ i and x1 “ ´i, and their complement. By choosing an arc connecting +the two points, we may identify the category of perverse sheaves with the category of diagrams +Φ +Ψ +Φ1 +f +g1 +g +f1 +of vector spaces such that both pf, gq and pf1, g1q satisfy the conditions from (5.0.2). Geometri- +cally, the automorphisms T “ id ´fg and T 1 “ id ´f1g1 can be interpreted as the monodromy +transformations on the space of nearby cycles Ψ corresponding to loops around the points x +and x1, respectively. +As mentioned above, the vector spaces Φ and Φ1 can be interpreted as the local vanishing +cycles of the given perverse sheaf F. We define the space of global vanishing cycles ΣpFq as +the fiber +fibpRΓpC, Fq Ñ RΓptℜz ă 0u, Fqq – Φ ‘ Φ1 +of the restriction map along the inclusion of the half-plane tℜz ă 0u Ă C. This vector space +comes equipped with natural maps +ΣpFq +Ψ +pf,f1q +pgT 1,g1q +(5.2.1) +and we compute +id ´pf, f1q +ˆgT 1 +g1 +˙ +“ id ´fgT 1 ´ f1g1 “ pid ´fgqpid ´f1g1q “ TT 1. +Therefore, the datum (5.2.1) defines a perverse sheaf on C with respect to the startification +given by t0u and its complement. +We refer to this sheaf as the amalgamate of F. +This +amalgamate likely describes the pushforward sheaf of F under the endomorphism of C, which +contracts a disc containing x and x1 to 0, but we do not verify this here. +43 + +As we will now explain, these geometric considerations can be lifted to analogous con- +structions for perverse schobers. To begin, a perverse schober on C, equipped with the above +stratification and auxiliary choices, can be interpreted as a pair +A +B +A1 +F +G1 +G +F 1 +of spherical adjunctions as indicated. As a categorical lift of (5.2.1), we define the functor +F +ñˆ +B F 1 : A +ñˆ +B A1 Ñ B . +(5.2.2) +Theorem 5.2.3. The functor F +ñˆ +B F 1 is spherical. The suspension of the cotwist functor of +the adjunction +F +ñˆ +B F 1 % +ˆ +F +ñˆ +B F 1 +˙R +is equivalent to the composite of the cotwist functors of the adjunctions F 1 % G1 and F % G. +In the following, we informally describe how Theorem 5.2.3 can be proven using further +ideas related to perverse schobers. The full and somewhat technical proof implementing these +ideas is given in §5.5. +Given a perverse sheaf F on pC, 0q, instead of considering the vector spaces of vanishing +and nearby cycles, we can equally well describe F in terms of its nearby cycles and its, as +it turns out, vector space of global sections Ψ2 with support on R Ă C or even its vector +space of global sections Ψn with support on any graph embedded in C with a single n-valent +vertex at 0. The resulting description states that the category of perverse sheaves on pC, 0q +is equivalent to the category of diagrams of vector spaces +fi : Φn ÐÑ Ψi :gi , +1 ď i ď n +(5.2.4) +satisfying that figi “ idΨi, fi`1gi is invertible and fjgi “ 0 for j ‰ i, i`1, with i, j considered +modulo n, see [KS16]. The vector spaces Ψi, with 1 ď i ď n, are all equivalent and may be +chosen for concreteness as the stalks of the n-th roots of unity. An ad-hoc categorification of +this local description is described in [Chr22a], based on Waldhausen’s relative S‚-construction. +Using these local descriptions, we can describe perverse sheaves or perverse schobers on any +surface S with 0-dimensional strata P and non-empty boundary by choosing a spanning graph +G Ă S. The inclusion of G into S is required to be a homotopy equivalence and each stratum +p P P is required to be the image of a vertex of G. A perverse sheaf or perverse schober on +S can be encoded as a constructible sheaf and cosheaf on G, which restricts at each vertex of +G to a diagram as in (5.2.4) or its categorification. The global sections of the constructible +sheaf on G describe the first cohomology of the derived global sections with support on G of +the perverse sheaf or perverse schober on S. +To prove Theorem 5.2.3, we consider the perverse schober F on pC, tx, x1uq defined by the +pair of spherical adjunctions F % G and F 1 % G1 and describe it as a constructible sheaf on +the ribbon graph G with two vertices at x, x1, depicted as follows (in blue): +44 + +x +x1 +G +q +p +C +The points p and q lie left or right of x and x1 as indicated, but are otherwise arbitrary. We +denote the global sections of F by RΓ1pG, Fq P Stk. Restriction to the two points p, q on G +defines a functor +pevp, evqq: RΓ1pG, Fq Ñ Bˆ2 , +The right adjoint is denoted pevpR, evqRq and the left adjoint is denoted pevpL, evqLq. These +functors turn out to be much easier to handle than the functor (5.2.2). +This is because +the functors evpR, evqR, evpL, evqL are all fully faithful and, as one can compute, there are +equivalences evqR » evpL ˝ T and evpR » evqL ˝ T 1, with T the cotwist functor of F % G +and T 1 the cotwist functor of F 1 % G1. From these facts, it also follows that pevp, evqq is a +spherical functor. +To show the sphericalness of the functor F +ñˆ +B F 1, we use the fact that this is equivalent to +the assertion that there exist a stable 8-category with a 4-periodic semiorthogonal decompo- +sition pA +ñˆ +B A1, Bq with gluing functor F +ñˆ +B F 1, see [HLS16, DKSS21]. We find this 4-periodic +semiorthogonal decompositions in the 8-category RΓ1pG, Fq. +Here, A +ñˆ +B A1 Ă RΓ1pG, Fq +arises as the full subcategory of global sections with support on G which vanish at p, i.e. the +kernel of evp : RΓ1pG, Fq Ñ B. The 8-category B Ă RΓ1pG, Fq arises as the image of evpR. +Theorem 5.2.3 is a special case of a more general phenomenon exhibited by perverse +schobers on a stratified surface S with non-empty boundary. We again choose a spanning +graph G of S. +By a G-parametrized perverse schober, we mean a constructible sheaf of +stable 8-categories on G encoding a perverse schober on S, as explained above and defined +in [Chr22a]. Given an edge e of G, we denote by Fpeq the stalk of F at any point on that +edge. +Theorem 5.2.5. Let F be a G-parametrized perverse schober on S. Let EB be the set of +external edges of G and +ź +ePEB +eve : RΓ1pG, Fq Ñ +ź +ePEB +Fpeq +the restriction functor of global sections with support on G to stalks at these boundary edges. +The functor ś +ePEB eve is spherical. +In the study of partially wrapped Fukaya categories, the functor ś +ePEB eve is also called +the cap functor, it is adjoint to the Orlov or cup fuctor, see for instance [Syl19]. We sketch +in §5.5 how the proof of Theorem 5.2.3 generalizes to a proof of Theorem 5.2.5. +45 + +5.3 +Spherical complexes and perverse schobers on Cn +Let n ě 1. We begin with a linear algebraic description of classical perverse sheaves on Cn +with respect to the stratification given by the hyperplane inclusions +t0u Ă C Ă C2 Ă C3 Ă ¨ ¨ ¨ Ă Cn , +(5.3.1) +each setting the last coordinate to zero. +Theorem 5.3.2. The category of perverse sheaves on Cn with respect to the stratification +(5.3.1) is equivalent to the category of diagrams +A0 +A1 +A2 +... +An´1 +An +d +δ +d +δ +d +δ +(5.3.3) +of vector spaces subject to the conditions +1. d2 “ 0, +2. δ2 “ 0, +3. for every 0 ď k ď n, the endomorphisms id ´dδ and id ´δd of Ak are invertible. +Proof. The stratification of Cn given by the coordinate hyperplanes, their intersections and +complements is a refinement of the stratification (5.3.1). By Theorem 5.4.1 below, a perverse +sheaf on Cn with the former stratification amounts to a certain cube of linear maps. One +readily finds that a perverse sheaf also defines a perverse sheaf on the latter stratification of +Cn if and only if all entries of this cube vanish, except for a sequence of entries as in (5.3.3). +Alternatively, Theorem 5.3.2 can also be directly deduced from iterated application of +Beilinson’s gluing formula for categories of perverse sheaves ([Bei87]). +Remark 5.3.4. We explain how to interpret the linear algebraic data from Theorem 5.3.2 +geometrically, in terms of the corresponding perverse sheaf as a constructible sheaf F valued +in the 8-category of cochain complexes of vector spaces, in analogy to Remark 5.1.1. As in +Remark 5.1.1, we assemble the linear algebraic data into a diagram +An +An´1 +. . . +A2 +A1 +A0 +öT1 +An +An´1 +. . . +A2 +A1 +0 +öT2 +An +An´1 +. . . +A2 +0 +0 +... +... +... +... +... +... +... +öTn´1 +An +An´1 +. . . +0 +0 +0 +öTn +An +0 +. . . +0 +0 +0 +δ +d +δ +δ +d +d +δ +d +δ +d +δ +δ +d +(5.3.5) +46 + +where +• the rows of (5.3.5) correspond to the stalks of Fi at the points x0, x1, x2, ..., xn P Cn +with +pxiqj “ +# +1 +for j ď i +0 +for j ą i +where the part of the complex depicted by the corresponding row of (5.3.5) is concen- +trated in degrees ´n, . . . , ´i. Thus, Ai lies in degree ´i. +• the restriction maps resi : Fi´1 Ñ Fi, 1 ď i ď n, correspond to the commutative +rectangles between the corresponding rows in (5.3.5) where all vertical maps are either +0 or id, +• we note without proof that the monodromies of each stalk about the “previous” hy- +perplane in (5.3.1) are induced by the chain automorphism Ti “ id‚ ´d‚δ‚ ´ δ‚d‚ of +Fi, +• we may interpret the maps d: A˚ Ñ A˚`1 as defining homotopies, as depicted in (5.3.5), +between resi and the composite Ti ˝ resi. +Remark 5.3.6. It is interesting to note here that the invertibility of the map T “ id ´dδ´δd +is equivalent to the invertibility of the maps id ´dδ and id ´δd by virtue of the formula +id ´dδ ´ δd “ pid ´dδqpid ´δdq “ pid ´δdqpid ´dδq +(5.3.7) +We leave the Verdier dual interpretation of the linear algebraic data in analogy to Remark 5.1.1 +to the reader, only noting that the roles of d and δ get swapped. This gives an explanation why +both d and δ need to square to 0. Theorem 5.3.2 shows that this data gives a full description +of F. +Inspired by Theorem 5.3.2, we introduce the following concept of spherical categorical +complexes. +Definition 5.3.8. A categorical complex A‚ P ChpStkq is called spherical if the differential +d : Ai Ñ Ai´1 is a spherical functor for all i P Z. +Spherical categorical complexes concentrated in degrees n, . . . , 0 can thus be regarded as +perverse schobers on Cn with respect to the stratification (5.3.1). We conclude this section +with some comments on the twist functors of spherical categorical complexes. +We fix a categorical complex A‚ P ChpStkq and i P Z. +The adjunctions di % δi and +di`1 % δi`1 give rise to the unit u : idAi Ñ δidi and the counit c : di`1δi`1 Ñ idAi. We can +compose these two natural transformations to obtain the following two commutative diagrams +in StkpAi, Aiq: +di`1δi`1 +idAi +di`1δi`1δidi +δidi +cu +u +di`1δi`1 +idAi +δididi`1δi`1 +δidi +cu +u +(5.3.9) +Note that di`1δi`1δidi » 0 and δididi`1δi`1 » 0 since δ2 » 0 and d2 » 0. +47 + +Lemma 5.3.10. Let Ti be the twist of di % δi and T 1 +i the cotwist of di`1 % δi`1. +The +totalization of the left square in (5.3.9) is equivalent to T 1 +iTi and the totalization of the right +square in (5.3.9) is equivalent to TiT 1 +i. +Proof. Immediate. +The totalization of the left diagram in (5.3.9) categorifies the expression id ´dδ ´ δd. +Since the totalization is T 1 +iTi, we find a categorification of the expression id ´T 1 +iTi “ dδ ` δd, +expressing that δ describes a homotopy between the chain maps T 1 +‚T‚ and idA‚. Similarly, the +right diagram in (5.3.9) expresses that δ is a homotopy between idA‚ and T‚T 1 +‚. The following +lemma shows that T‚T 1 +‚ » T 1 +‚T‚, categorifying the identity (5.3.7). +Lemma 5.3.11. Suppose that A‚ is spherical. The two diagrams in (5.3.9) are equivalent. +In particular, the resulting equivalence of their totalizations shows that the autoequivalences +Ti and T 1 +i commute. +Proof. Using that composition with an exact functor defines an exact functor between sta- +ble 8-categories of functors, we find that the two commutative squares in StkpAi, Aiq can +be extended to commutative diagrams where the horizontal sequences are fiber and cofiber +sequences: +T 1 +ir´1s +di`1δi`1 +idAi +T 1 +iδidir´1s +di`1δi`1δidi +δidi +T 1 +ir´1s u +cu +u +T 1 +ir´1s +di`1δi`1 +idAi +δidiT 1 +ir´1s +δididi`1δi`1 +δidi +u T 1 +ir´1s +cu +u +We begin by showing that the morphism T 1 +ir´1su and uT 1 +ir´1s in the upper diagrams are +equivalent. Consider the adjunction T 1 +iδi % dipT 1 +iq´1. Its unit is given by T 1 +iupT 1 +iq´1. Using +that T 1 +iδi » fibpdi`1δi`1δi Ñ δiq and δ2 » 0, we get T 1 +iδi » δir´1s and hence also dipT 1 +iq´1 » +dir1s. +It follows that the unit T 1 +iupT 1 +iq´1 of T 1 +iδi % dipT 1 +iq´1 is equivalent to the unit of +δir´1s % dir1s, which is equivalent to u. We thus obtain T 1 +ir´1su » uT 1 +ir´1s. +Since di`1δi`1δidi » δididi`1δi`1 » 0, it is now clear that the upper left squares in +the above two diagrams are equivalent. Hence the entire diagrams (which are recovered as +cofibers) are also equivalent. +5.4 +Spherical cubes and perverse schobers on Cn +We begin by recalling another classical description of the category of perverse sheaves on the +stratified space Cn with strata given by the coordinate hyperplanes and their (iterated) inter- +sections and complements. Let rr1ss be the 1-category with two objects 0, 1 and morphisms +freely generated by two morphisms 1 Ñ 0 and 0 Ñ 1. We identify the set of objects of rr1ssn +with the set Pprnsqop, by identifying J Ă rns with its characteristic function in rr1ssn. By a +cubical double diagram, we mean a functor rr1ssn Ñ Vectk, which amounts to the datum of +• a vector space VJ assigned to each J P Pprnsqop, +• a pair of linear maps fi : VJYtiu Ø VJ : gi assigned to each J P Pprnsqop and i P rnszJ, +• satisfying fifj “ fjfi, gigj “ gjgi and gifj “ fjgi for all i, j P rns, i ‰ j, whenever these +composites are defined. +48 + +With this terminology, we may then formulate the following classical description of perverse +sheaves on Cn: +Theorem 5.4.1 ([GGM85]). The category of perverse sheaves on Cn with respect to the +coordinate hyperplane stratification is equivalent to the category of cubical double diagrams +rr1ssn Ñ Vectk +(5.4.2) +satisfying that, for all pairs of maps fi : VIYtiu Ø VI :gi, the endomorphisms +gifi ´ idVIYtiu +and +figi ´ idVI +are invertible. +As in the 1-dimensional case, we ask the pairs of linear maps pfi, giq to become pairs of +adjoint functors upon categorification. Remarkably, the commutativity conditions gifj “ fjgi +then correspond to the Beck–Chevalley conditions already introduced and studied in §4.5 +(where the motivation for introducing them came from their effect on totalization). Inspired +by Theorem 5.4.1, we thus introduce the following: +Definition 5.4.3. A Beck-Chevalley categorical n-cube A˚ : In Ñ Stk is called spherical if +every rectilinear edge is a spherical functor. +A natural categorification of a perverse sheaf on Cn with the above stratification might +thus be a spherical categorical n-cube. We conclude this section, by showing that totalization +takes spherical categorical cubes to spherical categorical complexes. +Proposition 5.4.4. +(1) Consider a Beck-Chevalley chain map F : A‚ Ñ B‚ between spherical categorical com- +plexes, satisfying that Fi : Ai Ñ Bi is a spherical functor for all i P Z. Then FibpFq is +a spherical complex. +(2) Consider a spherical categorical n-cube A˚. Then the product totalization totˆpA˚q and +the coproduct totalization tot>pA˚q are equivalent and spherical categorical complexes. +Proof. We begin by proving part (1). We depict F as follows: +. . . +A2 +A1 +A0 +. . . +. . . +B2 +B1 +B0 +. . . +dA +2 +F2 +F1 +dA +1 +F0 +dB +2 +dB +1 +Using Theorem 5.2.3, we obtain a spherical functor +F1 +ñˆ +B1 +dB +2 : A1 +ñˆ +B1 +B2 ÝÑ B1 . +The functor +˜dA +1 :“ dA +1 +ñˆ +A1 +0 : A1 +ñˆ +B1 +B2 +πA1 +ÝÝÑ A1 +dA +1 +ÝÝÑ A0 , +49 + +where πA1 denotes the left adjoint of the inclusion of A1, is furthermore also spherical. To +see this, we first note that it is immediate by the fully faithfulness of pπA1qR that the cotwist +functor of the adjunction ˜dA +1 % p ˜dA +1 q +R is equivalent to the cotwist functor of the adjunction +dA +1 % pdA +1 q +R. To show that the twist is an autoequivalence, we apply [Dyc21, Lemma 4.13]. +Clearly, the twist acts on A1 Ă A1 +ñˆ +B1 +B2 as the twist of dA +1 % pdA +1 q +R and is thus invertible. +On B2 Ă A1 +ñˆ +B1 +B2, the twist clearly acts as the suspension functor, which is also invertible. +It thus remains to show that the twist preserves Cartesian edges, which are of the form +a ˚ÝÑ pdB +2 q +L ˝ F1paq. By the sphericalness of dB +2 we have pdB +2 q +L » T ˝ pdB +2 q +R with T the twist +functor of dB +2 +L % dB +2 . The Beck-Chevalley condition and d2 “ 0 thus imply +pdB +2 q +L ˝ F1 ˝ pdA +1 q +R ˝ dA +1 » 0 . +(5.4.5) +We get a cofiber sequence in A1 +ñˆ +B1 +B2 +a +pdA +1 q +R ˝ dA +1 paq +Tpaq +pdB +2 q +L ˝ F1paq +0 +pdB +2 q +L ˝ F1paqr1s +˚ +˚ +˚ +where (5.4.5) implies that the middle vertical arrow is Cartesian and the last vertical morphism +is Cartesian as the cofiber of Cartesian morphisms. +This shows that the twist preserves +Cartesian edge and is hence an equivalence. +The functor +A0 +p ˜dA +1 q +R +ÝÝÝÝÑ A1 +ñˆ +B1 +B2 +F1 +ñ +ˆ +B1 +dB +2 +ÝÝÝÝÝÑ B1 +is clearly equivalent to F1 ˝ pdA +1 q +R. It follows that A0 +ñ> +A1 +ñ +ˆ +B1 +B2 +B1 » A0 +ñ> +A1 B1 and the Beck- +Chevalley property further implies that A0 +ñ> +A1 B1 » A0 +ñˆ +B0 +B1. We find that the differential d1 +of FibpFq is equivalent to the the composite of these equivalences with the functor +˜dA +1 +ñ> +A1 +ñˆ +B1 +B2 +ˆ +F1 +ñˆ +B1 +dB +2 +˙ +: A1 +ñˆ +B1 +B2 ÝÑ A0 +ñ> +A1 +ñˆ +B1 +B2 +B1 . +This functor is again spherical by Theorem 5.2.3. The same argument applied to each degree +shows that all differentials of FibpFq are spherical functors, meaning that FibpFq is a spherical +categorical complex. This concludes the proof of part (1). +For part (2), we begin by noting that the equivalence of the product and coproduct total- +izations follows from repeated application of Proposition 4.3.12. We prove the sphericalness +of the product totalization via an induction on n. The case n “ 2 follows from part (1). Fix +n ě 3. We can consider the categorical n-cube A˚ as a morphism between two categorical +50 + +pn ´ 1q-cubes B˚ Ñ B1 +˚, whose totalizations are spherical categorical complexes by the in- +duction assumption. The arising morphism β : totˆpB˚q Ñ totˆpB1 +˚q is Beck-Chevalley by +Lemma 4.5.8. Repeated application of Lemma 5.4.6 shows that β is a spherical functor in +each degree. The sphericalness of totˆpA˚q thus follows again from part (1) applied to the +morphism of categorical complexes β, concluding the proof. +Lemma 5.4.6. Consider a commutative Beck-Chevalley diagram in Stk of the following form. +A +C +B +D +F +α +α1 +F 1 +If F and F 1 are spherical functors, then the induced functor +FˆÑF 1 : A +ш +α B ÝÑ C +ш +α1 D +is also spherical. +Proof. Let G and G1 be the right adjoints of F and F 1, respectively. Let TA and TB be the +twist functors of F % G and F 1 % G1, respectively. Adjoints on lax limits are determined +componentwise, see Remark A.2.5. We thus have FˆÑF 1 % GˆÑG1 and the twist functor of +this adjunction can be identified with the induced functor TA ˆÑ TB : A +ш +α B Ñ A +ш +α B. This +functor is invertible with inverse given by T ´1 +A ˆÑ T ´1 +B . An analogous argument shows that +the cotwist functor FˆÑF 1 % GˆÑG1 is invertible, showing the desired sphericalness of the +adjunction. +5.5 +The proof of Theorem 5.2.3 +Let F : A Ø B :G and F 1 : A1 Ø B :G1 be two spherical adjunction. +We denote C “ +A +ñˆ +B A1. Consider further the lax limits A +ш +F B and A1 ш +F 1 B. There are two canonical functors +evB, rcof : A +ш +F B Ñ B, acting on objects via evBpa Ñ bq “ b and rcofpa Ñ bq “ cofpFpaq Ñ +bq. There are similar functors ev1 +B, rcof1 : A1 ш +F 1 B Ñ B. Denote by D the limit of the following +diagram in Stk: +A +ш +F B +A1 ш +F 1 B +B +B +B +evB +rcof +ev1 +B +rcof1 +(5.5.1) +We further denote by B1, B2 : D Ñ B the functors contained in limit cone, going to the +leftmost and rightmost copy of B and B “ pB1, B2q: D Ñ Bˆ2. The 8-category D is a +concrete model for the 8-category RΓ1pG, Fq mentioned in §5.2 and the functor B describes +the functor pevp, evqq. +Lemma 5.5.2. The fiber of D +B2 +ÝÝÑ B is equivalent to C. The functor C ãÑ D +B1 +ÝÝÑ B is +equivalent to F +ñˆ +B F 1. +51 + +Proof. The first part follows from the observation that the fiber of rcof1 : A1 ш +F 1 B Ñ B is +equivalent to A1. +The second part can be checked for instance by describing all involved +functors using the universal properties of the lax limits. +Construction 5.5.3. We construct two functor A, C : Bˆ2 Ñ D, which we show in Lemma 5.5.5 +to be left and right adjoint to B “ pB1, B2q : D Ñ Bˆ2. +Consider the Grothendieck construction p of the diagram (5.5.1) and denote by L the +8-category of sections of p. The 8-category D can be identified with full subcategory of L +spanned by coCartesian sections. We denote +• by E1 the full subcategory of L spanned by p-relative left Kan extensions of their re- +striction to A +ш +F B. +• by E2 the full subcategory of L spanned by p-relative left Kan extensions of their re- +striction to the central copy of B. +• by E3 the full subcategory of L spanned by p-relative left Kan extensions of their re- +striction to A1 ш +F 1 B. +The restriction functor E1 Ñ A +ш +F B is a trivial fibration by [Lur09a, 4.3.2.15]. Choosing a +section, we can compose to a functor H1 : B +rcofR +ÝÝÝÑ A +ш +F B Ñ E1 Ă L. Similarly, restriction +defines a trivial fibration E2 Ñ B and we obtain a functor H2 : B Ñ E2 Ă L by choosing +a section. +The functor H2 is left adjoint to the restriction functor resB : L Ñ B at the +central copy of B, see [Lur09a, 4.3.2.17], so that we obtain a counit natural transformation +H2 ˝ resB Ñ idL. Precomposition with H1 yields a natural transformation +η1 : H2 ˝ evB ˝rcofR » H2 ˝ resB ˝H1 Ñ H1 . +Analogous to the above, we define a functor H3 : B +pev1 +BqR +ÝÝÝÝÑ A1 ш +F 1 B Ñ E3 Ă L and a natural +transformation +η2 : H2 » H2 ˝ ev1 +B ˝pev1 +BqR Ñ H3 . +The equivalence above arises from the fact that ev1 +B is a reflective localization. Finally, we +define the functor C1 : B Ñ L as the colimit of the diagram +H2 ˝ evB ˝rcofR +H1 +H3 ˝ evB ˝rcofR +η1 +η2˝evB ˝rcofR +(5.5.4) +in StkpB, Lq. It is straightforward to verify that C1 factors through D Ă L, and we consider +C1 as a functor B Ñ D in the following. +Exchanging the roles of F and F 1, or equivalently reflecting diagram (5.5.1) along the +vertical axis, and reperforming the above construction, we obtain a functor C2 : B Ñ D. We +denote +C “ pC1, C2q : Bˆ2 Ñ D . +52 + +Performing the above construction of C again, but replacing all right adjoints by left +adjoints, we obtain the functor A “ pA1, A2q : Bˆ2 Ñ D. +Lemma 5.5.5. The functor A is left adjoint to B and the functor C is right adjoint to B. +Proof. To determine the left adjoint of B C1 +ÝÑ D Ă L, we use that passing to adjoints defines +an exact functor p-qR : FunRpBˆ2, Lq Ñ FunLpL, Bˆ2qop between the functor categories of +right or left adjoint functors, and the description of C1 as the colimit of (5.5.4). The left +adjoint of H1 is by [Lur09a, 4.3.2.17] given by the composite L Ñ A +ш +F B +rcof +ÝÝÑ B of the +restriction functor to A +ш +F B and rcof. Composing this functor with D Ă L yields the functor +B1. A similar argument shows that the left adjoints of H2 ˝ evB ˝rcofR and H3 ˝ evB ˝rcofR +both restrict on D, up to equivalence, to the functor +E : D +resB +ÝÝÝÑ B +rcof ˝pev1 +BqL +ÝÝÝÝÝÝÝÝÑ B . +Using that restricting along D Ă L is also exact, we obtain that the left adjoint of C1 is given +by B1 » B1 ˆE E. +Similar arguments show B2 % C2, A1 % B1 and A2 % B2. From this it follows that A % B +and B % C. +Remark 5.5.6. The functors A1, A2, C1, C2 correspond to evpL, evqL, evpR, evqR from §5.2. +These functors are all fully faithful. This can be checked for instance by an explicit com- +putation of the derived Homs, using their pushout description in (5.5.4) and the facts that +H1, H2, H3 are fully faithful. +Proposition 5.5.7. The adjunctions A % B and B % C are spherical. +Proof. We begin by checking that the twist functor of A % B and the cotwist functor of +B % C are invertible. We have the following equivalences +idB » evB ˝rcofR +idB » ev1 +B ˝prcof1qR +TF%G » rcof ˝evBR +TF 1%G1 » rcof1 ˝pev1 +BqR +where TF%G and TF 1%G1 denote the cotwist functors of the adjunctions F % G and F 1 % G1, +see [Chr22a, Lemma 3.7] for a detailed verification of this. Unraveling the construction of C, +we thus find +BC » idBˆ2 ‘ +ˆ +0 +TF 1%G1 +TF%G +0 +˙ +as an endofunctor of Bˆ2. The second summand describes the cotwist functor T 1 of B % C +and is hence an autoequivalence. A similar description holds for the twist functor of A % B. +To deduce the sphericalness of A % B, we apply [Chr22c, Prop. 4.5], by which it suffices to +check that the twist functor T of A % B commutes pointwise with the unit of A % B and that +the essential image of A agrees with the essential image of C. The former is immediate, since +the unit is pointwise the inclusion of a direct summand. Tracing through the constructions, +we find equivalences A1 ˝ TF%Gpbq » C2pbq and A2 ˝ TF 1%G1 » C1pbq for all b P B, which shows +the latter. The sphericalness of B % C follows from the sphericalness of A % B, see [Chr22c, +Cor. 2.6], concluding the proof. +53 + +Proof of Theorem 5.2.3. We show that D has a 4-periodic semiorthogonal decomposition with +gluing functor F +ñˆ +B F 1, in the sense of [DKSS21]. We check that there is a semiorthogonal +decomposition pImpC2q, fibpB2qq of D, where ImpC2q Ă D denotes the stable subcategory given +by the essential image of C2. Given d P fibpB2q and C2pbq P ImpC2q, we have MapDpd, C2pbqq » +MapBpB2pdq, bq » ˚. Further, given d P D, the unit of B2 % C2 defines a morphism ud : d Ñ +C2B2pdq with B2pudq an equivalence, since C2 is fully faithful. This shows fibpudq P fibpB2q. +Similar arguments show the existence of semiorthogonal decompositions pfibpB2q, ImpA2qq +pImpC1q, fibpB1qq, pfibpB1q, ImpA1qq. Using that ImpC1q “ ImpA2q and ImpC2q “ ImpA1q, we +obtain the desired 4-periodic semiorthogonal decomposition of D. The gluing functors of the +involved semiorthogonal decompositions are hence spherical, see [DKSS21]. +The left gluing functor of the semiorthogonal decomposition pImpC2q, fibpB2qq is defined as +the composite fibpB2q ãÑ D πÝÑ ImpC2q, where π is right adjoint to the inclusion ImpC2q Ă D. +Under the equivalence C2 : B » ImpC2q, π identifies with TF%G ˝ B1. +Under the further +equivalences fibpB2q » C, the gluing functor identifies by Lemma 5.5.2 with TF%G ˝ F +ñˆ +B F 1. +This shows the sphericalness of F +ñˆ +B F 1. +It remains to describe the cotwist functor of F +ñˆ +B F 1 % +ˆ +F +ñˆ +B F 1 +˙R +. Let ι : C Ă D denote +the composite of C » fibpB2q with the inclusion. The equivalence B2C1 » TF 1%G1 induces by +passing to the adjoint C2 of B2 a natural transformation η : C1 Ñ C2 ˝TF 1%G1, satisfying that +fibpηq : B Ñ D factors through ι : C Ă D. We denote the factorization by fibpηq1 : B Ñ C. +Using that B2 ˝ ι » 0, we find +fibpηq1 » ιR ˝ ι ˝ fibpηq1 » ιR ˝ fibpηq » +´ +pfibpηqL ˝ ι +¯R +» +` +cofpT ´1 +F 1%G1 ˝ B2 Ñ B1q ˝ ι +˘R » pB1 ˝ ιqR » +ˆ +F +ñˆ +B F 1 +˙R +. +We thus have an equivalence +ˆ +F +ñˆ +B F 1 +˙ +˝ +ˆ +F +ñˆ +B F 1 +˙R +» B1˝fibpηq » fibpB1C1 Ñ B1C2TF 1%G1q » fibpidB Ñ TF%G˝TF 1%G1q . +One finds the counit of the adjunction F +ñˆ +B F 1 % +ˆ +F +ñˆ +B F 1 +˙R +to arise as the apparent natural +transformation fibpidB Ñ TF%G ˝ TF 1%G1q Ñ idB, which implies that the cotwist functor of +the adjunction is equivalent to TF%G ˝ TF 1%G1r´1s, as desired. This concludes the proof. +Proof sketch of Theorem 5.2.5. Let i be the number of boundary components of S and EB “ +š +1ďjďi EB +j the decomposition of the set EB of external edges of G into the sets of edges ending +on a given boundary component of S. For each 1 ď j ď i, the functor ś +xPEB +j eve : RΓpG, Fq Ñ +ś +xPEB +j Fpxq is spherical. This follows from adapting the proof of Proposition 5.5.7 if |EB +j | ě +2 or the proof of Theorem 5.2.3 if |EB +j | “ 1. +The main difficulties in spelling out these +adaptations is the required involved notation and we omit these details in favor of a shorter +exposition. +54 + +Let now 1 ď j1, j ď i with j1 ‰ j. It is easy to see that ś +xPEB +j eve ˝ ś +xPEB +j1 eveR » 0. +Hence, the twist functors of the adjunctions ś +xPEB +j eve % ś +xPEB +j eveR and ś +xPEB +j1 eve % +ś +xPEB +j1 eveR commute. The twist functor of the adjunction ś +xPEB eve % ś +xPEB eveR is equiv- +alent to the composite of the commuting twist functors of the adjunctions ś +xPEB +l eve % +ś +xPEB +l eveR, 1 ď l ď i, and hence an equivalence. +The cotwist functor of ś +xPEB eve % +ś +xPEB eveR acts on Fpeq, with e P EB +l an external edge, as the cotwist functor of the adjunc- +tion ś +xPEB +l eve % ś +xPEB +l eveR and is hence also invertible. This shows the sphericalness of +the functor ś +xPEB eve. +6 +Calabi-Yau complexes +The goal of this section is to introduce a notion of Calabi–Yau structure on a categorical +complex. In §6.1, we recall the definitions of Hochschild homology and negative cyclic ho- +mology of k-linear 8-categories and introduce the total Hochschild homology of a categorical +complex. In §6.2, we define left Calabi–Yau structures on categorical complexes. In §6.3, we +discuss some ways in which Calabi–Yau structures arise on lax limits. In the final §6.4, we +introduce Calabi–Yau structures on categorical cubes and show that they induce Calabi–Yau +structures on their totalizations. This will be our main technique to construct examples of +categorical Calabi–Yau complexes in §7. +6.1 +Total Hochschild and negative cyclic homology +Let C be a compactly generated k-linear 8-category. As explained in §2.1, the dual of C with +respect to the monoidal structure on Stk can be described as +C_ “ IndpCop +0 q, +where C0 Ă C denotes the subcategory of compact objects. We thus have +StkpD, C_q » StkpD bk C, Modkq +for any D P Stk. In particular, we have StkpC, Cq » StkpC bk C_, Modkq and can define the +diagonal bimodule ∆C : C bk C_ Ñ Modk as the image of the identity functor idC under this +equivalence. The k-linear Hochschild homology of C is defined as the trace +HHpCq :“ p∆C ˝ φp∆Cqqpkq P Modk, +where φ denotes the equivalence StkpC bk C_, Modkq » StkpModk, C bk C_q. When k is the +sphere spectrum, we obtain topological Hochschild homology and when k is a field, we obtain +the usual Hochschild complex. Hochschild homology comes equipped with an action of the +circle group S1, see [HSS17], and we denote its fixed points by HHS1pCq. We call HHS1pCq +the negative cyclic homology of C. Hochschild homology and negative cyclic homology form +functors +HH, HHS1 : Stcpt +k +Ñ Modk +on the subcategory Stcpt +k +Ă Stk of compactly generated 8-categories and compact objects +preserving functors, see [HSS17]. There is a canonical natural transformation HHS1 Ñ HH. +55 + +The k-linear stable 8-category C is called smooth if it is compactly generated and ∆C +admits a bimodule left dual ∆! +C P StkpModk, Cbk C_q. In this case, we sometimes consider ∆! +C +as an object in StkpCbkC_, Modkq using the equivalence φ. In this case, the k-linear Hochschild +homology of C is equivalent to RHomStkpCbkC_,Modkqp∆! +C, ∆Cq. +Given a k-linear, compact +objects preserving functor F : C Ñ D, we denote HHpD, Cq “ cof HHpFq and HHS1pD, Cq “ +cof HHS1pFq. Supposing that C and D are smooth, the arising morphism +RHomStkpCbkC_,Modkqp∆! +C, ∆Cq » HHpC, Cq +HHpFq +ÝÝÝÝÑ HHpD, Dq » RHomStkpDbkD_,Modkqp∆! +D, ∆Dq +admits a concrete description, as noted in [BD21, Prop. 4.4]. It maps a degree m-morphism +α: ∆! +C Ñ ∆Crms to the morphism ∆! +D Ñ ∆Drms obtained as the composite +∆! +D +F˚p∆! +Cq +F˚p∆Cqrms +∆Drms +u +F˚pαq +curms +where F˚ is the functor +StkpC bk C_, Modkq » StkpC, Cq +F˝-˝G +ÝÝÝÝÑ StkpD, Dq » StkpD bk D_, Modkq +(6.1.1) +with G the k-linear right adjoint of F. Above, the natural transformation cu is the counit of +F % G and u is the defined as the composite +∆! +D Ñ ∆! +D ˝ ∆C ˝ ∆! +C Ñ ∆! +D ˝ ∆D ˝ pF bk Ind fopq ˝ ∆! +C Ñ pF bk Ind fopq ˝ ∆! +C » F˚p∆! +Cq , +with Ind fop : C_ Ñ D_ obtained by restricting F to compact objects, passing to opposite +categories and Ind-completing. +Definition 6.1.2. A categorical n-complex A˚ P ChnpStkq is called smooth if it consists of +smooth k-linear stable 8-categories and all differentials preserve compact objects. +Definition 6.1.3. Let A‚ P ChpStkq be a bounded, smooth categorical complex. The total +negative cyclic homology HHS1,totpA‚q is defined as the total cofiber of the negative cyclic +homology cube, obtained by applying HHS1 to the corresponding categorical cube of A‚, see +§2.2 (see [DJW19][A.2] for terminology and basic results on total fibers and cofibers). The +total Hochschild homology HHtotpA‚q is defined similarly. We have normalized the suspensions +in the totalization such that, if A‚ “ Ar0s is concentrated in degree 0, then HHS1,totpA‚q » +HHS1pAq and HHtotpA‚q » HHpAq. +Definition 6.1.4. Let A‚ P ChpStkq. +• The lower truncation τěipA‚q at i P Z of A‚ is defined as the complex +A‚ +. . . +Ai`1 +Ai +Ai´1 +Ai´2 +. . . +τěipA‚q +. . . +Ai`1 +Ai +0 +0 +. . . +which is identical to A‚ in degrees i or larger and vanishes in degrees less than i. Lower +truncation forms a functor τěi : ChpStkq Ñ ChpStkq and is equipped with a canonical +natural transformation idChpStkq Ñ τěi. +56 + +• The upper truncation τ ďipA‚q at i P Z of A‚ is defines as the complex +A‚ +. . . +Ai`2 +Ai`1 +Ai +Ai´1 +. . . +τ ďipA‚q +. . . +0 +0 +Ai{Ai`1 +Ai´1 +. . . +x +which vanishes in degrees larger than i, is in degree i given by the cofiber of Ai`1 +dÝÑ Ai +and in the other degrees identical to A‚. Upper truncation forms a functor τ ďi : ChpStkq Ñ +ChpStkq and is equipped with a canonical natural transformation idChpStkq Ñ τ ďi. Note +that τ ďi preserves smooth categorical complexes, as smoothness of categories is pre- +served under quotients along compact objects preserving functors. +• The two-sided truncation τ ďi +ějpA‚q is defined as τ ďi ˝ τějpA‚q. Note that the upper and +lower truncation commute. +Lemma 6.1.5. Let A‚ P ChpStkq be a bounded, smooth categorical complex. +There is a +canonical biCartesian square in Modk: +HHS1,totpτěipA‚qq +HHS1,totpτ ďi`1 +ěi +pA‚qq +HHS1pAi, Ai`1{Ai`2qris +HHS1,totpτěi`1pA‚qq +HHS1,totpτ ďi`1 +ěi`1 pA‚qq +HHS1pAi`1{Ai`2qri ` 1s +˝ +» +» +Proof. This follows from the pasting law for biCartesian squares and the commutative diagram +HHS1,totpτěipA‚qq +HHS1pAi, Ai`1{Ai`2qris +0 +HHS1,totpτěi`1pA‚qq +HHS1pAi`1{Ai`2qri ` 1s +HHS1pAiqri ` 1s +˝ +in which the right square and the outer square are biCartesian. +Repeatedly applying Lemma 6.1.5, we obtain that HHS1,totpA‚q is equivalent to the limit +of the diagram +HHS1pAi´1, Ai{Ai`1qri ´ 1s +. . . +HHS1pAi, Ai`1{Ai`2qris +HHS1pAi{Ai`1qris +HHS1pAi`1, Ai`2{Ai`3qri ` 1s +HHS1pAi`1{Ai`2qri ` 1s +. . . +The limit of the above diagram is equivalent to an equalizer as follows: +57 + +Proposition 6.1.6. Let A‚ P ChpStkq be a bounded, smooth categorical complex. +Then +HHS1,totpA‚q is equivalent to the equalizer of the diagram in Modk +À +iPZ HHS1pAi, Ai`1{Ai`2qris +À +iPZ HHS1pAi{Ai`1qris , +À δi +À πi +where +δi : HHS1pAi, Ai`1{Ai`2qris Ñ HHS1pAi`1{Ai`2qri ` 1s +is the fiber map and +πi : HHS1pAi, Ai`1{Ai`2qris Ñ HHS1pAi{Ai`1qris +is the map induced by the quotient functor Ai Ñ Ai{Ai`1. +The above statement also holds when replacing HHS1 with HH and HHS1,tot with HHtot. +Remark 6.1.7. Proposition 6.1.6 expresses, that a Hochschild or total negative cyclic ho- +mology class consists of a compatible family of relative Hochschild or relative negative cyclic +homology classes of the functors Ai`1{Ai`2 Ñ Ai, shifted into the appropriate degrees. +6.2 +Left Calabi–Yau structures +Let F : C Ñ D be a compact objects preserving, k-linear functor between smooth k-linear +8-categories. As explained in §6.1, a k-linear Hochschild class σ: krns Ñ HHpD, Cq defines a +diagram +∆! +D +F˚p∆! +Cq +F˚p∆Cqrn ´ 1s +∆Crn ´ 1s +together with a choice of null-homotopy. It hence gives rise to a diagram with horizontal fiber +and cofiber sequences as follows: +∆! +D +F˚p∆! +Cq +cof +fib +F˚p∆Cqrn ´ 1s +∆Crn ´ 1s +We call the Hochschild class σ non-degenerate if all vertical maps in the above diagram are +equivalences. +Definition 6.2.1. +(1) A weak left n-CY structure on the functor F consists of a non- +degenerate Hochschild class σ: krns Ñ HHpD, Cq. +(2) A left n-CY structure on the functor F consists of a negative cyclic class η: krns Ñ +HHS1pD, Cq, whose image under HHS1pD, Cq Ñ HHpD, Cq defines a non-degenerate +Hochschild class. +We refer to Example 6.3.2, Theorem 7.1.4 and Theorem 7.4.1 for examples of left Calabi– +Yau structures. +58 + +Definition 6.2.2. Let A‚ P ChpStkq be a bounded, smooth categorical complex. +(1) A weak left n-Calabi–Yau structure on A‚ consists of a class σ: krns Ñ HHtotpA‚q, +whose composite with the morphism from Proposition 6.1.6 +HHtotpA‚q ÝÑ +à +iPZ +HHpAi, Ai`1{Ai`2qris +defines a collection of weak left pn ´ iq-Calabi–Yau structures on the functors +Ai`1{Ai`2 Ñ Ai. +(2) A left n-Calabi–Yau structure on A‚ consists of a class η: krns Ñ HHS1,totpA‚q, whose +composite with the morphism from Proposition 6.1.6 +HHS1,totpA‚q ÝÑ +à +iPZ +HHS1pAi, Ai`1{Ai`2qris +defines a collection of left pn´iq-Calabi–Yau structures on the functors Ai`1{Ai`2 Ñ Ai. +Remark 6.2.3. There is an apparent analogue of Definition 6.2.2 for bounded complexes +of proper k-linear 8-categories, obtained by replacing left Calabi–Yau structures with right +Calabi–Yau structures. +6.3 +Calabi–Yau structures and lax limits +Consider a colimit diagram in Stk +A2 +A +B1 +A1 +B +C +x +where all appearing 8-categories are smooth and all functors preserve compact objects. Sup- +pose further that the functors AˆA1 Ñ B and AˆA2 Ñ B1 carry left n-Calabi–Yau structures, +which are compatible at A. +Theorem 6.3.1. The functor A1 ˆ A2 Ñ C inherits a left n-Calabi–Yau structure. +Proof. If k is a field, this is [BD19, Thm. 6.2]. For k an arbitrary E8-ring spectrum, a proof +will appear in [Chr22b]. +Theorem 6.3.1 admits a refinement, in the form of a relative Calabi–Yau structure on the +directed pushout. Before stating that result in Lemma 6.3.3, we need to consider the following +class of examples of relative left Calabi–Yau structures. +Example 6.3.2. Let A be a smooth k-linear 8-category with a left pn ´ 1q-Calabi–Yau +structure η: krn ´ 1s Ñ HHS1pAq. Consider the functor +GA2 “ pG1, G2, G3q :“ pev0, cof, ev1q: Funp∆1, Aq ÝÑ Aˆ3 , +59 + +with components given by the evaluations at i P t0, 1u and the cofiber, respectively. The +notation A2 comes from regarding Funp∆1, Aq as the 8-category of A-valued representations +of the A2-quiver. The left adjoint +FA2 “ pF1, F2, F3q: Aˆ3 Ñ Funp∆1, Aq +of GA2 admits a left n-Calabi–Yau structure which restricts to the given left pn ´ 1q-Calabi– +Yau structure η‘3 : k‘3 Ñ HHS1pAq‘3 » HHS1pAˆ3q on Aˆ3. +If k is a field, this can be seen as follows. For A “ Modk » Dpkq with the apparent +left 0-Calabi–Yau structure, this left 1-Calabi–Yau structure on FA2 is described in [BD19, +Thm. 5.14]. For a general k-linear 8-category A, we have Funp∆1, Modkq b A » Funp∆1, Aq +and Modˆ3 +k +bA » Aˆ3 using the symmetric monoidal structure on Stk. +The desired left +Calabi–Yau structure now arises as the image of the above pair of Calabi–Yau structures +under the canonical map +HHS1 +1 pFunp∆1, Modkq, Modˆ3 +k q ˆ HHS1 +n´1pAq Ñ HHS1 +n pFunp∆1, Aq, Aˆ3q . +If k is not a field, a slightly more careful argument is required, this case will be treated in +[Chr22b]. +Lemma 6.3.3. The functor +A ˆ A1 ˆ A2 Ñ B +ñ> +A B1 +(6.3.4) +inherits a left n-Calabi–Yau structure which induces the left n-Calabi–Yau structure on A1 ˆ A2 Ñ +C from Theorem 6.3.1 when forming the pushout along the functor A Ñ 0. +Proof. We have the following colimit diagram in Stk: +A +A ˆ A +Funp∆1, Aq +A1 ˆ A2 +B ˆ B1 +B +ñ> +A B1 +F2 +pF1,F3q +x +Applying Theorem 6.3.1, we find that the left n-Calabi–Yau structures on FA2 and on A ˆ +A ˆ A1 ˆ A2 Ñ B ˆ B1 glue to a left n-Calabi–Yau structure on the functor (6.3.4). +We end this section with the following further observations concerning left Calabi–Yau +structures on lax limits. +Lemma 6.3.5. Let F : A Ñ B be a k-linear, compact objects preserving functor between +smooth k-linear stable 8-categories. +(1) Let π: A1 Ñ A be a k-linear functor which preserves compact object and admits a fully +faithful right adjoint. A class +η: krns ÝÑ HHS1pB, A1q +60 + +determines a left n-Calabi–Yau structure on F ˝ π if and only if its composite with +HHS1pB, A1q ÝÑ HHS1pB, Aq +determines a left n-Calabi–Yau structure on F. +(2) Assume that there are k-linear functors Fi : A Ñ Bi with 1 ď i ď n, such that +B » ppB1 +ñ> +A B2q +ñ> +A . . .q +ñ> +A Bn +is an iterated directed pushout and F is equivalent to the iteratively induced functor into +the directed pushout. Compatible left n-Calabi–Yau structures on the functors Fi for all +1 ď i ď n canonically determine a left n-Calabi–Yau structure on F. +(3) Conversely, in the setting of (2), a left n-Calabi–Yau structure on F determines com- +patible left n-Calabi–Yau structures on the functors Fi, 1 ď i ď n; these assignments +are inverse to each other. +Proof. Part (1) follows from the observation that the functor π˚, see (6.1.1), maps the diagonal +bimodule ∆A1 to ∆A, by the fully faithfulness of πR +Part (2) follows from repeated application of Lemma 6.3.3 (with A1 “ A2 “ 0). +For part (3), we note that the projection map πi : B ↠ Bi admits a fully faithful right +adjoint for each 1 ď i ď n. Again pπiq˚ maps ∆B to ∆Bi and the arising map +HHS1pB, Aq ÝÑ HHS1pBi, Aq +thus left n-Calabi–Yau structures to left n-Calabi–Yau structures. +6.4 +Cubical Calabi–Yau structures +Recall that I “ r1sop “ t1 Ñ 0u. Let +A˚ : IN ÝÑ Stk +be a smooth, cubical diagram of k-linear 8-categories. Applying HHS1yields a cubical diagram +HHS1pA˚q: IN ÝÑ Modk . +We denote by HHS1,totpA˚q P Modk the total cofiber of HHS1pA˚q. To clarify the chosen +grading, note that if A˚ assigns 0 to all vertices of the cube except p0, . . . , 0q, then we have +HHS1,totpA˚q » HHS1pAp0,...,0qq. For L P Modk, we refer to a map η: L Ñ HHS1,totpA˚q as +an L-class. Using that the total cofiber is the N-fold suspension of the total fiber, we may +identify an L-class η with a natural transformation in FunpIN, Modkq +η: Lr´Ns0 ÝÑ HHS1pA˚q +where, for E P Modk, we denote by E0 the cubical diagram +E0 : IN ÝÑ Modk, +x ÞÑ +# +E +for x “ p1, 1, ..., 1q, +0 +else. +From this perspective, we obtain from η the following data: +61 + +1. for every 1 ď i ď N, a morphisms in FunpIN´1, Modkq +Biη: Lr´Ns0 ÝÑ HHS1pABi˚q +where ABi˚ denotes the restriction of the cube A˚ to the face IN´1 Ñ IN obtained by +setting the ith coordinate to 1, and hence a class +Lr´1s ÝÑ HHS1,totpABi˚q, +(6.4.1) +2. a diagram +Lr´1s +0 +colimăp0,...,0q HHS1pA˚q +HHS1pcolimăp0,...,0q A˚q +HHS1pAp0,...,0qq, +and hence a relative class +L ÝÑ HHS1,totpA˚q » HHS1pAp0,...,0q, colimăp0,...,0q A˚q, +(6.4.2) +where above colimăp0,...,0q denotes the colimit over the punctured cube IN +ăp0,...,0q “ +IN ztp0, . . . , 0qu. +We recursively define an n-Calabi–Yau structure on cubical diagrams as follows. +Definition 6.4.3. Let A˚ : IN ÝÑ Stk be a smooth cubical diagram in Stk. A class η: krns Ñ +HHS1,totpA˚q is called an n-Calabi–Yau structure on A˚, if +(1) for every 1 ď i ď N, the class +krn ´ 1s Ñ HHS1,totpABi˚q +from (6.4.1) defines a left pn ´ 1q-Calabi–Yau structure on the pN ´ 1q-cube ABi˚, +(2) the class krns Ñ HHS1pAp0,...,0q, colimăp0,...,0q A˚q from (6.4.2) defines a left n-Calabi– +Yau structure on the functor +colimăp0,...,0q A˚ ÝÑ Ap0,...,0q. +(6.4.4) +Example 6.4.5. Let A‚ P ChpStkq be categorical complex concentrated in degrees N, N ´ 1, +..., 0 so that A‚ gives rise to a categorical N-cube A˚ (see (2.2.4) in §2.2). Then HHS1,totpA‚q » +HHS1,totpA˚q and a class η: krns Ñ HHS1,totpA‚q describes a left n-Calabi-Yau structure on +A‚ if and only if it describes a left n-Calabi-Yau structure on A˚. +Theorem 6.4.6. Let A˚ be a smooth cubical diagram in Stk equipped with a left n-Calabi-Yau +structure. Then its coproduct totalization inherits a canonical left n-Calabi–Yau structure. +62 + +Before proving Theorem 6.4.6, we collect some preparatory results. We begin with an anal- +ysis of the differentials in the totalization of a categorical cube. For notational convenience, +we will use the reparameterization of the poset IN`1 “ pr1sopqN`1 by the poset PprNsqop +of subsets of the set rNs “ t0, 1, ..., Nu (by means of associating to a subset its character- +istic function). For instance, p1, . . . , 1q is identified with rNs P PprNsqop, p1, 0, . . . , 0q with +t1u P PprNsqop and p0, . . . , 0q with H P PprNsqop. +Let A˚ : PprNsqop Ñ Stk be a smooth categorical N-cube with N ě 2. When taking +product and coproduct totalizations, we equip the subset of PprNsqop of subset of cardinality +i with the canonical total order, where I ă J if minpIzJq ă minpJzIq, for I, J Ă rNs. +Lemma 6.4.7. There exist equivalences in Stk +totˆpA˚qi » +ñ +ą +totˆpA˚qi´1 +AJ +and +tot>pA˚qi » +ñ +ž +tot>pA˚qi`1 +AJ +where the iterated directed pullback and the iterated directed pushout run over all J Ă rNs +with |J| “ i, with the order as specified above. +Proof. We only verify the formula for the product totalization, the coproduct totalization +can be treated analogously. The proof is by induction on N using the iterative definition +of the product totalization. The case N “ 2 is clear. We proceed with the induction step. +Given a categorical pN ` 1q-cube A˚, we can consider it as describing a chain map f˚ : +B˚ Ñ C˚ between N-cubes. The induction step now follows from combining the following +two observations. +• We have that totˆpB˚qi » Śñ +totˆpA˚qi´1 BJ and a similar statement for C˚. This follows +from the fact that the functor totˆpA˚qi´1 Ñ totˆpB˚qi´1 is a reflective localization. +• Spelling out the definition of the iterated directed pullback, one finds that the order of +the bracketing is irrelevant (i.e. satisfies associativity), implying that +¨ +˝ +ñ +ą +totˆpA˚qi´1 +BJ +˛ +‚ˆñ +totˆpA˚qi´1 +¨ +˝ +ñ +ą +totˆpA˚qi´1 +CJ +˛ +‚» +ñ +ą +totˆpA˚qi´1 +AJ . +Let J Ă rNs with |J| “ i. Our next goal is to find a description of the functor +tot>pA˚qi`2 +dÝÑ tot>pA˚qi`1 » +ñ +ž +tot>pA˚qi`1 +AJ1 ↠ AJ . +Definition 6.4.8. Fix J Ă rNs with |J| “ i. +63 + +(1) We define the stable 8-category P J as the full subcategory +PJ :“ +ñ +ą +totˆpA˚qi´2, jPJ +AJztju Ă +ñ +ą +totˆpA˚qi´2 +AJ1 , +meaning the stable subcategory generated by the components AJztju of the iterated lax +product. We further define kerpdqJ Ă P J as the fiber in Stk of +P J Ă totˆpA˚qi´1 +d +ÝÑ totˆpA˚qi´2 . +(2) The dual version PJ is defined as the full subcategory +PJ :“ +ñ +ž +tot>pA˚qi`2, jPrNszJ +AJYtju Ă +ñ +ž +tot>pA˚qi`2 +AJ1 » tot>pA˚qi`1 . +The left adjoint of this inclusion is denoted by tot>pA˚qi`1 ↠ PJ. We further define +cokerpdqJ as the cofiber in Stk of +tot>pA˚qi`2 +d +ÝÑ tot>pA˚qi`1 ↠ PJ . +Lemma 6.4.9. Let J Ă rNs. +(1) The k-linear stable 8-category kerpdqJ is equivalent to the limit in Stk of the restriction +of A˚ to the full subcategory PprNsqop +ąJ of PprNsqop spanned by objects J1 ‰ J satisfying +J1 Ă J. +(2) The k-linear 8-category cokerpdqJ is equivalent to the colimit in Stk of the restriction +of A˚ to the full subcategory PprNsqop +ăJ of PprNsqop spanned by objects J1 ‰ J satisfying +J Ă J1. +Proof. Part (2) is dual to part (1), i.e. follows from passing to right adjoints. +Part (1) +follows from an induction over |J| and by decomposing the limit over PprNsqop +ąJ via [Lur09a, +4.2.3.10]. +Via the universal properties of these limits and colimits, we obtain functors dJ : AJ Ñ +kerpdqJ and dJ : cokerpdqJ Ñ AJ. +Lemma 6.4.10. Let A˚ be a categorical n-cube and let J Ă rNs with |J| “ i. +(1) There is a commutative diagram in Stk +AJ +kerpdqJ +totˆpA˚qi +totˆpA˚qi´1 +totˆpA˚qi´2 +dJ +0 +d +d +with fully faithful vertical functors. Furthermore, if A˚ takes values in limit preserving +functors, then all functors in the above diagram also preserve limits. +64 + +(2) There is a commutative diagram in Stk +tot>pA˚qi`2 +tot>pA˚qi`1 +tot>pA˚qi +cokerpdqJ +AJ +d +0 +d +dJ +such that the vertical functors admit fully faithful right adjoints. Furthermore, if A˚ +takes values in compact objects preserving functors, then all functors in the above dia- +gram also preserve compact objects. +Proof. We only prove part (1), part (2) is dual. Using the notation of Definition 6.4.8, we +have a commutative diagram in Stk: +kerpdqJ +0 +AJ +P J +totˆpA˚qi +totˆpA˚qi´1 +totˆpA˚qi´2 +{ +d +d +The top right square is pullback, the dotted arrow arises via the universal property of this +pullback. +Unraveling the definition of d, one can further show that the dotted arrow is +described by the functor dJ. If A˚ takes values in limit preserving functors, the pullback is +also equivalent to the pushout of the k-linear left adjoint diagram in Stk. Passing again to +right adjoints, we find that all functors in the diagram preserve limits. +Example 6.4.11. Consider a categorical 3-cube A˚. Then +cokerpdqtju » Atj,j1u >Ar3s Atj,j2u +with j, j1, j2 P r3s pairwise distinct. The 8-category cokerpdqH is equivalent to the colimit in +Stk of the diagram +Ar3s +At1,2u +At1,3u +At1u +At2,3u +At2u +At3u +which is also equivalent to the iterated (usual, not directed) pushout +´ +At1u >At1,2u At2u +¯ +>At1,3u>Ar3sAt2,3u At3u . +65 + +Proposition 6.4.12. Let A‚ P ChNpStkq be a smooth cube. A class +η: krns Ñ HHS1,totptot>pA‚qq , +which induces left pn ´ |J|q-Calabi–Yau structures on the functors dJ : cokerpdqJ Ñ AJ, for +all J Ă rNs, describes a left n-Calabi–Yau structure on tot>pA‚q. +Proof. Combine parts (1) and (2) of Lemma 6.3.5 with Lemma 6.4.7 and Lemma 6.4.10. +Proof of Theorem 6.4.6. There is a canonical equivalence HHS1,totpA˚q » HHS1,totptot>pA˚qq, +which can be obtained using the recursive definition of the coproduct totalization. Consider +a left n-Calabi-Yau structure η: krns Ñ HHS1,totpA˚q of A˚. Under the above equivalence, +the arising top non-degeneracy condition of HHS1,totpABi...Bj˚q implies that the functor dJ : +cokerpdqJ Ñ AJ with J “ ti, . . . , ju inherits a left pn´|J|q-Calabi–Yau structure from η. The +Theorem thus follows from Proposition 6.4.12. +Remark 6.4.13. In view of part (3) of Lemma 6.3.5, the proof of Theorem 6.4.6 also shows +the ’converse’ of Theorem 6.4.6, namely that any left n-Calabi–Yau structure of tot>pA˚q +canonically determines a left n-Calabi-Yau structure on A˚. +7 +Examples +7.1 +Normal crossings divisors and categorical intersection complexes +As a standing assumption in this section, we require that all schemes are separated, reduced +and of finite type over a field k of characteristic zero. Morphisms between schemes are assumed +to be separated. The category of such schemes is denoted Schk. +Let k be a field of characteristic 0, Z a smooth scheme over k and i: D Ă Z a normal +crossing divisor, by which we mean a union D “ Ť +1ďiďN Di of smooth divisors Di, intersecting +transversely. We may organize the intersections +DI :“ +č +iPI +Di, +for the various subsets I Ă t1, ..., Nu, along with their inclusions DI Ď DJ for I Ě J, into a +cubical diagram D˚ of smooth schemes. Here, we interpret the empty intersection DH as the +ambient scheme Z. For example, for N “ 2, 3, the resulting cube can be depicted as follows: +D1 X D2 +D1 +D2 +Z +D1 X D2 X D3 +D1 X D3 +D2 X D3 +D3 +D1 X D2 +D1 +D2 +Z +66 + +Consider the functor IndCoh: NpSchkq Ñ Stk, which assigns to a scheme Y its k-linear 8- +category IndCohpY q of Ind-coherent sheaves and to a morphism f : Y Ñ Y 1 of schemes the +k-linear functor f˚ : IndCohpY q Ñ IndCohpY 1q, see [GR17] for details. If f is proper, then +the functor f˚ admits a right adjoint f!, which preserves colimits, thus defining a morphism +in Stk. Applying IndCoh to the cube D˚, we obtain a categorical N-cube IndCohpD˚q, for +example in the case N “ 2 given by: +IndCohpD1 X D2q +IndCohpD1q +IndCohpD2q +IndCohpZq +(7.1.1) +This categorical cube is Beck-Chevalley by base change, see [GR17, Cor. 3.1.4]. +Definition 7.1.2. We call the coproduct totalization tot>pIndCohpD˚qq of IndCohpD˚q the +categorical intersection complex of tDiu1ďiďN. +For the square (7.1.1), the categorical intersection complex has 3 nontrivial terms given +by +IndCohpD1 X D2q ÝÑ IndCohpD1q +ñ> +IndCohpD1XD2q IndCohpD2q ÝÑ IndCohpZq . +The 8-category +IndCohpD1q +ñ> +IndCohpD1XD2q IndCohpD2q +describes the “lax gluing” of the two schemes D1 and D2 along D1 Y D2. We proceed with +describing situations in which the categorical intersection complex is spherical or allows a +Calabi–Yau structure. +Proposition 7.1.3. Suppose that Z is a smooth, projective variety. Then IndCohpD˚q is a +spherical categorical cube. In this case, the categorical intersection complex is hence a spherical +categorical complex. +Proof. The sphericalness of the cube follows from the fact that the inclusion of any smooth +divisor into a smooth projective variety induces a spherical functor when passing to Ind- +coherent sheaves, see for instance [Add16]. The statement about the totalization follows from +Proposition 5.4.4. +Recall the following Theorem from [BD19]: +Theorem 7.1.4 ([BD19, Thm. 5.13]). Suppose that Z is a Gorenstein scheme of dimension +n and i: D Ă Z an anticanonical divisor. Then IndCohpDq admits a canonical left pn ´ +1q-Calabi–Yau structure and the functor i˚ : IndCohpDq Ñ IndCohpZq admits a canonical +compatible left n-Calabi–Yau structure. +In terms of categorical complexes, this result equips the categorical 2-term complex +IndCohpDq +i˚ +ÝÑ IndCohpZq +associated to the anticanonical divisor D Ă Z with an n-dimensional Calabi–Yau structure. +We will now show that, if D “ Ť +1ďiďN Di is an anticanonical normal crossings divisor, +67 + +then the statement can be refined to provide an n-dimensional Calabi–Yau structure on the +categorical intersection complex of tDiu1ďiďN. For the proof, we require the sphericalness of +the cube, and to this end we assume that Z is smooth projective. +Theorem 7.1.5. Let Z be a smooth projective scheme of dimension n and let D “ Ť +1ďiďN Di +be an anticanonical normal crossings divisor. Then the categorical N-cube +IndCohpD˚q +is spherical and admits a canonical left n-Calabi–Yau structure. +Corollary 7.1.6. Let Z be a smooth projective scheme of dimension n and let D “ Ť +1ďiďN Di +be an anticanonical normal crossings divisor. Then the categorical intersection complex +tot>pIndCohpD˚qq +is spherical and admits a canonical left n-Calabi–Yau structure. +Proof. Combine Theorem 7.1.5, Proposition 5.4.4 and Theorem 6.4.6. +Remark 7.1.7. Theorem 7.1.5 and Corollary 7.1.6 provide possible answers to the question +raised by Katzarkov-Kontsevich-Pantev, as to what kind of Calabi–Yau and spherical cate- +gorical structures arise from an anticanonical normal crossings divisor, see Remark 4.36 in +[KKP08]. +Before proving Theorem 7.1.5, we recall some results from [BD19, Section 5.2]. Given a +scheme Z, we denote by RHomZp-, -q the k-linear derived Hom in IndCohpZq. We denote by +ω‚ +Z the dualizing complex of Z, defined as π!pkq with π: Z Ñ ˚. Using the diagonal map +∆: Z Ñ Z ˆ Z, one finds a canonical morphism +RHomZpOZ, ω‚ +Zq Ñ RHomZˆZp∆˚OZ, ∆˚ω‚ +Zq » HHpIndCohpZqq , +natural in Z. +A scheme Z is called Cohen-Macaulay of dimension d if ωX “ ω‚ +Xr´ds is a coherent sheaf. +In this case, there exists an isomorphism Hd HHpIndCohpZqq » Hd HHS1pIndCohpZqq. In the +setting of Theorem 7.1.4, we further have i!ωZ » ωDr´1s and a cofiber sequence +OZ Ñ i˚OD Ñ ωZr1s . +(7.1.8) +A straightforward computation shows that the morphism i˚OD Ñ ωZr1s is adjoint to an +equivalence OD » ωD, see [BD19]. Furthermore, the morphism i˚OD Ñ ωZr1s factors through +the equivalence i˚OD » i˚ωD via the counit cu of i˚ % i!: +i˚OD » i˚ωD » i˚i!ωZr1s cu +ÝÑ ωZr1s . +It follows that the cofiber sequence (7.1.8) gives rise to a relative negative cyclic homology +class describing a Calabi–Yau structure on i˚. +Recall our notation DJ :“ Ş +jPJ Dj, for J Ă rNs. We further denote ˜DJ :“ Ť +jPrNszJ DJYtju Ă +DJ. The inclusions DJ Ă Z is denoted fJ. +68 + +Each component Di Ď Z is cut out by a section si : OZ Ñ OpDiq. We dualize si to obtain +a map s_ +i : Op´Diq Ñ OZ which is part of a cofiber sequence +Op´Diq +OZ +0 +pfiq˚ODi . +The tensor product of the morphisms s_ +i , 1 ď i ď n, yields a cubical diagram +p: PprNsqop Ñ IndCohpZq, J ÞÑ Op´ +ÿ +iPJ +Diq. +Further, by passing to iterated cofibers, or equivalently directly tensoring the cofiber mor- +phisms, we obtain a “reflected” cubical diagram +q: PprNsq Ñ IndCohpZq, J ÞÑ pfJq˚ODJ . +(7.1.9) +By construction, the total fiber of the cube q is ωZ so that its total cofiber is ωZrNs. Applying +the functor RHomZp´, ω‚ +Zq to the cube q, we obtain the cube +qω : PprNsqop Ñ Modk, J ÞÑ RHomZppfJq˚ODJ, ω‚ +Zq . +Finally, composing with the pushforward along the diagonal map ∆: Z Ñ Z ˆ Z, we obtain +the cube +Qω : Pprnsqop Ñ Modk, J ÞÑ RHomZˆZp∆˚pfJq˚ODJ, ∆˚ω‚ +Zq . +Lemma 7.1.10. There is a natural equivalence of cubes +Qω » HHpIndCohpD˚qq . +(7.1.11) +Proof. We have equivalences +RHomZˆZp∆˚pfJq˚ODJ, ∆˚ω‚ +Zq » RHomDJˆDJp∆˚ODJ, ∆˚ω‚ +DJq +arising from the various adjunctions p´q˚ % p´q! and base change associated to the pullback +square +DJ +DJ ˆ DJ +Z +Z ˆ Z +∆ +fJ +fJˆfJ +∆ +along with the equivalence f! +Jω‚ +Z » ω‚ +DJ. These can be combined with the natural equivalences +RHomDJˆDJp∆˚ODJ, ∆˚ω‚ +DJq » HHpIndCohpDJqq +to the desired equivalences of cubes. The remainder of the proof justifies that these equiva- +lences coherently assemble into an equivalence of cubes. +In the cases N “ 1, 2, the involved coherence issues can be deal with directly. +For +N ą 2, we show jointly the equivalence (7.1.11) and the statement that HH preserves the +69 + +colimit over IN +ăp0,...,0q by induction. +The fact that HH preserves the colimit in the case +N “ 2 follows from the argument presented in the proof of Lemma 7.1.12 below. We ap- +ply the decomposition of colimits [Lur09a, 4.2.3.10] and the induction assumption, to find +that colimăp0,...,0q HHpIndCohpD˚qq is equivalent to the pushout of the diagram +HHpIndCohpŤ +1ďiďN´1 Di X D1qq +HHpIndCohpŤ +1ďiďN´1 Diqq +HHpIndCohpDt1uq , +and thus given by HHpInd Cohp ˜DrNsqq. Applying the induction assumption once more, we find +an equivalence between the cubes Qω and HHpIndCohpD˚qq punctured at the final vertices. +Via the universal property of the colimit, we can extend this equivalence to the entire cubes +by virtue of the commutative diagram +RHomZˆZp∆˚i˚O ˜DrNs, ∆˚ω‚ +Zq +HHpInd Cohp ˜DrNsq +RHomZˆZp∆˚OZ, ∆˚ω‚ +Zq +HHpIndCohpZqq . +» +» +Here i : ˜DrNs Ă Z denotes the inclusion. +We have constructed the equivalence of cubes +(7.1.11), concluding the induction step. +Lemma 7.1.12. +(1) The total Hochschild homology HHtotpIndCohpD˚qq is equivalent to +the cofiber of +RHomZˆZp∆˚i˚OD, ∆˚ω‚ +Zq +˝∆˚u +ÝÝÝÝÑ RHomZˆZp∆˚OZ, ∆˚ω‚ +Zq , +with u the canonical morphism OZ Ñ i˚OD. +(2) The cofiber sequence in IndCohpZq +OZ Ñ i˚OD Ñ ωZ +determines a total negative cyclic homology class η: krns Ñ HHS1,totpIndCohpD˚qq. +Proof. Given any categorical N-cube A˚, one has an equivalence +HHtotpInd CohpA˚qq » cofpcolimJăp0,...,0q HHpInd CohpAJqq Ñ HHpInd CohpAqqq . +Using Lemma 7.1.10, part (1) thus follows from the equivalence +colimăp0,...,0q HHpInd CohpD˚qq » colimăp0,...,0q Qω +» colimJăp0,...,0q RHomZˆZp∆˚pfJq˚ODJ, ∆˚ω‚ +Zq +» RHomZˆZp∆˚ limpIN +ăp0,...,0qqoppfJq˚ODJ, ∆˚ω‚ +Zq +» RHomZˆZp∆˚i˚OD, ∆˚ω‚ +Zq +» HHpInd CohpDqq . +70 + +We proceed with part (2). Pushing the given cofiber sequence forward along ∆˚, we obtain +a cofiber sequence +∆˚OZ +∆˚i˚OD +0 +∆˚ωZ +which determines by part (1) a class σ: krns Ñ HHtotpIndCohpD˚qq. This class lifts uniquely +to HHS1,totpIndCohpD˚qq, as follows from the above mentioned equivalences +Hn´|J| HHS1pIndCohpDJqq » Hn´|J| HHpIndCohpDJqq . +Remark 7.1.13. There is an apparent analogue of Lemma 7.1.12 with the essentially same +proof: +Let J Ă rNs and denote by PprNsqop +ěJ the subposet of PprNsqop consisting of those J1 Ă rNs +satisfying that J Ă J1. Denote by IndCohpD˚ěJq the restriction of IndCohpD˚q to PprNsqop +ěJ. +The total Hochschild homlogy +HHtotpInd CohpDěJq +is equivalent to the cofiber of +RHomDJˆDJp∆˚piJq˚O ˜DJ, ∆˚ω‚ +DJq +˝∆˚u +ÝÝÝÝÑ RHomDJˆDJp∆˚ODJ, ∆˚ω‚ +DJq , +with iJ : ˜DJ Ă DJ the inclusion. Any commutative diagram +ODJ +piJq˚pO ˜DJq +0 +ωDJ +thus determines a total Hochschild class η: krn ´ |J|s Ñ HHS1,totpIndCohpD˚ěJqq. +Proof of Theorem 7.1.5. We show that the negative cyclic class η from part (2) of Lemma 7.1.12 +describes the desired left n-Calabi-Yau structure of IndCohpD˚q. +The recursive nondegeneracy conditions can be checked before pushing forward along the +diagonal morphism ∆. They amount to the following statements: +(1) As in the argument for the proof of Theorem 7.1.4 recalled above, the first nondegeneracy +condition amounts to the fact that the commutative diagram +OZ +i˚OD +0 +ωZ +giving rise to η is a cofiber sequence. +71 + +(2) Let J “ tj1, . . . , jlu Ă rNs. +Consider pN ` 1q-cube ˜q, with qaug|PprN`1sqop +ątN`1u “ q +(using that PprN ` 1sqop +ątN`1u » PprNsq), qaugpHq “ ω‚ +Z and qaugpxq “ 0 for all other +x P PprN `1sqop, exhibiting ω‚ +Z as the total cofiber of q. We obtain an pN ´|J|`1q-cube +˜qaug +J +by restricting qaug to PprN ` 1sqop +ąJYtN`1u. Passing to partial colimits, we obtain +the commutative diagram +pfJq˚ODJ +pfJq˚piJq˚O ˜DJ +0 +ωZr|J|s +(7.1.14) +with iJ : ˜DJ Ă DJ the inclusion. The diagram (7.1.14) is adjoint under pfJq˚ % f! +J to +the following diagram: +ODJ +piJq˚O ˜DJ +0 +ωDJ , +(7.1.15) +This diagram encodes as in Remark 7.1.13 the induced Hochschild homology class +Bj1 ¨ ¨ ¨ Bjlη: krn ´ |J|s Ñ HHS1,totpIndCohpD˚ěJqq . +The inductive argument below shows that the square (7.1.15) is biCartesian, yielding +the non-degeneracy condition of this Hochschild class. +Let j P J and assume that the non-degeneracy of the class Bj1 ¨ ¨ ¨ Bjl´1η has been shown. +The restriction of q to PprNsqěJztjlu :“ pPprNsqop +ěJztjluqop is given by the pushforward +pfJztjluq˚ of a cube denoted qJztjlu. The cube qJztjlu can be extended to a colimit cube +qaug +Jztjlu , exhibiting ω‚ +DJztjlu as the total cofiber of qJztjlu; this colimit cube is adjoint +under pfJztjluq˚ % f! +Jztjlu to ˜qaug +Jztjlu. +The cube qJztjlu arises from applying the unit of the adjunction i˚ +J,jl % piJ,jlq˚, with +iJ,jl : DJ Ă DJztjlu, to its face PprNsqěJYtjuzPprNsqěJ. We thus consider the pN´J`1q- +cube qJztjlu as a morphism between this face and the opposite face, and can pass to the +cofiber morphism to obtain another pN ´ J ` 1q-cube cofqJztjlu. The cube cofqJztjlu in +turn arises from applying the counit of the adjunction piJ,jq˚ % i! +J,j to its face given +by the cofiber pN ´ |J|q-cube, as follows from the sphericalness of piJ,jq˚ % i! +J,j, see +[DKSS21, Lemma 2.5.15]. Partially totalizing cofqJztjlu, we produce from cofqJztjlu the +restriction of qaug +Jztjlu to one of its faces, and thus upon passing to a partial colimit the +commutative square +piJ,jlq˚ODJ +piJ,jlq˚piJq˚O ˜DJ +0 +ωDJztjlur1s +α +which expresses a counit map cofpαq Ñ ωDJztjlur1s. The adjoint square under piJ,jlq˚ % +i! +J,jl, given by (7.1.15), is thus biCartesian, concluding the argument. +72 + +Remark 7.1.16. The proof of Theorem 7.1.5 shows that the divisor ˜DJ Ă DJ is anticanonical +for any J Ă rNs. This can also be checked directly using the adjunction formula. +Remark 7.1.17. Note, that in the context of Theorem 7.1.5, the datum that defines the +Calabi–Yau structure on the categorical intersection complex +tot>pIndCohpD˚qq +of the normal crossings divisor D “ Ť +1ďiďN Di is identical to the datum that defines the +relative Calabi–Yau structure on the functor +IndCohpDq +i˚ +ÝÑ IndCohpZq. +Namely, in both cases this is a class in the relative Hochschild homology of i˚. This may +be regarded as a noncommutative analog of the fact that an orientation on a manifold with +corners induces a compatible system of orientations on all boundary strata. +We conclude the section by noting that the above construction of the categorical inter- +section complex of a normal crossings divisor can be generalized to the context of cubical +resolutions of schemes (see e.g. [GNAPGP88]). +Definition 7.1.18. Let S˚ be a In-scheme, i.e., a functor In Ñ Schk. A 2-resolution of S˚ +consists of a commutative diagram of In-schemes +S1,1,˚ +S1,0,˚ +S0,1,˚ +S0,0,˚ +a˚ +f˚ +b˚ +with S0,0,˚ “ S˚ and satisfying for all i P r1sn that ai and bi are closed immersions, fi is proper, +S1,0,i is smooth and fi restricts to an isomorphism of schemes between S1,0,izf´1 +i +pS0,0,iq and +S0,0,izS0,1,i. +Example 7.1.19. Consider a scheme S with smooth blowup BLZpSq at a closed subvariety +Z: +E +BLZpSq +Z +S +{ +Then the above diagram defines a 2-resolution of S. +Let S be any scheme. Then there exists a 2-resolution S2 of S obtained as the pullback +square of any resolution of S, see [GNAPGP88, Thm 2.6]. We may thus choose a 2-resolution +of S2. We consider the first column of this 2-resolution as a I-scheme, which in turn admits a +further 2-resolution. We choose one such, denoted S3, which we consider as a I3-scheme. We +proceed in this way for all n ă N, choosing Sn`1 as a 2-resolution of the In´1-scheme obtained +73 + +from restricting Sn along In´1 ˆt1u ãÑ In. From this, we extract an apparent IN-scheme S˚ +satisfying +S0,...,0 “ S , +S1,0,...,0 “ S2 +1,0 , +... +S˚,1,0,...,0 “ Sj`2 +˚,1,0 for ˚ P Ij , +... +S˚,1 “ SN +˚,1 . +Definition 7.1.20. Let S be a scheme an S˚ an IN-scheme obtained as above. We call S˚ +an S-augmented cubical hyperresolution if Si is smooth for all i P IN ztp0, . . . , 0qu. +Example 7.1.21. For S “ X1 YX2 YX3 the union of three smooth subschemes with smooth +intersections, a cubical hyperresolution arises from restricting the following diagram consisting +of two 2-resolutions, to the ’outer’ 3-cube. +X1 X X2 X X3 +X1 X X3 +X2 X X3 +X3 +X1 X X2 +X1 X pX2 Y X3q +X1 +X2 +X2 Y X3 +X1 Y X2 Y X3 +More generally, given a scheme S, written as the union of smooth subschemes with smooth +intersections, there is an apparent IN-scheme S˚ with SJ “ Ş +jPJ Xj for all J P PprNsqop » +IN. The IN-scheme S˚ is a cubical hyperresolution of S. The IN-scheme associated to a +normal crossing divisor inside a scheme Z at the beginning of the section can be obtained +from this hyperresolution by changing the value at the terminal vertex of the N-cube from +S “ Ť +jPrNs Xj to Z. +Given an IN-scheme S˚, we obtain a functor IndCohpS˚q: IN Ñ Stk by composing with +IndCoh. The diagram IndCohpS˚q describes a categorical cube, which is in general neither +spherical, nor does its totalization admit a Calabi–Yau structure. +In the case where S˚ +is an S-augmented cubical hyperresolution, it seems an intriguing question to explore the +implications of considering the truncation τě1 tot> IndCohpS˚q as a resolution of the stable +8-category IndCohpSq in terms of smooth k-linear stable 8-categories. +7.2 +Picard-Lefschetz theory and Fukaya-Seidel complexes +We begin by explaining how classical Picard–Lefschetz theory can be used to construct certain +cell complexes modelling the cohomology of affine varieties. +74 + +Let X Ă CN be an n-dimensional smooth affine subvariety. Let X1 Ă X be a generic +hyperplane section of X in CN. The pair pX, X1q induces an exact triangle +S‚pXq +S‚pX, X1q +S‚´1pX1q +B +`1 +(7.2.1) +in singular homology. In this context, the classical Lefschetz hyperplane theorem, due to, in +the given affine setup, to Andreotti-Frankel, implies that the complex S‚pX, X1q +1. has homology concentrated in degree n, and +2. HnpX, X1q is generated by vanishing thimbles. +In particular, the exact triangle (7.2.1) provides a means of computing the homology +H‚pXq in terms of the group HnpX, X1q and the homology H‚pX1q. We may iterate this +construction, choosing a sequence of generic hyperplane sections +X Ą X1 Ą X2 Ą ¨ ¨ ¨ Ą Xn +so that the the corresponding exact triangles of consecutive pairs combine to give a description +of the singular homology H‚pXq as the homology of the complex +HnpX, X1q Ñ Hn´1pX1, X2q Ñ ... Ñ H0pXnq +(7.2.2) +Under suitable technical assumptions and with suitable choices of symplectic structures (cf. +[Sei08], P. Seidel showed that a categorical variant of the above discussion can be implemented +to provide an effective means for computing Fukaya categories of affine varieties in terms of +Lefschetz fibrations. Namely, suppose that the hyperplane section X1 “ π´1 +1 pt1uq is given +as the fiber of a Lefschetz fibration π1 : X Ñ C (such as the restriction of a generic linear +function on CN). Then we have a “left exact” sequence of categories +FukpXq ãÑ FSpπ1q +B +ÝÑ FukpX1q +i.e., the category FukpXq is the kernel of the functor B. As shown in [Sei08], the category +FSpπ1q admits an exceptional collection of objects given by Lagrangian vanishing thimbles. +Again, we may iterate this consideration to produce a categorical complex +FSpπ‚q :“ FSpπ1q Ñ FSpπ2q Ñ ... Ñ FukpXnq +(7.2.3) +where FSpπkq is the Fukaya–Seidel category of a Lefschetz fibration πk : Xk´1 Ñ C. +Remark 7.2.4. Note that, since each of the Fukaya–Seidel categories FSpπkq has an excep- +tional collection given by Lagrangian vanishing thimbles of πk, we have, by the Lefschetz +hyperplane theorem, an isomorphism of abelian groups K0pFSpπkqq – HnpXk´1, Xkq. Fur- +ther, the complex K0pFSpπ‚qq reproduces the complex (7.2.2). +Note that in the original reference [Sei08] all Fukaya categories are described in the frame- +work of A8-categories. We may turn them into k-linear 8-categories by virtue of the coherent +nerve construction REF . Further, in order to obtain presentable 8-categories, we pass to Ind- +completions: +Definition 7.2.5. The categorical complex IndpFSpπ‚qq is called the (Ind-completed) Fukaya– +Seidel complex of the family of Lefschetz fibrations tπ‚u. +Remark 7.2.6. As will be seen in the examples below, the categories of cycles of the complex +IndpFSpπ‚qq correspond to wrapped variants of the Fukaya categories considered in [Sei08] (the +Fukaya categories in [Sei08] are always generated by compact Lagrangian submanifolds). +75 + +7.3 +Mirror symmetry +In the context of homological mirror symmetry, we expect equivalences between the categorical +complexes from §7.1 and §7.2, respectively. +In this section, we illustrate this “mirror symmetry for categorical complexes” in a some- +what familiar context and then formulate a conjecture as to what to expect in greater gener- +ality. To this end, we start recalling a well-known example of homological mirror symmetry +(see [Sei01, AKO08] for details): +We consider the affine hypersurface +X “ txyz “ 1u Ă C3 +and note that X – pC˚q2. Consider the Lefschetz fibration π1 : X Ñ C given by the restriction +of the linear function π1 “ x`y`z. The general fiber of π1 is an elliptic curve with 3 punctures +which degenerates to a nodal cubic over the 3 critical values of π1. As a regular fiber, we +may take X1 :“ π1p0q “ Eztp1, p2, p3u. A depiction of the 3 vanishing spheres in X1 that +correspond to the critical values can be found in [Sei01, §3B]: +This example does not directly fit into the context discussed in [Sei08], but it is explained in +[Sei01] how to associate to π1 a Z-graded Fukaya-Seidel category which can be described as +follows: The category FSpπ1q is equivalent to the derived category of the quiver +‚ +‚ +‚ +a1 +a2 +a3 +b1 +b2 +b3 +subject to the relations biaj “ bjai, for i ‰ j, and biai “ 0. Thus, we further have +FSpπ1q » CohpP2q +by [Bei84]. +As a next step, we choose the Lefschetz fibration π2 : X1 Ñ C obtained by restricting the +linear function π2 :“ x ´ y on C3. This is a ramified covering map of degree 3, the fibers +of π2 move in the pencil generated by x ´ y and w on E whose general fiber consists of 3 +distinct points degenerating over the 6 nondegenerate critical values. Again, this setup does +not directly fit into [Sei08], but the necessary modifications are explained in [Sei01]. We may +take the fiber X2 :“ π´1 +2 p0q “ ta, b, cu as a regular fiber. Then the 6 critical values correspond +to 6 vanishing cycles (0-spheres = pairs of points) in X2: +ta, bu, ta, bu, ta, cu, ta, cu, tb, cu, tb, cu +76 + +The directed subcategory on these can be described by the quiver +‚ +‚ +‚ +‚ +‚ +‚ +a +b +a +b +a +c +c +b +c +with zero relations given by the rule that the composite of composable arrows is zero iff they +are labelled by different letters. We omit the discussion of grading choices (which is explained +in [Sei01]). +Alternatively, we may describe this category as a topological Fukaya category of the Rie- +mann surface Eztp1, p2, p3u with one stop at each puncture (or rather the cylindrical end +corresponding to it). The corresponding spine is given by the Ribbon graph +Γ “ +so that the formalism of [Dyc17] yields the diagram of Ribbon graphs depicted in Figure 7.3.1 +This latter diagram yields an equivalence between the topological Fukaya category and the +colimit of the corresponding diagram of Figure 7.3.2. Here, ReppA2q denotes the bounded +derived category of representations of the quiver +A2 “ +‚ +‚, +the functors of the form Cohpptq Ñ ReppA2q are induced by the inclusion of source (resp. +target) of the quiver. The functors of the form Cohpptq Ñ CohpP1q are given by pushforward +along the inclusion of the points 0 and 8, respectively, into P1. Again, there are auxiliary +choices to be made to determine the Z-grading, such as a trivialization of the tangent bundle +(or rather its square) of Eztp1, p2, p3u, which we do not discuss here. +Informally, the colimit of the diagram from Figure 7.3.2 can thus be described by starting +with three disjoint copies of CohpP1q and freely adjoining three arrows connecting the various +skyscraper sheaves of the projective lines at 0 and 8, respectively. We keep this intuition in +mind by denoting the resulting category by +Cohp +q +(7.3.1) +describing it as an amalgamate of commutative geometry (the projective lines) and noncom- +mutative geometry (the quiver arrows). In terms of the constructions of §2.4, this means that +the category is described by the directed pushout of three copies of CohpP1q along three copies +of Cohpptq. +77 + +Figure 7.3.1: Decomposition of Γ. +CohpP1q +Cohpptq +CohpP1q +CohpP1q +Cohpptq +Cohpptq +Cohpptq +Cohpptq +Cohpptq +ReppA2q +ReppA2q +ReppA2q +Figure 7.3.2: Diagram computing the topological Fukaya category of Γ. +78 + +The above choices of potentials π1, π2 in total yield a Fukaya-Seidel complex of the form +FSpπ1q ÝÑ FSpπ2q ÝÑ FukpX2q +(7.3.2) +with +(1) FSpπ1q » CohpP2q +(2) FSpπ2q » FukpΓq » Cohp +q +(3) FukpX2q » Cohppt > pt > ptq +On the other hand, consider P2 with the normal crossings divisor given by three lines +L1, L2, L3 in general position: +L1 +L2 +L3 +Ă P2 +The corresponding categorical cubical diagram induced by the various push-forward func- +tors takes the following form: +CohpHq +CohpL2 X L3q +CohpL1 X L3q +CohpL3q +CohpL1 X L2q +CohpL2q +CohpL1q +CohpP2q +Its coproduct totalization yields the categorical intersection complex +Cohppt>3q +Cohp +q +CohpP2q +(7.3.3) +where we use the above notation (7.3.1) to denote the middle term. In particular, all terms +of the complexes (7.3.3) and (7.3.2) agree. We do not verify here the expectation that the +differentials in both complexes are in fact adjoint to one another (this can probably be ex- +tracted with some effort from the existing literature) but rather formulate the more general +conjecture: +79 + +Conjecture 7.3.4. Let X “ pC˚qn Ă Cn`1. Then there exists a family of iterated Lefschetz +fibrations π‚ on X such that the corresponding Fukaya–Seidel complex FSpπ‚q is equivalent +to the (right adjoint of the) categorical intersection complex of the normal crossings divisor +given by n ` 1 hyperplanes in Pn in general position. +The conjecture has its natural generality in some context of normal crossings divisors in +toric Fano varieties, where Hori-Vafa mirror symmetry provides a prediction of the mirror +potential , but for the sake of concreteness, we leave it as stated. +While presenting some of this material during a talk in the Edinburgh Hodge Seminar, we +learned about some work in progress on wrapped Fukaya categories of “multi-potentials” which +seems to be closely related to the perspective on homological mirror symmetry via categorical +complexes. Indeed, once the multi-potential approach is implemented, one expects that this +directly yields cubical diagrams of wrapped Fukaya categories which are mirror to the cubical +diagrams of coherent sheaves from which we build the categorical intersection complex via +totalization. +This “mirror symmetry conjecture for cubes” has for example been described in the recent +article [Lee22]. Once this type of cubical mirror symmetry is established, our Conjecture 7.3.4 +then essentially reduces to a statement that our Fukaya-Seidel complexes from Definition +7.2.5 arise as totalization of the cubical wrapped Fukaya-category diagrams that one expects +to associate to a multi-potential Landau–Ginzburg model (cf. [AGHJ22]). This statement, +as well as its relevance for higher–dimensional perverse schobers (cf.§5.4 and §5.3), seems +interesting in its own right and will be investigated in future work. +7.4 +Manifolds with corners and complexes of 8-local systems +We fix a field k. Given a space X, we denote by LocpXq “ FunpX, Dpkqq the stable 8-category +of Dpkq-valued local systems on X. Given a morphism f : X Ñ Y between spaces, we denote +by f˚ : LocpY q Ñ LocpXq the pullback functor. It admits left and right adjoints f! and f˚, +given by left Kan extension and right Kan extension, respectively, see [Lur09a, 4.3.3.7]. Recall +the following Theorem from [BD19]. +Theorem 7.4.1. Let X be a compact oriented manifold of dimension n with boundary +f : BX Ă X. Then the functor +f! : LocpBXq ÝÑ LocpXq +admits a canonical left n-Calabi–Yau structure. +The goal of this section is to show that Theorem 7.4.1 admits an extension to categorical +complexes arising from oriented bordisms. In the simplest case, an oriented n-dimensional +bordisms between two closed, oriented pn ´ 1q-dimensional manifolds M, M1 consists of an +oriented n-dimensional manifold N with boundary BN “ M >M1, with M and M1 carrying the +induced orientation. If instead M and M1 are not closed and not disjoint in N, but instead +overlap on their boundary, one can further ask that M and M1 again define two pn ´ 1q- +dimensional bordisms between pn ´ 2q-dimensional manifolds, and so on. This perspective +can be formalized by organizing oriented bordisms up to dimension n in an p8, nq-category of +oriented bordisms Bordor +n , whose m-cells are m-dimensional oriented bordisms for m ď n, see +80 + +for instance [Lur09b, CS19]. In the following, we associate to a functor from the n-simplex to +the p8, nq-category Bordor +n a categorical n-cube whose totalization is left Calabi–Yau. +We fix an integer n ě 1. Given an n-simplex ∆n, we denote by Sd ∆n its barycentric +subdivision, considered as a poset. We denote by Mfd the 1-category with objects compact +oriented manifolds of any dimension with boundary and morphisms oriented inclusions into the +boundary. A functor from the n-simplex to Bordor +n amounts to a functor A˚ : Sd ∆n Ñ Mfd, +mapping each m-simplex x P p∆nqm to an m-dimensional manifold Ax (possibly empty), such +that the boundary of Ax is given by +BAx “ +m +ď +i“0 +Adipxq +for any x P p∆nqm. We further ask that the intersection of Adipxq and Adjpxq is given by +Adjdipxq for any j ă i. +There is an equivalence of posets φ: Sd ∆n » In ztp1, . . . , 1qu, i.e. the barycentric subdi- +vision of ∆n is a cube with the initial vertex removed. Given an n-simplex A˚ in Bordor +n , we +hence find a categorical cube +LocpA˚q: In Ñ Stk , +with LocpA˚qφpxq “ LocpAφpxqq and LocpAqpp1,...,1qq “ 0. We will omit the usage of the equiva- +lence φ in the following, and write for instance AJ and LocpAJq for Aφ´1pJq and LocpAφ´1pJqq, +with J Ă rns. +Theorem 7.4.2. Let A˚ be an n-simplex in Bordor +n . Then the categorical n-cube LocpA˚q +admits a canonical left n-Calabi–Yau structure. +Corollary 7.4.3. Let A˚ be an n-simplex in Bordor +n . Then the coproduct totalization tot>pLocpA˚qq +admits a canonical left n-Calabi–Yau structure. +Proof. Combine Theorem 7.4.2 and Theorem 6.4.6. +Before proving Theorem 7.4.2, we recall some details from the proof of Theorem 7.4.1 +from [BD19]. Given a space X, we denote by C‚pXq P Dpkq its singular chain complex and +by LX its free loop space. There are natural morphisms in Dpkq +C‚pXq Ñ C‚pLXq » HHS1pLocpXqq , +(7.4.4) +where the first morphisms arises from the inclusion of the constant loops X Ñ LX. If X +is a closed n-dimensional manifold, any orientation, considered as an element of HnC‚pXq, +gives via the above map rise to a left n-Calabi–Yau on LocpXq. If Y is an n-dimensional +manifold with boundary X, then a relative orientation of Y is a suitable n-th homology class +of C‚pY, Xq :“ cofpC‚pXq Ñ C‚pY qq and its image in HHS1pLocpY q, LocpXqq gives rise to a +relative Calabi–Yau structure on the functor LocpXq +pXĂY q! +ÝÝÝÝÝÑ LocpY q. +Lemma 7.4.5. There exists a natural map +C‚pAp0,...,0q, BAp0,...,0qq ÝÑ HHS1,totpLocpA˚qq . +81 + +Proof. Consider the diagram C‚pA˚q: In Ñ Modk. The colimit of its restriction to In +ăp0,...,0q “ +In ztp0, . . . , 0qu is by Mayer-Vietoris equivalent to C‚pBAp0,...,0qq. Consider the following dia- +gram in Dpkq: +C‚pBAp0,...,0qq +C‚pAp0,...,0qq +C‚pAp0,...,0q, BA0,...,0qq +colimJăp0,...,0q HHS1,totpAJq +HHS1pAp0,...,0qq +HHS1,totpAq +The left square commutes by the naturality of the morphisms (7.4.4). Both the upper and +lower sequence in this diagram are cofiber sequences. It follows that there exists a dotted +arrow as indicated, making the diagram commutative. +Proof of Theorem 7.4.2. Let x P ∆n be the top cell. Any choice of orientation of Ax rel- +ative to BAx determines by Lemma 7.4.5 a total negative cyclic homology class η: krns Ñ +HHS1,totpLocpA˚qq. Its restriction to HHS1,totpLocpABi...Bj˚qq arises via an analogue of Lemma 7.4.5 +for the cube ABi...Bj˚ from an induced orientation of Adi...djpxq relative to BAdi...djpxq. +Using that passing to local systems preserves colimits, we find for any J Ă rns a commu- +tative diagram +LocpBAJqq +LocpŤm +jRJ AJYtjuq +LocpAJq . +colimIn +ăJ LocpA˚q +“ +» +The total Hochschild class of LocpA˚q thus restricts by Theorem 7.4.1 to a Calabi-Yau struc- +ture on the lower horizontal morphism above. This shows that η defines a left n-Calabi-Yau +structure for LocpA˚q, as desired. +Example 7.4.6. Consider the pn ` mq-simplex A in Bordn`m, with +Arn`ms “ Dn ˆ Dm +the product of the n- and m-dimensional real unit discs and +Ad0rn`ms “ pBDnq ˆ Dm “ Sn´1 ˆ Dm , +Ad1rn`ms “ Dm ˆ Sm , +Ad2 +0rn`ms “ Sn´1 ˆ Sm´1 +and Ax “ H otherwise. The totalization tot>pLocpAqq is given by the complex +LocpSn´1 ˆ Sm´1q ÝÑ LocpDn ˆ Sm´1q +Ð> +LocpSn´1ˆSm´1q LocpSn´1 ˆ Dmq ÝÑ LocpDn ˆ Dmq +concentrated in degrees 2 to 0. Except for being left pn ` mq-Calabi–Yau, this complex is +furthermore spherical as it arises from totalizing the following spherical square: +LocpSn´1 ˆ Sm´1q +LocpDn ˆ Sm´1q +LocpSn´1 ˆ Dmq +LocpDn ˆ Dmq +82 + +The above square arises from a pullback square of Kan fibrations and is hence Beck-Chevalley, +see [Cis19, Prop. 4.4.11, Thm. 6.4.13]. These Kan fibrations are furthermore spherical fibra- +tions, meaning that their fibres are spheres, the sphericalness of the functors was thus shown +in [Chr22c]. We note that in general, tot>pLocpAqq is not a spherical complex. +A version of this example producing an n-term complex arises by starting with the product +of n ` 1 discs. +8 +Lax additivity +In this section we develop the general framework of lax additive (or 2-additive) p8, 2q- +categories. +8.1 +Additive 1-categories +To explain our philosophy, let us first remind the reader of the classical story for ordinary +additive categories. +We start by recalling the definition. +Definition 8.1.1. A category A is called additive if: +(1) The category A is enriched in abelian monoids; i.e., each hom-set Apx, yq has an asso- +ciative, commutative addition ` with neutral element 0 such that composition +Apx, yq ˆ Apy, zq Ñ Apx, zq +preserves ` and 0 in each argument. +(2) Each commutative monoid pApx, yq, `, 0q admits negatives, hence is an abelian group. +(3) The category A admits finite products and coproducts (including empty ones). +(4) For each finite set of objects x1, . . . , xn P A, the natural map +n +ž +s“1 +xs ÝÑ +n +ź +t“1 +xt, +(8.1.2) +whose components xs Ñ xt are +# +1: xs Ñ xs +, if s “ t +0 P Apxs, xtq +, otherwise , +(8.1.3) +is an isomorphism. +A category only satisfying (1), (3) and (4) is called semiadditive. +One typically identifies finite products and coproducts via the canonical map (8.1.2) and +uses the symbol ‘ (called direct sum or biproduct) for both. +The use of the phrase “is called additive if” implies that being additive is a property of +the category A rather than extra structure. This is justified by the that the addition on +83 + +the hom-sets of an additive category is uniquely determined. Explicitly, it is given by the +following formula: Given two maps f, g: x Ñ y, their sum is the composite +x Ñ x ‘ x +f‘g +ÝÝÑ y ‘ y Ñ y +(8.1.4) +where the first map is the diagonal x Ñ x ˆ x and the last map is the codiagonal y > y Ñ y. +In this sense, the biproduct structure ‘ determines the addition structure ` on the hom- +sets. The converse is also true, as explained by the following lemma: +Lemma 8.1.5. Let A be a category enriched in abelian monoids. Let x1, . . . , xn be a finite +set of objects in A. +(1) Let x be an object of A equipped with a cone P “ pps : x Ñ xsqn +s“1 and a cocone +I “ pis : xs Ñ xqn +s“1 satisfying the two equations +(a) +nÿ +s“1 +is ˝ ps “ 1 P Apx, xq +(8.1.6) +(b) +pt ˝ is “ +# +1 P Apxs, xtq, if s “ t +0, otherwise +(8.1.7) +Then P and I exhibit x as the product śn +s“1 xn and as the coproduct šn +s“1 xn, respec- +tively. Morover, the canonical comparison map (8.1.2) is the identity 1: x Ñ x. +(2) Assume the product x “ śn +s“1 exists and let P “ pps : x Ñ xsqn +s“1 be the product cone. +Then there exists a unique cocone I “ pis : xs Ñ xqn +s“1 satisfying conditions (a) and (b) +above. +(3) Dually, for every coproduct cocone I “ pis : xs Ñ xqn +s“1 there exists a unique cone +P “ pps : x Ñ xsqn +s“1 satisfying (a) and (b). +Since ‘ and ` determine each other, we have the following corollary: +Corollary 8.1.8. +(1) Let A be a category enriched in abelian monoids. If A admits finite +products (equivalently, finite coproducts) then it is automatically semiadditive. +(2) Let F : A Ñ A1 be a functor between additive categories. The following are equivalent: +(a) the functor F preserves products; +(b) the functor F preserves coproducts; +(c) the functor F preserves the addition on the hom-sets. +Lemma 8.1.5 is well known. However, its proof is very instructive for categorification, so +we shall explain it here: +84 + +Proof of Lemma 8.1.5. +(1) We show that P is a product cone; the statement about I is +dual. We need to show that for each t P A the natural map +P˚ : Apt, xq Ñ +n +ź +s“1 +Apt, xsq; +f ÞÑ pps ˝ fqn +s“1 +(8.1.9) +is a bijection. Using I we can produce an explicit inverse via the formula +I˚ : pfsqn +s“1 ÞÑ +nÿ +s“1 +isfs. +(8.1.10) +It satisfies +pP˚ ˝ I˚qpfsqn +s“1 “ P˚p +nÿ +s“1 +isfsq +(8.1.11) +“ ppu ˝ +nÿ +s“1 +isfsqn +u“1 +(8.1.12) +“ p +nÿ +s“1 +puisfsqn +u“1 +(8.1.13) +“ pfuqn +u“1 +(8.1.14) +(using equation (b) in the last step) and +pI˚ ˝ P˚qpfq “ I˚pps ˝ fqn +s“1 +(8.1.15) +“ +ÿ +s“1 +pispsfq +(8.1.16) +“ p +ÿ +s“1 +ispsq ˝ f +(8.1.17) +“ 1 ˝ f “ f +(8.1.18) +(using equation (a) in the last step), as desired. +Moreover, equation (b) says precisely that the identity 1: x Ñ x satisfies the defining +equation to be the map (8.1.2). +(2) By the univeral property of the product cone P, there are unique maps is : xs Ñ x +satisfying equation (b). These maps then assemble into the desired cocone I. To verify +equation (a) it suffices to postcompose with all the product projections pu and compute +pu ˝ +nÿ +s“1 +isps “ +nÿ +s“1 +puisps +(8.1.19) +“ pu “ 1 ˝ pu +(8.1.20) +(using equation (b) in the second step). +85 + +There is one last aspect of additive categories which is going to be useful to categorify: +matrix calculus. +This is the observation that in any category A any map +f : +n +ž +s“1 +xs ÝÑ +m +ź +t“1 +yt +(8.1.21) +from a coproduct to a product can be encoded through the bijection +A +˜ n +ž +s“1 +xs, +m +ź +t“1 +yt +¸ +– +m +ź +t“1 +n +ź +s“1 +Apxs, ytq +(8.1.22) +as an m ˆ n-matrix pftsqm,n +t“1,s“1 whose entry fts is a map xs Ñ yt. +The special thing about semiadditive categories is that it makes sense to consider the +composite +h: +n +ž +s“1 +xs +fÝÑ +m +ź +t“1 +yt – +n +ž +t“1 +yt +gÝÑ +lź +u“1 +zu +(8.1.23) +of two such maps by using the identification (8.1.2). This composite corresponds to a matrix +phusql,n +u“1,s“1 P +lź +u“1 +n +ź +s“1 +Apxs, zuq. +(8.1.24) +It is not hard to verify that the matrix corresponding to the composite h arises from the +matrices of f and g by the usual rule for matrix multiplication: +hus “ +m +ÿ +t“1 +guthts +(8.1.25) +From this perspecive, the identification (8.1.2) is just the identity matrix which has identities +on the diagonal an zeroes everywhere else. +8.2 +p8, 2q-categories +In this paper, we think of p8, 2q-categories as categories enriched in the 8-category Cat8 +of 8-categories. For a general treatment of enriched 8-categories, we refer to the work of +Gepner and Haugseng [GH15]. +Our goal is not to develop any p8, 2q-categorical foundations but rather to develop the +theory of lax additivity while assuming that such foundations are already laid. In practice, +this means that none of our arguments and constructions are performed explicitly in a model, +but only using the general high-level features which any theory of p8, 2q-categories is expected +to share. We treat these ingredients axiomatically: +Let C be an p8, 2q-category. +• It has an underlying 8-category C1, and underlying 8-groupoid C» “ pC1q». +86 + +• It has a hom-functor +Cp´, ´q: Cop ˆ C Ñ Cat8, +(8.2.1) +which takes values in the p8, 2q-category of 8-categories. Occasionally, it is convenient +to consider the hom-functor +Cp´, ´q: Cop +1 ˆ C1 Ñ Cat8 +(8.2.2) +as a functor of the underlying 8-categories, and its associated cartesian fibration +Tw˚pCq “ +ż ˚ +Cp´, ´q Ñ C1 ˆ Cop +1 . +(8.2.3) +• There are composition functors +CpX, Y q ˆ CpY, Zq Ñ CpX, Zq, +(8.2.4) +functorial in X, Y, Z : C». Composition is coherently associative; this is formalized in +[GH15] by encoding the p8, 2q-category C as an algebra in the monoidal 8-category +pCat8, ˆq of a certain generalized nonsymmetric operad ∆op +C» Ñ ∆op. +• More generally, the composition map (8.2.4) is also natural in X : Cop +1 , Z : C1 and di- +natural in Y : C1 (and not just in their groupoid cores). Thinking in terms of fibrations, +this means that composition can be written as the dashed functor +şX +Y CpX, Y q ˆpY :C1q +şY +Z CpY, Zq +şX +Z CpX, Zq +pX : C1q ˆ pY : C1q ˆ pZ : C1q +pX : C1q ˆ pZ : C1q +(8.2.5) +of mixed (cartesian, cocartesian) fibrations. +• It makes sense to talk about adjunctions f % fR : X Ñ Y in C. These are characterized +by the fact that +pf˝q % pfR˝q: CpT, Xq Ñ CpT, Y q +and +p˝fRq % p˝fq: CpX, Tq Ñ CpY, Tq +(8.2.6) +are adjunctions of 8-categories for all T : C. +For the purpose of developing the theory of lax additivity we do not need the full coher- +ent associativity of the composition law, but only its incoherent shadow. More precisely, it +suffices to postcompose the enrichment with the symmetric monoidal functor pCat8, ˆq Ñ +phoCat8, ˆq, and think of C as enriched in the homotopy category of 8-categories up to +equivalence. +Remark 8.2.7. Throughout this section we use the notation “x : A” borrowed from homotopy +type theory to say that x is a term/inhabitant/element/object of the (8-)groupoid, (8- +)category, or even p8, 2q-category A. When we construct an object “Fpxq : B for each x : A”, +it is understood that Fpxq is supposed to be functorial in x in the appropriate sense. This +allows us to unambiguosly write formulas such as colimx:A Fpxq or pFpxqqx:A, which of course +only make sense with the additional functoriality in mind. +We reserve the notation x P A for the case when A is discrete, i.e. (equivalent to) a set; +in this case, the question of functoriality is vacuous. +87 + +8.3 +Lax limits and colimits +We start by recalling the definition of a lax limits and colimits in a p8, 2q-category. Let S be +an 8-category. +First, let X: S Ñ Cat8 be a diagram of 8-categories. Let laxlim X be the 8-category +of sections of the (covariant) Grothendieck construction +ş +˚ X Ñ S associated to functor X. +Informally, objects of laxlim X consist of +(1) for each object s of S, an object xs in Xs, +(2) for each edge f : s Ñ t in S, a morphism xf : Xfpxsq Ñ xt in Xt, +(3) for each 2-simplex s +fÝÑ t +gÝÑ u (with the composite gf implicit) in S a 2-simplex +Xgpxtq +Xgfpxsq +xu +xg +Xgpxfq +xgf +(8.3.1) +in Xu, +(4) and so on for higher simplices of S. +We will denote an object of laxlims:S X as a tuple pxsqs:S. +Now, let C be an arbitrary p8, 2q-category and X: S Ñ C a diagram. A lax cone over the +diagram X with vertex L is an object pφsqs:S : laxlims:S CpL, Xsq, where s ÞÑ CpL, Xsq is the +S-shaped diagram in Cat8 obtained as the composite +S XÝÑ C +CpL,´q +ÝÝÝÝÑ Cat8 +(8.3.2) +Informally, such a cone consists of +(1) for each object s : S, a structure map φs : L Ñ Xs, +(2) for each arrow f : s Ñ t in S, a lax cone +L +Xs +Xt +φt +φs +Xf +(8.3.3) +i.e. a map φf : Xfφs Ñ φt in CpL, Xtq, +(3) together with coherent pasting identifications, φg ˝ Xgφf » φgf for composable arrows +s +fÝÑ t +gÝÑ u in S. +88 + +For each other object L1 : C we have a canonical composition map +CpL1, Lq ˆ laxlims:S CpL, Xsq +laxlims:S CpL1, Xsq +laxlims:S CpL1, Lq ˆ laxlims:S CpL, Xsq +laxlims:S CpL1, Lq ˆ CpL, Xsq +´˝´ +∆ˆid +» +(8.3.4) +which informally sends a cone pφsqs:S with vertex L and a morphism l: L1 Ñ L to the cone +pφs ˝ lqs:S. +Dually we can define the 8-category of lax cocones on X with vertex L as laxlims:Sop CpXs, Lq. +Explicitly, a such a cocone pψsqs:S has structure maps ψs : Xs Ñ L and lax triangles +L +Xs +Xt +ψs +Xf +ψt +φf +(8.3.5) +over each arrow f : s Ñ t of S. +Definition 8.3.6. A cone P “ ppsqs:S : laxlims:S CpL, Xsq is called a lax limit cone if for each +object L1 : C the functor +´ ˝ P : CpL1, Lq ÝÑ laxlims:S CpL1, Xsq +(8.3.7) +is an equivalence of 8-categories; in this case we call the object L a lax limit of the dia- +gram X: S Ñ C and write L “ laxlims:S Xs. Dually, we say that a cocone I “ pisqs:Sop : +laxlims:Sop CpXs, Lq is a lax colimit cone if for each L1 : C the functor +I ˝ ´: CpL, L1q ÝÑ laxlims:Sop CpXs, L1q +(8.3.8) +is an equivalence; in this case we call L a lax colimit of X and write L “ laxcolims:S Xs. +Example 8.3.9. Let C “ Cat8 be the p8, 2q-category of 8-categories. Let X: S Ñ Cat8 be +a diagram. +(1) As the notation suggests, the lax limit of X is the 8-category L :“ laxlims:S Xs :“ +FunSpS, +ş +s:S Xsq of sections of the corresponding Grothendieck construction. Indeed, +naturally in L1 : Cat8 we have the equivalence +Cat8pL1, Lq “ FunpL1, FunSpS, +ż +˚ +Xqq +(8.3.10) +» FunSpS ˆ L1, +ż +˚ +Xq +(8.3.11) +» FunSpS, +ż +s:S +FunpL1, Xsqq +(8.3.12) +“ laxlims:SpCat8pL1, Xsqq, +(8.3.13) +89 + +which is induced by composition with the canonical cone +P “ pps : L “ laxlimS X Ñ Xsqs:S +(8.3.14) +given by evaluation of sections. +(2) The lax colimit of the diagram X is the contravariant Grothendieck construction +laxcolims:S Xs “ +ż s:S +Xs, +(8.3.15) +exhibited by the canonical cocone +I “ +ˆ +is : Xs Ñ +ż ˚ +X +˙ +s:Sop +(8.3.16) +that includes the individual fibers. +Assume now that the diagram X takes values in stable 8-categories, +(3) The 8-category laxlims:S Xs “ FunSpS, +ş +˚ Xq is again stable because limits and colim- +its of sections are computed pointwise. For the same reason, every functor F : L1 Ñ +laxlims:S X is exact if and only if each composite ps ˝ F is exact. It follows that the +cone P exhibits the 8-category laxlims:S Xs also as a lax limit in the p8, 2q-category of +stable 8-categories and exact functors. +(4) The 8-category +ş˚ X, which is the lax colimit of X in Cat8, is typically not stable; to +compute the lax colimit of X in the p8, 2q-category of stable 8-categories one therefore +has to stabilize this 8-category, which is a rather tricky operation. However, we it will +follow from the theory of lax matrices that this lax colimit indeed just agrees with the +lax limit which can be computed in Cat8. +8.4 +Lax matrices +Analogously to the case of ordinary coproducts and products (which corresponds to the case +where the category S is just a set), we can interpret maps from a lax colimit to a lax limit as +a sort of matrices: By the defining property we have +Cplaxcolims:S Xs, laxlimt:T Ytq » laxlimt:T Cplaxcolims:S, Ytq +(8.4.1) +» laxlimt:T laxlims:Sop CpXs, Ytq +(8.4.2) +» laxlimpt,sq:TˆSop CpXs, Ytq +(8.4.3) +so that we can interpret a map α: laxcolims:S Xs Ñ laxlimt:T Yt as a tuple pαt,sqpt,sq:TˆSop +which we think of as a matrix whose rows are indexed by T and whose columns are indexed +by Sop. We define +laxMatCpX, Yq :“ laxlimpt,sq:TˆSop CpXs, Ytq. +(8.4.4) +Note that this is a well defined 8-category even when laxcolim X and/or laxlim Y does not +exist. When X: t˚u Ñ C is just an object X “ X˚, we still use the notation laxMatCpX, Yq “ +laxMatCpX, Yq and observe that it is precisely the 8-category of lax cones on Y with vertex +X; and analogously for lax cocones +90 + +Example 8.4.5. Let S “ T “ ∆1 “ t0 +fÝÑ 1u be the walking arrow and consider two diagrams +X: S Ñ C and Y: T Ñ C. Then we can compactly describe objects of +laxMatCpX, Yq “ laxlim +¨ +˚ +˚ +˚ +˝ +CpX0, Y0q +CpX1, Y0q +CpX0, Y1q +CpX1, Y1q +˛ +‹‹‹‚ +(8.4.6) +as T ˆ Sop-indexed diagrams in the Grothendieck construction, which we depict as follows: +¨ +˚ +˚ +˝ +α00 +α01 +α10 +α11 +˛ +‹‹‚ +(8.4.7) +Explicitly unpacking this notation, such a matrix consists of: +(1) four 1-morphisms +α00 : X0 Ñ Y0 +α01 : X1 Ñ Y0 +α10 : X0 Ñ Y1 +α11 : X1 Ñ Y1 +(8.4.8) +(2) four 2-morphisms +Yf ˝ α00 +α00 +α01 ˝ Xf +α10 +Yf ˝ α01 +α10 +α11 ˝ Xf +α11 +αf0 +α0f +αf1 +α1f +(8.4.9) +(3) assembling into a commutative square +Yf ˝ α00 +Yf ˝ α01 ˝ Xf +α10 +α11 ˝ Xf +αf1 +Yf˝α0f +αf1˝Xf +α1f +(8.4.10) +We now introduce the lax matrix multiplication which categorifies the classical formula +(8.1.25). The classical formula involves a finite sum of elements in some hom-set Apxs, zuq of +the category A. Our categorical analog of these sums will be categorical colimits. +For the remainder of this section, let C be an p8, 2q-category enriched in 8-categories +with colimits, i.e., +• each hom-category CpX, Y q has all colimits and +• and each composition functor CpX, Y q ˆ CpY, Zq Ñ CpX, Zq preserves colimits in each +variable separately. +91 + +Construction 8.4.11. Let S be an 8-category, and X: S Ñ C a diagram. Passing to the +cartesian fibrations classifying the respecive hom-functors, we obtain a commutative square +Tw˚pSq “ +ş˚ Sp´, ´q +ş˚ Cp´, ´q “ Tw˚pCq +S ˆ Sop +C1 ˆ Cop +1 +p +α +q +XˆXop +(8.4.12) +which amounts to the dashed section +şps,tq CpXs, Xtq +Tw˚pCq +Tw˚pSq +S ˆ Sop +C1 ˆ Cop +1 +{ +q +p +α +XˆXop +(8.4.13) +of the pullback-fibration pX ˆ Xopq˚pqq which informally sends an arrow pf : s Ñ tq : Tw˚pSq +to Xf : CpXs, Xtq. +We can now construct the composite functor +laxlims:S CpL, Xsq ˆ laxlimt:Sop CpXt, L1q +(8.4.14) +“ FunSˆSop +˜ +S ˆ Sop, +ż +ps,tq +CpL, Xsq ˆ CpXt, L1q +¸ +(8.4.15) +ÝÑ FunSˆSop +˜ +Tw˚pSq, +ż +ps,tq +CpL, Xsq ˆ CpXt, L1q +¸ +(8.4.16) +ÝÑ FunSˆSop +˜ +Tw˚pSq, +ż ps,tq +CpXs, Xtq ˆSˆSop +ż +ps,tq +CpL, Xsq ˆ CpXt, L1q +¸ +(8.4.17) +“ FunSˆSop +˜ +Tw˚pSq, +ż +s +CpL, Xsq ˆS +ż ps,tq +CpXs, Xtq ˆSop +ż +t +CpXt, L1q +¸ +(8.4.18) +ÝÑ Fun +` +Tw˚pSq, CpL, L1q +˘ colim +ÝÝÝÑ CpL, L1q, +(8.4.19) +where +• the first arrow is pullback of sections along p: Tw˚pSq Ñ S ˆ Sop, +• the second arrow adds the section α in the first component of the fiber product, +• the third arrow is given by composition with the composition map +ż +s +CpL, Xsq ˆS +ż ps,tq +CpXs, Xtq ˆSop +ż +t +CpXt, L1q ÝÑ CpL, L1q, +(8.4.20) +• the last arrow is just the colimit functor in the 8-category CpL, L1q. +On objects, this functor takes a lax cone and a lax cocone on X, +Φ : laxlims:S CpL, Xsq +and +Ψ : laxlimt:Sop CpXt, L1q, +(8.4.21) +92 + +and sends them to the map Ψ ˝ Φ: L Ñ L1 defined by the formula +Ψ ˝ Φ :“ +colim +pf:sÑtq:Tw˚pSqpΨt ˝ Xf ˝ Φsq. +(8.4.22) +Remark 8.4.23. When S “ t1, 2, . . . , nu is a finite set the (cartesian) twisted arrow category +Tw˚pSq Ñ S ˆ S can be canonically identified with the diagonal ∆: S Ñ S ˆ S. Under this +identification the formula (8.4.22) simplifies to +pΨ ˝ Φq “ pΨ ˝S Φq “ +ž +s,tPS +s“t +Ψt ˝ id ˝Φs “ +n +ž +s“1 +Ψs ˝ Φs +(8.4.24) +which is just the usual multiplication (8.1.25) of the row vector Ψ with the column vector Φ. +We assemble our categorified analog of row-column multiplication to the lax version of +matrix multiplication: +Construction 8.4.25. Let S, U be 8-categories, and X: S Ñ C and Z: U Ñ C two diagrams. +For each object Y : C we consider the functor +laxlims:Sop CpXs, Y q ˆ laxlimu:U CpY, Zuq +(8.4.26) +“ laxlimpu,sq:UˆSop CpXs, Y q ˆ CpY, Zuq +(8.4.27) +ÝÑ laxlimpu,sq:UˆSop CpXs, Zuq, +(8.4.28) +induced by composition of C. On objects it takes a lax cocone and a lax cone, +Ψ : laxlims:Sop CpXs, Y q +and +Φ : laxlimu:S CpY, Zuq, +(8.4.29) +and sends them to the matrix Φ ˝ Ψ : laxlimpu,sq:UˆSop described by the formula +pΦ ˝ Ψqus “ Φu ˝ Φs. +(8.4.30) +More generally, we can replace the object Y : C by a diagram Y: T Ñ C and consider the +functor +laxMatCpX, Yq ˆ laxMatCpY, Zq +(8.4.31) +“ laxlimpu,sq:UˆSop plaxlimt:T CpXs, Ytq ˆ laxlimt1:T op CpYt1, Zuqq +(8.4.32) +laxlimu,sp´˝T ´q +ÝÝÝÝÝÝÝÝÝÝÑ laxlimpu,sq:UˆSop CpXs, Zuq “ laxMatCpX, Zq +(8.4.33) +which is given in componentwise in u, s by the composition functor from Construction 8.4.11 +(applied to lax cones and cocones on Y). Explicitly, this functor is given by the formula +Φ ˝ Ψ “ Φ ˝T Ψ “ +ˆ +colim +pf:tÑt1q:Tw˚pTqpΦut1 ˝ Yf ˝ Ψtsq +˙ +pu,sq:UˆSop +. +(8.4.34) +This is what we call the lax matrix multiplication. +Finally, we can assemble all lax matrices of different shapes into a category hoLaxMatC: +93 + +• Objects are equivalence classes of diagrams X: S Ñ C, where S is any small 8-category. +• Morphisms from X: S Ñ C to Y: T Ñ C are equivalence classes of matrices Φ : +laxMatCpX, Yq. +• Composition is given by lax matrix multiplication of Construction 8.4.25. +Note that the lax matrix multiplication is functorial by construction, making it in partic- +ular well defined on equivalence classes. To prove that hoLaxMatC is indeed a category, we +will thus only need to construct the identity matrix and prove that lax matrix multiplication +is associative up to equivalence. +In , we shall prove a slightly stronger statement: +Lemma 8.4.35. The lax matrix multiplication of Construction 8.4.25 is +(1) associative up to natural equivalence, i.e. for diagrams W: R Ñ C and X: S Ñ C and +Y: T Ñ C and Z: U Ñ C we have +p´ ˝T ´q ˝S ´ » ´ ˝T p´ ˝S ´q +(8.4.36) +as functors +laxMatCpW, Xq ˆ laxMatCpX, Yq ˆ laxMatCpY, Zq ÝÑ laxMatCpW, Zq +(8.4.37) +(2) unital up to natural equivalence, i.e., for each diagram X: S Ñ C there is a matrix +IX : laxMatCpX, Xq with components +IX +ts “ colim +f:Sps,tq Xf : CpXs, Xtq +(8.4.38) +such that +IX ˝S ´ » id +and +´ ˝S IX » id +(8.4.39) +as endofunctors of laxMatCpY, Xq and laxMatCpX, Yq, respectively (for each other dia- +gram Y: U Ñ C). +Remark 8.4.40. The category hoLaxMatC is of course only the truncation of an p8, 2q- +category of lax matrices, which one could construct with more effort. Even Lemma 8.4.35 +only shows that lax matrix multiplication is associative up to natural equivalence, but does +not exhibit any sort of coherence such as the pentagon. We will not be needing this additional +layer of coherence in this article so Lemma 8.4.35 will suffice. +The proof of associativity is relatively straightforward: +Proof of Lemma 8.4.35 (1). For matrices +F : laxlimSˆRoppWr, Xsq, +G : laxlimTˆSoppXs, Ytq, +H : laxlimUˆT oppYt, Zuq +(8.4.41) +94 + +we compute +ppH ˝T Gq ˝S Fqur » colim +f : sÑs1pH ˝T Gqus1XfFsr +(8.4.42) +» colim +f : sÑs1pcolim +g : tÑt1 Hut1YgGts1qXfFsr +(8.4.43) +“ colim +f : sÑs1 colim +g : tÑt1pHut1YgGts1XfFsrq +(8.4.44) +“ +colim +pf,gq:Tw˚pSqˆTw˚pTqpHut1YgGts1XfFsrq +(8.4.45) +» ¨ ¨ ¨ » pH ˝T pG ˝S Fqqur +(8.4.46) +naturally in F, G, H and u : U, r : Rop; where the third step uses that composition in C +preserves colimits in each variable. +Before we can prove part (2) of Lemma 8.4.35 we need to construct the unit matrices +IX : laxMatCpX, Xq. +Construction 8.4.47. Consider the commutative square +Tw˚pSq “ +ş +˚ Sp´, ´q +ş +˚ Cp´, ´q “ Tw˚pCq +Sop ˆ S +Cop +1 ˆ C1 +p +α +q +XopˆX +(8.4.48) +induced by a diagram X: S Ñ C. Here the vertical maps are the cocartesian fibrations classi- +fying the respective hom-functors of S and C. The that the p8, 2q-category C is enriched in +8-categories with colimits means that there exists an (essentially unique) left q-Kan extension +of α along p, giving rise to the dashed lift +Tw˚pSq +Tw˚pCq +Sop ˆ S +Cop +1 ˆ C1 +p +α +q +I1 +XopˆX +(8.4.49) +Since the pullback of the cocartesian fibration q along XopˆX is, by definition, the cocartesian +fibration +ş +ps,tq:SopˆS CpXs, Xtq Ñ Sop ˆS, this lift I1 corresponds to a section of this fibration, +i.e., an object +IX “ I : laxlimps,tq:SopˆS CpXs, Xtq +(8.4.50) +By the pointwise formula, we can explicitly compute the value of IX at ps, tq : Sop ˆ S as the +colimit of the composite +Tw˚pSq{ps, tq αÝÑ Tw˚pCq{pXs, Xtq Ñ CpXs, Xtq, +(8.4.51) +which is the functor that informally maps +pf1 : s1 Ñ t1, g: s Ñ s1, h: t1 Ñ tq ÞÑ Xh ˝ Xf1 ˝ Xs. +(8.4.52) +Since the inclusion Sps, tq » Tw˚pSqps,tq ãÑ Tw˚pSq{ps, tq has a left adjoint (because q is a +cocartesian fibration), it is homotopy terminal; thus we can compute the components IX +ts via +the desired explicit formula (8.4.39) +95 + +Remark 8.4.53. Since all pointwise colimits 8.4.39 are taken over spaces Sps, tq (as opposed +to arbitrary 8-categories), we see that for the construction of the unit matrix we could +have relaxed our assumption on C and only required it to be enriched in 8-categories with +groupoidal colimits. +Remark 8.4.54. When S is a set this formula simplifies to +Its “ colim +f:Sps,tqpXfq “ +# +idXs, if s “ t +H, if s ‰ t +(8.4.55) +which is the direct analog of formula (8.1.3), with the initial object H of Cpxs, xtq taking the +role of the zero object of a commutative monoid. +Example 8.4.56. Continuing Example 8.4.5, we consider a diagram X: ∆1 “ t0 10 +ÝÑ 1u Ñ C. +The unit ∆1 ˆ p∆1qop-matrix then looks as follows: +¨ +˚ +˚ +˝ +idX0 +H +X10 +idX1 +˛ +‹‹‚: laxMatCpX, Xq +(8.4.57) +since the indexing space of the colimit colimf:Sps,tq Xf is either empty in the case s “ 1, t “ 0 +or a singleton otherwise. +We now finish the proof of Lemma 8.4.35. +Proof of Lemma 8.4.35 (2). We only treat the case of postcomposition with IX; the other +statement is dual. We need to show that for every diagram Y: R Ñ C, the functor +IX ˝S ´: laxMatCpY, Xq Ñ laxMatCpY, Xq +(8.4.58) +is naturally equivalent to the identity. Without loss of generality, we may assume that R “ t˚u +so that Y “ Y˚ is a single object. Naturally in F : laxMatCpY, Xq and u : S we compute (in +the 8-category CpY, Xsq) +pIX ˝S Fqu » +colim +pf : sÑtq:Tw˚pSqp colim +g:Spt,uq XgqXfFs +(8.4.59) +“ +colim +pf : sÑtq:Tw˚pSq +g:Spt,uq +XgfFs +(8.4.60) +where the shape of the second colimit is the category +ż +pf : sÑtq:Tw˚pSq +Spt, uq “ Tw˚pSq ˆSop pS{uqop +(8.4.61) +Note that the diagram pf, gq ÞÑ XgfFs over which we are taking the colimit arises as the +pullback of the diagram +S{u Ñ CpY, Xuq, +ph: s Ñ uq ÞÑ XhFs +(8.4.62) +96 + +along the functor +γ : Tw˚pSq ˆSop pS{uqop Ñ S{u, +pf : s Ñ t, g: t Ñ uq ÞÑ pgf : s Ñ uq. +(8.4.63) +This functor γ has a left adjoint +S{u Ñ Tw˚pSq ˆSop pS{uqop, +pf : s Ñ uq ÞÑ pf : s Ñ u, idu : u Ñ uq, +(8.4.64) +and is therefore homotopy terminal. This allows us to finish the computation: +pIX ˝S Fqu » +colim +pf : sÑtq:Tw˚pSq +g:Spt,uq +XgfFs +(8.4.65) +» +colim +ph: sÑuq:S{u XhFs +(8.4.66) +» XiduFu “ Fu, +(8.4.67) +using in the last step that idu : u Ñ u is a terminal object of the comma category S{u. +We can characterize lax limit and colimits purely in terms of the matrix calculus encoded +in the category hoLaxMatC: +Lemma 8.4.68. Let X: S Ñ C be a diagram. +(1) A lax cone P : laxMatCpL, Xq is a lax limit cone if and only if it is an isomorphism in +the category hoLaxMatC. +(2) A lax cocone I : laxMatpX, Lq is a lax colimit cone if and only if it is an isomorphism +in the category hoLaxMatC. +Proof. We prove the statement about lax cones; the other one is dual. +First assume that P is a lax limit cone. This implies that the functor +P ˝ ´: laxMatCpX, Lq “ laxlims:Sop CpXt, Lq Ñ laxlimpt,sq:SˆSop CpXs, Xtq “ laxMatCpX, Xq +(8.4.69) +is an equivalence. In particular the map +P ˝ ´: hoLaxMatCpX, Lq Ñ hoLaxMatCpX, Xq +(8.4.70) +is a bijection which implies that P is an isomorphism in hoLaxMatC; the inverse map is the +unique lax cocone I with P ˝ I “ IX. +Conversely, assume that P has an inverse in hoLaxMatC, i.e., a lax cocone I : laxMatCpX, Lq +satisfying P ˝ I » IX and I ˝S P » idL. Then for each L1 : C we have equivalences +pP ˝ ´q ˝ pI ˝S ´q “ pP ˝ pI ˝S ´qq » pP ˝ Iq ˝S ´ » IX ˝S ´ » id +(8.4.71) +and +pI ˝S ´q ˝ pP ˝ ´q “ I ˝S pP ˝ ´q » pI ˝S Pq ˝ ´ » idL ˝´ “ id +(8.4.72) +as endofunctors of +laxMatCpL1, Xq +and +CpL1, Lq, +(8.4.73) +respectively (using Lemma 8.4.35), showing that P ˝ ´ is an equivalence of 8-categories, as +required. +97 + +Corollary 8.4.74. A diagram X: S Ñ C admits a lax limit if and only if it admits a lax +colimit. When they exist, the unit matrix IX : laxMatCpX, Xq corresponds to an equivalence +IX : laxcolimS X » +ÝÑ laxlimS X. +(8.4.75) +Proof. The diagram X admits a lax (co)limit if and only if it is isomorphic in hoLaxMatC to +an object L: t˚u Ñ C. In this case L is both the lax limit and the lax colimit, exhibited by +mutually inverse lax (co)cones I : X Ñ L and P : L Ñ X. By definition, the map IX : L Ñ +L corresponding to the matrix IX is determined (up to equivalence) by the property that +P ˝ IX ˝ I » IX. Since the identity idL satisfies this property, we conclude that IX » idL; in +particular this map is an equivalence. +Remark 8.4.76. The comparison map (8.4.75) does not just depend on the objects +L “ laxlim X +and +L1 “ laxcolim X +(8.4.77) +but on the implicit lax (co)limit cones P : laxMatCpL, Xq and I : laxMatCpX, L1q. Specifically, +the map IX is characterized up to equivalence by the relation +P ˝ IX ˝ I » IX +(8.4.78) +or equivalently +IX » pI ˝S Pq´1 +(8.4.79) +(since P and I are isomorphisms in hoLaxMatC and IX is the identity on X : hoLaxMatC). +Having described lax (co)limits via lax matrix formulas in hoLaxMatC, we can immediately +deduce that all lax (co)limits are absolute with respect to the colimit-enrichment. +Proposition 8.4.80. Let C, C1 be p8, 2q-categories enriched in 8-categories with colimits. +Let F: C Ñ C1 be a functor which preserves colimits on hom-cateogories. Then F preserves +all lax colimits and lax limits. +Proof. Since F preserves colimits on hom-categories, it induces a well defined functor +hoLaxMatF: hoLaxMatC Ñ hoLaxMatC1, +(8.4.81) +given by applying F pointwise to diagram and matrices. Since this functor necessarily sends +isomorphisms to isomorphisms, Lemma 8.4.68 implies that F sends lax (co)limit cones to lax +(co)limit cones. +Finally, we deduce that lax matrix multiplication corresponds to composition of maps +between lax colimits/limits in the case where those lax (co)limits exist. +Proposition 8.4.82. Let X: S Ñ C, Y: T Ñ C, Z: U Ñ C be diagrams indexed by 8- +categories and admitting lax limits/colimits. +Then there is a commutative square of 8- +categories +CplaxcolimS X, laxlimT Yq ˆ CplaxcolimT Y, laxlimU Zq +laxMatCpX, Yq ˆ laxMatCpY, Zq +CplaxcolimS X, laxlimU Zq +laxMatCpX, Zq +´˝pIYq´1˝´ +» +´˝T ´ +» +(8.4.83) +In other words, after identifying lax colimits and lax limits via the canonical unit matrix, lax +matrix multiplication corresponds precisely to function composition. +98 + +Proof. Denote by IX : laxMatCpX, laxcolim Xq and PX : laxMatCplaxlim X, Xq two lax (co)limits +cones for the diagram X (and similarly for Y and Z). The implicit identification +Cplaxcolim X, laxlim Yq » +ÝÑ laxMatCpX, Yq +(8.4.84) +is given explicitly as PY ˝ ´ ˝ IX, and similarly for the other horizontal maps. Therefore the +desired commutative square is just the natural equivalence +pPZ ˝ ´ ˝ IYq ˝T pPY ˝ ´ ˝ IXq » PZ ˝ ´pIY ˝T PYq ˝ ´ ˝ IX » PZp´ ˝ pIYq´1 ˝ ´q ˝ IX. (8.4.85) +using the equivalence (8.4.79) in the last step. +Example 8.4.86. Continuing Example 8.4.56, we consider two maps +X FÝÑ +lax +à +∆1 +pY0 Ñ Y1q G +ÝÑ Z +(8.4.87) +corresponding to matrices +F “ +¨ +˚ +˚ +˚ +˝ +F0 +F1 +˛ +‹‹‹‚ +and +G “ pG0 Ð G1q +(8.4.88) +The cartesian twisted arrow category Tw˚p∆1q is the poset +00 +01 +11 +(8.4.89) +The composite GF : X Ñ Z is therefore the pushout of the diagram +G0F0 +G1Y01F0 +G1F1 +G01F0 +G1F01 +(8.4.90) +More general ∆1 ˆp∆1qop matrices can then be multiplied in the usual row-by-column fashion +since each entry pGFqus only depends on row u of G and column s of F. For example, we +can compute (with X “ Y and F “ I) +G ˝ I “ +¨ +˚ +˝ +G00 +G01 +G10 +G11 +˛ +‹‚ +¨ +˚ +˚ +˝ +idX0 +H +X10 +idX1 +˛ +‹‹‚ +(8.4.91) +» +¨ +˚ +˚ +˚ +˝ +colim +` +G00 Ð G01X10 +“ +ÝÑ G01X10 +˘ +colim +` +H +“ +ÐÝ H Ñ G01 +˘ +colim +` +G10 Ð G11X10 +“ +ÝÑ G11X10 +˘ +colim +` +H +“ +ÐÝ H Ñ G11 +˘ +˛ +‹‹‹‚» G +(8.4.92) +99 + +8.5 +Lax additivity +Classical semi-additivity of a category A manifests itself on two levels: +(1) Each hom-set of A has a commutative monoid structure which allows to take sums +ř +sPS fs indexed by arbitary finite sets S. +(2) The category A allows direct sums À +sPS xs indexed by finite sets S which are both +products and coproducts. +We categorify these notions by replacing (discrete) addition ΣsPS on the hom-sets by +colimits colims:S on the hom-categories and (discrete) coproducts/products š +sPS » ś +sPS by +lax bilimits laxcolims:S » laxlims:S, which are now indexed by arbitary small 8-categories S +rather than finite sets. +Definition 8.5.1. Let C be an p8, 2q-category enriched in 8-categories with groupoidal +colimits. Let X: S Ñ C a diagram indexed by an 8-category S. A lax bilimit of X consists +of a lax colimit cone I : laxMatCpX, L1q and a lax limit cone P : laxMatCpL, Xq such that the +canonical map +IX : L1 Ñ L +(8.5.2) +corresponding to the unit matrix IX : laxMatCpX, Xq is an equivalence. We identify L and L1 +via IX and write +lax +à +S +X +or +lax +à +s:S +Xs +(8.5.3) +for both/either of them. +When X: S Ñ t˚u X +ÝÑ C is a constant diagram, we write S bX or XS for the constant lax +bilimit Àlax +s:S X “ Àlax +S X. When convenient we drop the typographical distinction between a +matrix F : laxMatCpX, Yq and the associated map F : Àlax X Ñ Àlax Y. +We can now finally define the notion of lax semiadditivity. +Definition 8.5.4. An p8, 2q-category A is called (finitely) lax semiadditive if +(1) it is enriched in 8-categories with (finite) colimits (with functors preserving them), +(2) each diagram S Ñ A indexed by a (finite) small 8-category S admits a lax bilimit. +Remark 8.5.5. We have seen in Corollary 8.4.74 that in the presence of sufficently many +colimits in the hom-categories, every lax limit or colimit is automatically a lax bilimit. Thus +the second condition could be weakened to just require the existence of lax limits or lax +colimits. +The final step is to categorify the passage from semiadditve to additive categories which +amounts to requiring the hom-monoids to be abelian groups. Following our philosophy of +§1.1, abelian groups should be replaced by stable 8-categories leading to the following easy +definition: +Definition 8.5.6. A (finitely) lax semiadditive p8, 2q-category A is called (finitely) lax addi- +tive if every hom-8-category ApX, Y q is stable. +100 + +Remark 8.5.7. Denote by Cop the p8, 2q-category obtained from C by reversing the directions +of the 1-morphisms, i.e., CoppX, Y q “ CpY, Xq. +If C is enriched in (stable) 8-categories +with colimits, then so is Cop. Moreover lax limits/colimits/bilimits in Cop correspond to lax +colimits/limits/bilimits in C. Thus an p8, 2q-category A is (finitely) lax (semi)additive if and +only if Aop is (finitely) lax (semi)additive. +Example 8.5.8. Lax limits in the p8, 2q-category PrL of presentable 8-categories exist +and are computed as underlying 8-categories. Since PrL is enriched in 8-categories with +colimits it follows that it is a lax semiadditive p8, 2q-category. The full p8, 2q-category St of +presentable stable 8-categories is closed under lax colimits and enriched in stable 8-categories, +thus it is lax additive. The p8, 2q-category StR of presentable stable 8-categories and right +adjoint functors is only finitely lax additive, since composition with a right adjoint functor +is exact but does not preserve arbitrary colimits. The p8, 2q-category of stable 8-categories +and exact functors is finitely lax additive. +Note that one can replace all presentability assumptions by just requiring the relevant +8-categories to have colimits (or limits, in the case of StR) and for the functors between them +to preserve them. +Construction 8.5.9. Let T, S be small 8-categories and let +F : T ˆ Sop Ñ Spaces » PrLpSpaces, Spacesq +(8.5.10) +be a matrix of spaces. Let A be a lax semiadditive p8, 2q-category. For each X : A, denote by +F X :“ F b idX the matrix +F X : T ˆ Sop FÝÑ Spaces ´bidX +ÝÝÝÝÑ ApX, Xq, +(8.5.11) +where the second functor arises from the tensoring by Spaces on the 8-category ApX, Xq with +colimits. In this way, the space-valued matrix can be interpreted as a map XS Ñ XT , which +we call the action of F on X. Similarly, when the hom-8-categories are pointed/stable, we can +interpret in A every matrix of pointed spaces/spectra, by using the corresponding tensoring. +Lemma 8.5.12. Action of matrices is compatible with matrix multiplication, i.e., we have +equivalences F X ˝ GX » pF ˝ GqX whenever F, G are composable matrices of spaces/pointed +spaces/spectra and X is an object in a correspondingly enriched p8, 2q-category. +Proof. We have, naturally in u : U, s : Sop: +pF ˝ GqX +us » +ˆ +colim +f : tÑt1 Fut1 b Gts +˙ +b idX +(8.5.13) +» +ˆ +colim +f : tÑt1pFut1 b idXq ˝ pGts b idXq +˙ +(8.5.14) +» pF X ˝ GXqus +(8.5.15) +using that the tensoring preserves colimits and that idX ˝ idX » idX. +Example 8.5.16. The universal identity ∆1 ˆ p∆1qop-matrix is +I “ +¨ +˚ +˚ +˝ +t˚u +H +t˚u +t˚u +˛ +‹‹‚: ∆1 ˆ p∆1qop Ñ Spaces. +(8.5.17) +101 + +Indeed, every p8, 2q-category C whose hom-8-categories have initial objects, the unit matrix +IX for the constant diagram X: ∆1 Ñ t˚u X +ÝÑ C is given by IX “ IX :“ I b idX. +More generally, for any 8-category S the universal idenity S ˆ Sop-matrix is the just the +transpose of the hom-functor +I⊺ “ Sp´, ´q: Sop ˆ S Ñ Spaces. +(8.5.18) +Example 8.5.19. Consider the matrices +Cof :“ +¨ +˚ +˝ +S0 +S0 +0 +S0 +˛ +‹‚: ∆1 ˆ p∆1qop Ñ Spaces˚ +(8.5.20) +and +Fib :“ +¨ +˚ +˝ +0 +Sr´1s +S +0 +˝ +˛ +‹‚: ∆1 ˆ p∆1qop Ñ Sp, +(8.5.21) +defined in pointed spaces and spectra, respectively. If A is a pointed 8-category with colimits, +then the matrix Cof acts as the cofiber map +CofA “ +¨ +˚ +˝ +idA +idA +0 +idA +˛ +‹‚: A∆1 Ñ A∆1, +(8.5.22) +Indeed, for every +¨ +˝ a +b +˛ +‚: B Ñ A∆1 +(8.5.23) +we can compute the matrix product +¨ +˚ +˝ +idA +idA +0 +idA +˛ +‹‚˝ +¨ +˝ a +b +˛ +‚» +¨ +˚ +˚ +˝ +colimpa Ð a Ñ bq +colimp0 Ð a Ñ bq +˛ +‹‹‚» +¨ +˚ +˝ +b +cofpa Ñ bq +˛ +‹‚“ Cofpa Ñ bq. +(8.5.24) +If A is also stable, then a similar calculation shows that the matrix Fib is inverse to Cof and +acts as the fiber map. +For every matrix F : T ˆ Sop Ñ ApX, Xq corresponding to an arrow XS Ñ XT in A, we +denote by F⊺ the “transposed” matrix +F⊺ : Sop ˆ pT opqop » T ˆ Sop FÝÑ ApX, Xq. +(8.5.25) +describing the dual map XT op Ñ XSop. +102 + +Lemma 8.5.26. In the setting of Construction 8.5.9 there are commutative diagrams +ApX, Y Sq +ApX, Y qS +ApX, Y T q +ApX, Y qT +» +ApX,F Y q +F ApX,Y q +» +and +ApS b X, Y q +ApX, Y qSop +ApT b X, Y q +ApX, Y qT op +» +ApF X,Y q +F ApX,Y q +⊺ +» +(8.5.27) +Proof. We do the second computation; the first one is similar. Functorially in M : ApX, Y qT op » +ApT b X, Y q and s : Sop we compute +ApF X, Y qpMqs “ pM ˝ F Xqs “ +colim +pf : tÑt1q:Tw˚pTq Mt1 ˝ F X +ts +(8.5.28) +“ +colim +f:Tw˚pTq Mt1 ˝ pFts b idXq +(8.5.29) +» +colim +pf : tÑt1q:Tw˚pTq Fts b Mt1 +(8.5.30) +» +colim +pg : t1Ñtq:Tw˚pT opq Fts b Mt1 +(8.5.31) +» +colim +pg : t1Ñtq:Tw˚pT opqpFts b idApX,Y qqpMt1q +(8.5.32) +“ +colim +pg : t1Ñtq:Tw˚pT opqppF⊺qst b idApX,Y qqpMt1q +(8.5.33) +“ F ApX,Y q +⊺ +pMqs. +(8.5.34) +Apart from expanding the various definitions, we have used +• that Mt1 ˝ ´: ApX, Xq Ñ ApX, Y q preserves colimits and hence the tensoring Fts b ´, +• the canonical identification Tw˚pTq » Tw˚pT opq which reverses source and target, +• that colimits of functors ApX, Y q Ñ ApX, Y q are computed pointwise, hence also the +tensoring Fts b ´. +Lemma 8.5.35. Let A be a lax semiadditive p8, 2q-category. Then A is lax additive if and +only if each hom-8-category ApX, Y q is pointed and the matrix Cof from Example 8.5.19 acts +invertibly on each X : A. If this is the case, then the inverse is given by the action of Fib. +Proof. Assuming that all hom-8-categories ApX, Y q of the lax semiadditive p8, 2q-category +A are pointed, they are stable if and only if the cofiber functor +CofApX,Y q : ApX, Y q∆1 Ñ ApX, Y q∆1 +(8.5.36) +is invertible. Using Lemma 8.5.26, we can identify this functor with ApX, CofY q. Thus A is +lax additive if and only if all ApX, Y q are stable, if and only if all ApX, CofY q are invertible, +if and only if all CofY are invertible, as claimed. +8.6 +Oplax additivity +So far we have focused our discussion exclusively on lax limits and colimits, as opposed to +oplax ones. We could have of course passed to the 2-morphism dual everywhere (obtained +103 + +from an p8, 2q-category C by replacing each hom-8-category CpX, Y q by its opposite) and told +an analogous story using oplax colimits/limits/bilimits. This would lead to what we might +call oplax (semi)additive p8, 2q-categories A, which are enriched in (stable) 8-categories with +limits and allow the formation of oplax bilimits +oplax +à +S +X :“ oplaxcolimS X » oplaxlimS X +(8.6.1) +of any diagram X: S Ñ A indexed by a small 8-category (note the direction of the little +arrow above the symbol À). +For the convenience of the reader, we summarize the main formulas of this dual theory; +all the constructions and proofs are dual to the ones we saw earlier. +(1) For two diagrams X: S Ñ A and Y: T Ñ C, we define +oplaxMatCpX, Yq :“ oplaxlimpt,sq:TˆSop CpXs, Ytq +(8.6.2) +as the category of oplax matrices from X to Y. Explicitly, such matrices are sections of +the contravariant Grothendieck construction +ż pt,sq:TˆSop +ApXs, Ytq ÝÑ T op ˆ S. +(8.6.3) +(2) The oplax matrix multiplication +oplaxMatCpX, Yq ˆ oplaxMatCpY, Zq Ñ oplaxMatCpX, Zq +(8.6.4) +is given by the formula +pΦ ˝ Ψqus :“ +lim +pf : tÑt1q:Tw˚pTq Φut1YfΨts. +(8.6.5) +(3) The oplax unit matrix for a diagram X: S Ñ C is +J X “ ppt, sq ÞÑ +lim +f:Sps,tq Xfq : oplaxMatCpX, Xq +(8.6.6) +Example 8.6.7. The p8, 2q-category St is enriched in stable 8-categories and admits oplax +limits; thus it is finitely oplax additive. +It is not oplax additive because composition of +functors does not preserve arbitrary limits. +8.7 +Coordinate change for ∆1-matrices +We have seen that any lax semiadditive p8, 2q-category admits a nicely behaved calculus of +lax matrices. However, if we apply K0 componentwise to the lax matrix multiplication for lax +∆1-bilimits (see Example 8.4.56 and Example 8.4.86) we obtain the very unusual formula +p a0 a1 q ˝ +´ +b0 +b1 +¯ +“ a0b0 ´ a1b0 ` a1b1 +(8.7.1) +104 + +or, more generally A ˝ B “ AI´1B, where I “ K0pI∆1q “ p 1 0 +1 1 q is the new unit matrix. +The goal of this section is to introduce a convenient “coordinate change” in the lax additive +(as opposed to merely lax semiadditive) setting, which up to a sign recovers the usual matrix +multiplication on K0. +The key ingredient is the cofiber-fiber-equivalence +Cof : Funp∆1, Aq +» +ÐÝÑ Funp∆1, Aq :Fib +(8.7.2) +pfibpu1q “ b uÝÑ aq Ø pa u1 +ÝÑ b1 “ cofpuqq +(8.7.3) +for every stable 8-category A. +More precisely, we make use of the following dependent version of the cofiber-fiber- +equivalence which identifies the oplax limit over an arrow with the lax limit. +Lemma 8.7.4 (Lemma 1.3 in [DJW21]). Let f : A Ñ B be a diagram of stable 8-categories. +Then there is a natural equivalence +Cof : oplaxlim∆1pB +fÐÝ Aq +» +ÐÑ laxlim∆1pA +fÝÑ Bq :Fib +(8.7.5) +described by the formula +pb “ fibpu1q, a, b uÝÑ faq Ø pa, b1 “ cofpuq, fa vÝÑ b1q. +(8.7.6) +While not strictly necessary, it is convenient to implement this equivalence by explicit +matrices using a combination of the lax and oplax matrix calculus. +For the remainder of the section, let A be a lax additive p8, 2q-category. Then A is in +particular enriched in 8-categories with finite limits, so that we have available the finite oplax +matrix calculus (dual to the one in §8.4) as long as we restrict to diagrams indexed by finite +8-categories S. +Construction 8.7.7. Let X: ∆1 Ñ A be a diagram, X “ pX0 +FÝÑ X1q. Consider the oplax +cone and cocone +Fib :“ +Àlax +∆1 X +X0 +X1 +fib +p0 +F +and +Fib_ :“ +Àlax +∆1 X +X0 +X1 +F +i1 +fib_ +(8.7.8) +described by the p∆1qop ˆ p∆1qop-matrix and ∆1 ˆ ∆1-matrix +Fib :“ +¨ +˚ +˝ +p0 +fib +˛ +‹‚“ +¨ +˚ +˚ +˝ +idX0 +0 +0 +r´1sX1 +˝ +˛ +‹‹‚ +and +Fib_ :“ pfib_ ÝÑ i1q “ +¨ +˚ +˚ +˝ +r´1sX1 +0 +0 +idX0 +˝ +˛ +‹‹‚, +(8.7.9) +respectively. Here P “ ppsq and I “ pisq (indexed by s : ∆1) are the lax limit/colimit cone +exhibiting the lax bilimit Àlax +∆1 X i.e., the rows and columns of the unit matrix IX. +The following lemma explains the name of the cones Fib and Fib_ in terms of the maps +they represent/corepresent. +105 + +Lemma 8.7.10. Let Y : A. +(1) The induced map +laxlims:∆1 ApY, Xsq +» +ÐÝ ApY, +lax +à +∆1 +Xq Fib˝´ +ÝÝÝÝÑ oplaxlims:∆1 ApY, Xsq +(8.7.11) +is precisely the dependent fiber functor of Lemma 8.7.4 for the ∆1-diagram ApY, X0q Ñ +ApY, X1q. +(2) The induced map +laxlims:p∆1qop ApXs, Yq +» +ÐÝ Ap +lax +à +∆1 +X, Yq ´˝Fib_ +ÝÝÝÝÝÑ oplaxlims:p∆1qop ApXs, Yq +(8.7.12) +is precisely the dependent fiber functor of Lemma 8.7.4 for the p∆1qop “ ∆1-diagram +ApX0, Yq Ð ApX1, Yq. +Proof. A quick matrix computation for each x “ +¨ +˚ +˝ +x0 +x1 +˛ +‹‚: laxlims ApY, Xsq shows +Fib ˝ x “ +¨ +˚ +˝ +p0 ˝∆1 x +fib ˝∆1 x +˛ +‹‚“ +¨ +˚ +˝ +x0 +fibpFx0 Ñ x1q +˛ +‹‚: oplaxlims ApXs, Yq, +(8.7.13) +as required. Similarly, for each x_ “ px_ +0 ÐÝ x_ +1 q : laxlims:p∆1qop ApXs, Yq we have +x_ ˝ Fib_ “ pcofpx_ +1 F Ñ x_ +0 q ÝÑ x_ +1 q : oplaxlims ApXs, Yq, +(8.7.14) +as desired. +As an immediate application of Lemma 8.7.4 we therefore get that the oplax cone/cocones +Fib and Fib_ exhibit the lax bilimit Àlax +∆1 X also as an oplax limit and colimit. The following +lemma makes a more precise statement, showing that Fib and Fib_ are inverse up to a shift. +Lemma 8.7.15. The oplax cone Fib and cocone Fib_ from Construction 8.7.7 are, up to +negative shift r´1s, mutually inverse with respect to the oplax matrix multiplication. In par- +ticular, Fib and Fib_r1s (or Fibr1s and Fib_) exhibit the lax bilimit Àlax +∆1 X also as an oplax +bilimit of the diagram X: ∆1 Ñ A. +Proof. An explicit computation with the oplax matrix multiplication shows: +Fib ˝ Fib_ “ +¨ +˚ +˝ +p0fib_ +p0i1 +fibfib_ +fibi1 +˛ +‹‚» +¨ +˚ +˚ +˝ +idX0r´1s +0 +Fr´1s +idX1r´1s +˛ +‹‹‚“ J Xr´1s +(8.7.16) +106 + +and +Fib_ ˝∆1 Fib » limpfib_p0 Ñ i1Fp0 Ð i1fibq +(8.7.17) +» lim +¨ +˚ +˝ +¨ +˚ +˝ +r´1s +0 +0 +0 +˛ +‹‚Ñ +¨ +˚ +˝ +0 +0 +F +0 +˛ +‹‚Ð +¨ +˚ +˝ +0 +0 +0 +r´1s +˛ +‹‚ +˛ +‹‚ +(8.7.18) +» +¨ +˚ +˚ +˝ +r´1s +0 +Fr´1s +r´1s +˛ +‹‹‚» IXr´1s, +(8.7.19) +where in the second computation we omit the straightforward verification that the unnamed +arrows appearing in the last matrix are indeed those of IXr´1s. +Remark 8.7.20. While there is a distinguished choice for the cofiber-fiber equivalence (8.7.5), +the Lemma 8.7.15 provides two (but equally distinguished) ways to identify the lax bilimit +Àlax +∆1 X and the oplax bilimit Àoplax +∆1 +X, depending on whether we look at the represented map +(using Fib and treating them as (op)lax limits) or the corepresented map (using Fib_ and +treating them as (op)lax colimits). These two ways are not equivalent: they differ precisely +by a suspension. +We now have several different ways to represent maps X0 +lax +‘ X1 +αÝÑ Y0 +lax +‘ Y1, with the +passage between them implemented by applying the cofiber-fiber equivalence to rows and/or +columns of a matrix. +¨ +˚ +˝ +α00 +α01 +αv +10 +αv +11 +˛ +‹‚ +´˝Fib_ +ÝÝÝÝÝÑ +¨ +˚ +˚ +˝ +αh +00 +α01 +αhv +10 +αv +11 +˛ +‹‹‚ +Fib˝´ +ݧ§ +Fib˝´ +ݧ§ +¨ +˚ +˝ +α00 +α01 +α10 +α11 +˛ +‹‚ +´˝Fib_ +ÝÝÝÝÝÑ +¨ +˚ +˚ +˝ +αh +00 +α01 +αh +01 +α11 +˛ +‹‹‚ +(8.7.21) +Remark 8.7.22. Consider two composable maps +X +βÝÑ Y0 +lax +‘G Y1 +αÝÑ Z. +(8.7.23) +Each of the two maps β and α can be represented by a matrix in two ways, depending on +whether we treat the middle term as a lax or oplax bilimit. The following table shows the +four corresponding possible row-column-multiplications with the standard lax multiplication +in the lower left. General 2 ˆ 2 matrices describing maps between (op)lax bilimits over ∆1 +can then be multiplied in the usual row-by-column fashion. +107 + +˝ +pα0 ÐÝ α1q +` +αh +0 ÝÑ α1 +˘ +¨ +˚ +˚ +˝ +β0 +βv +1 +˛ +‹‹‚ cofpα1βv +1 Ñ α1Gβ0 Ñ α0β0q +lim +¨ +˚ +˝ +αh +0β0 +α1Gβ0 +α1βv +1 +˛ +‹‚r1s +¨ +˚ +˝ +β0 +β1 +˛ +‹‚ +colim +¨ +˚ +˝ +α0β0 +α1Gβ0 +α1β1 +˛ +‹‚ +cofpαh +0β0 Ñ α1Gβ0 Ñ α1β1q +(8.7.24) +Observe how the entry in top right differs from the standard oplax multiplication by a shift +r1s. The reason for this is that we used the canonical cofiber-fiber-equivalence (8.7.5) both +horizontally and vertically, which amounts to using the identification Fib: Àlax Y Ñ Àoplax Y +when discussing maps into the (op)lax bilimit but the identification Fib_ : Àoplax Y Ñ Àlax Y +when discussing maps from the (op)lax bilimit; we have seen in Lemma 8.7.15, that these two +identifications are only inverse up to shift. +The following table depicts the unit matrix with respect to each of the four multiplications; +they are just obtained from the standard lax unit matrix (lower left) by passing to horizontal +and/or vertical fibers. +lax +oplax +oplax +¨ +˚ +˝ +id +0 +0 +r´1s +˝ +˛ +‹‚ +¨ +˚ +˚ +˝ +r´1s +0 +r´1s +r´1s +˛ +‹‹‚ +lax +¨ +˚ +˝ +id +0 +id +id +˛ +‹‚ +¨ +˚ +˝ +r´1s +0 +0 +id +˝ +˛ +‹‚ +(8.7.25) +From the matrix multiplication formulas of (8.7.24) we can immediately see the advantage +of this change of coordinates. By working with lax-oplax or oplax-lax matrices, we obtain, on +K0 the formulas +p a0 a1 q +´ +b0 +b1 +¯ +“ ˘pa0b0 ´ a1b1q. +(8.7.26) +The that matrix multiplication now involves an alternating sum rather than an ordinary sum +is a feature, rather than a bug. In the next section we will see, for example, how we can +express the differential of the mapping cone of a chain map f : pA‚, αq Ñ pB‚, βq by directly +categorifying the canonical matrix δ “ +` α 0 +f β +˘ +without having to introduce any signs; the signs +are already part of the matrix multiplication. +Another convenient feature is that the identification between lax-oplax and oplax-lax +matrices is compatible with the passage to adjoints in the following sense: +Construction 8.7.27. Assume that G: Y0 Ñ Y1 has a right adjoint G % GR. Then Corol- +lary A.1.3, applied to the adjunctions pG˝q % pGR˝q and p˝GRq % p˝Gq yields equivalences +Ap´, Y0 +lax +‘G Y1q » Ap´, Y1 +oplax +‘GR Y0q +and +ApY1 +lax +‘GR Y0, ´q » ApY0 +oplax +‘G Y1, ´q +(8.7.28) +108 + +given explicitly by passing to vertical and horizontal transposes +¨ +˚ +˝ +Gy0 +y1 +u +˛ +‹‚Ø +¨ +˚ +˝ +GRy1 +y0 +u +˛ +‹‚ +and +´ +y_ +1 +vÐÝ y_ +0 GR¯ +Ø +´ +y_ +0 +vÝÑ y_ +1 G +¯ +, +(8.7.29) +where we have added the application of the gluing functor in the matrix to make the effect of +the transposition more apparent (usually we would just write something like py_ +0 Ñ y_ +1 q). +Lemma 8.7.30. For each X, Z : A, we have a commutative square +ApX, Y0 +lax +‘G Y1q ˆ ApY0 +oplax +‘G Y1, Zq +ApX, Zq +ApX, Y1 +oplax +‘GR Y0q ˆ ApY1 +lax +‘GR Y0, Zq +ApX, Zq +» +(8.7.31) +where the horizontal maps are the oplax-lax and lax-oplax matrix multiplication, respectively, +and the left vertical map is the equivalence of Construction 8.7.27. +Proof. For each +´ +y_ +0 +vÝÑ y_ +1 +¯ +: Y0 +oplax +‘G Y1 Ñ Z +and +¨ +˚ +˝ +y0 +y1 +u +˛ +‹‚: X Ñ Y0 +lax +‘G Y1, +(8.7.32) +the two different row-column products are the cofiber in ApX, Zq of the two composite maps +y_ +0 y0 +vy0 +ÝÝÑ y_ +1 Gy0 +y_ +1 u +ÝÝÑ y_ +1 y1 +and +y_ +0 y0 +y_ +0 u +ÝÝÑ y_ +0 GRy1 +vy1 +ÝÝÑ y_ +1 y1. +(8.7.33) +A straightforward computation using the triangle identities for G % GR shows that these two +maps are in canonically identified; hence so are their cofibers. +8.8 +Chain complexes and chain maps +Throughout this section, let A be a finitely lax additive p8, 2q-category. +Let Z “ pZ, ďq be the standard poset of integers. A chain complex in A is a functor +Zop Ñ A, depicted as +. . . +A2 +A1 +A0 +. . . , +α +α +α +α +(8.8.1) +with the conditions that each αk is a zero object in ApAn, An´1´kq for k ě 2. +There are various notions of chain maps, corresponding to different notions of natural +transformations of diagrams Zop Ñ A in the 2-categorical context (see also §A.2). +• A chain map (without further qualifier) is a commutative diagram of the form +. . . +A2 +A1 +A0 +. . . +. . . +B2 +B1 +B0 +. . . +α +α +f2 +α +f1 +α +f0 +β +β +» +β +» +β +(8.8.2) +109 + +Chain complexes and chain maps in A assemble into an p8, 2q-category ChpAq, defined +as a full sub-2-category of FUNpZop, Aq. +• A lax chain map is a diagram of the form, commuting only up to possibly noninvertible +2-cells. +. . . +A2 +A1 +A0 +. . . +. . . +B2 +B1 +B0 +. . . +α +α +f2 +α +f1 +α +f0 +β +β +β +β +(8.8.3) +Chain complexes and chain maps in A assemble into an p8, 2q-category ChlaxpAq, defined +as a full sub-2-category of FUNlaxpZop, Aq. +• Dually, we define the full sub-2-category ChoplaxpAq Ă FUNoplaxpZop, Aq of chain com- +plexes and oplax chain maps, which explicitly look as follows: +. . . +A2 +A1 +A0 +. . . +. . . +B2 +B1 +B0 +. . . +α +α +f2 +α +f1 +α +f0 +β +β +β +β +(8.8.4) +Given two chain complexes pA‚, αq and pB‚, βq, we write +Maplax +0 pA, Bq Ðâ MappA, Bq ãÑ Mapoplax +0 +pA, Bq +(8.8.5) +for the three corresponding 8-categories of lax chain maps, chain maps and oplax chain maps +A Ñ B. More generally, we write +Maplax +k pA‚, B‚q :“ Maplax +0 pA‚, Bk`‚q +and +Mapoplax +k +pA‚, B‚q :“ Mapoplax +0 +pA‚, Bk`‚q +(8.8.6) +for the (stable) 8-category of lax/oplax degree-k-maps from A to B. Abstractly, these various +8-categories are just the hom-categories in the p8, 2q-categories FUNpZop, Aq, FUNlaxpZop, Aq +and FUNoplaxpZop, Aq. For us, a more useful description/definition will be as certain sections +of certain tautological fibrations. +For the rest of this section, fix two chain complexes pA‚, αq and pB‚, βq and an integer +k P Z. +Construction 8.8.7. Consider the functor +Zop ˆ Z BˆAop +ÝÝÝÝÑ A ˆ Aop Aopp´,´q +ÝÝÝÝÝÝÑ St; +pm, nq ÞÑ ApAn, Bmq +(8.8.8) +and its two mixed Grothendieck constructions +q : +ż m:Zop +n:Z +ApAn, Bmq Ñ Z ˆ Z +and +q1 : +ż n:Z +m:Zop ApAn, Bmq Ñ Zop ˆ Zop, +(8.8.9) +(contravariant,covariant) and (covariant,contravariant), respectively. We can identify oplax +and lax chain maps A Ñ B with sections of q and q1 on the diagonal. More precisely, we +define +Mapoplax +k +pA, Bq » FunZˆZ +ˆ +Zpkq, +ż m:Zop +n:Z +ApAn, Bmq +˙ +(8.8.10) +110 + +and +Maplax +k pA, Bq » FunZopˆZop +ˆ +Zoppkq, +ż n:Z +m:Zop ApAn, Bmq +˙ +(8.8.11) +where Zpkq ãÑ Z ˆ Z is the k-shifted diagonal n ÞÑ pn ` k, nq. Concretely, such a section +consists of objects fn : ApAn, Bn`kq and morphisms +fn Ñ fn`1 +or +fn`1 Ñ fn +(8.8.12) +in the corresponding Grothendieck construction, amounting to morphisms +fnα Ñ βfn`1 +or +βfn`1 Ñ fnα, +(8.8.13) +in ApAn`1, Bn`kq respectively. +The full subcategories of Mapoplax +0 +pA, Bq and Maplax +0 pA, Bq spanned by those sections +where the corresponding maps (8.8.13) are equivalences are canonically equivalent to each +other by passing to inverses; we define this common full subcategory to be MappA, Bq. +Remark 8.8.14. We shall not unravel the definition of FUNlax and FUNoplax and show +that the mapping categories therein do indeed agree with the 8-categories constructed in +Construction 8.8.7. For the purpose of this paper, the reader may take this construction as +the definition. +It will be useful to study more general sections of the fibrations (8.8.9). +Construction 8.8.15. Denote by Zpěkq ãÑ Z ˆ Z the full shifted triangular subposet of +those pm, nq satisfying m ě k ` n. We write +Mapoplax +ěk +pA, Bq :“ FunZˆZ +ˆ +Zpěkq, +ż m:Zop +n:Z +ApAn, Bmq +˙ +. +(8.8.16) +for the 8-category of shifted upper triangular sections; see Figure 8.8.1 for a depiction of such +sections. For each k we have the obvious restriction functors +Mapoplax +k +pA, Bq +|k +ÐÝ Mapoplax +ěk +pA, Bq +|ěk`1 +ÝÝÝÑ Mapoplax +ěk`1pA, Bq. +(8.8.17) +The following lemma states that we can “crop” redundant zeroes in a section f : Mapoplax +ěk +pA, Bq. +Lemma 8.8.19. Denote by +U r +k :“ tpm, nq | k ď m ´ n ď k ` ru Ă Zpěkq +(8.8.20) +the k-shifted diagonal strip of width r. The canonical restriction functor along U r +k ãÑ Zpěkq +induces an equivalence +Mapoplax +ěk +pA, Bq|ěk`r“0 +» +ÝÑ FunZˆZ +ˆ +U r +k, +ż m:Zop +n:Z +ApAn, Bmq +˙ +|ěk`r“0 +, +(8.8.21) +where on both sides we are only considering those sections which are zero on the r-th off +diagonal and beyond. +111 + +¨ ¨ ¨ +A2 +A1 +A0 +A´1 +¨ ¨ ¨ +... +... +Bk`2 +fk +2 +fk`1 +1 +fk`2 +0 +fk`3 +´1 +Bk`1 +fk +1 +fk`1 +0 +fk`2 +´1 +Bk +fk +0 +fk`1 +´1 +Bk´1 +fk +´1 +... +... +α +α +α +α +α +β +β +β +β +β +(8.8.18) +Figure 8.8.1: +A section F “ pFmnqměk`n as in Construction 8.8.15 with with fr +n “ Fr`n,n. +The complexes A‚ and B‚ are drawn for reference to indicate how the section spreads across +the fibers of the fibration. +Proof. First of all, we claim that the restriction functor (8.8.21) admits a fully faithful left ad- +joint given by left q-Kan extension (q is the fibration (8.8.9)). The pointwise q-Kan extension +formula trivializes, since for each m ě n ` k ` r the overcategory U r +k{pm, nq has a terminal +object given by the vertical edge pn ` k ` r, nq Ñ pm, nq. We thus only have to argue that +there are sufficiently many cocartesian edges over these vertical edges pn ` k ` r, nq Ñ pm, nq. +Since we are, by definition, only considering sections whose value at pn ` k ` r, nq is zero, +this is automatic; the resulting Kan extended diagram is zero on Zpěk`rq. +The result +follows since by construction the essential image of this left q-Kan extension is precisely +Mapoplax +ěk +pA, Bq|ěk`r“0. +Remark 8.8.22. Even when the r-th off-diagonal is zero, we cannot crop the diagram any +further without losing information. In other words, the restriction functor +Mapoplax +ěk +pA, Bq|ěk`r“0 +» +ÝÑ FunZˆZ +ˆ +U r´1 +k +, +ż m:Zop +n:Z +ApAn, Bmq +˙ +, +(8.8.23) +112 + +is not typically an equivalence because the commutative squares +βfr´1 +n +0 +fr´2 +n +fr´1 +n´1α +(8.8.24) +at the edge of the strip U r +k carry more data than just the composable arrows +fr´1 +n´1α Ñ fr´2 +n +Ñ βfr´1 +n +, +(8.8.25) +namely a trivialization of their composite. +Remark 8.8.26. In the special case r “ 2, Lemma 8.8.19 says that a section f : Mapoplax +ěk +pA, Bq +satisfying f|ěk`2 “ 0 amounts to the following data: +• objects fn :“ fk +n : ApAn, Bk`nq, +• objects hn :“ fk`1 +n +: ApAn, Bk`n`1q, +• commutative squares +βhn +0 +fn +hn´1α +(8.8.27) +in ApAn, Bk`nq. +Lemma 8.8.28. Restriction to the discrete k-shifted diagonal (i.e., from the poset Zpkq ãÑ +Zpěkq to its underlying discrete set) induces the dashed equivalence +Mapoplax +ěk +pA, Bq +t|ěk`1 “ 0u +ś +nPZ ApAn, Bk`nq. +» +Y +(8.8.29) +when restricted to the kernel of the restriction functor. +Mapoplax +ěk +pA, Bq +|ěk`1 +ÝÝÝÑ Mapoplax +ěk`1pA, Bq. +(8.8.30) +Proof. By Lemma 8.8.19 (with r “ 1), we may restrict our sections to the strip U 1 +k Ă Zpěkq, +which, as a poset, is simply isomorphic to Z via pm, nq ÞÑ m ` n. Therefore a diagram of +shape U 1 +k just amounts to a sequence of objects and arrows. If such a diagram is zero on +odd-indexed objects 2n`1 fi pn`1, nq, then all arrows are uniquely determined and the only +relevant data are the values at the even-indexed objects 2n fi pn, nq. +Lemma 8.8.31. The restriction functor +Mapoplax +ěk +pA, Bq Ñ Mapoplax +ěk`1pA, Bq +(8.8.32) +113 + +admits fully faithful adjoint on both sides, given by left / right q-Kan extension. A section +(8.8.18) lies in the essential image if and only each leftmost horizontal / bottommost vertical +edge is cocartesian / cartesian, i.e. induces an equivalence +fk`1 +n´1α » +ÝÑ fk +n +/ +fk +n +» +ÝÑ βfk`1 +n +. +(8.8.33) +Proof. The pointwise left q-Kan extension formula at pn`k, nq along the inclusion Zpěk`1q ãÑ +Zpěkq trivializes, since the overcategory Zpěn`k`1q{pn ` k, nq has a terminal object pn ` +k, n ´ 1q. Therefore the desired left q-Kan extension exists if and only if each horizontal edge +pn ` k, nq Ð pn ` k, n ´ 1q admits a cocartesian lift. Since the fibration +q: +ż m:Zop +n:Z +ApAn, Bmq ÝÑ Z ˆ Z +(8.8.34) +is (by construction) cocartesian in the second variable, this is always the case. +The argument for the right adjoint is dual. +Corollary 8.8.35. The restriction functor |ěk`1 is part of a recollement +ś +nPZ ApAn, Bk`nq +Mapoplax +ěk +pA, Bq +Mapoplax +ěk`1pA, Bq +|ěk`1 +j1 +j +(8.8.36) +with gluing functor f‚ +‚ ÞÑ +´ +fibpfk`1 +n´1α Ñ βfk`1 +n +q +¯ +n. +Proof. The existence of the recollement follows from Lemma 8.8.31 with the kernel being +identified by Lemma 8.8.28. From the pointwise formulas for the relative Kan extensions we +see that for each f‚ +‚ : Mapoplax +ěk`1pA, Bq the transformation jpfq Ñ j1pfq is given on the main +diagonal by the structure map +jpfqk +n “ fk`1 +n´1α Ñ βfk`1 +n +“ j1pfqk +n +(8.8.37) +(for n P Z); passing to fibers yields the desired formula for the gluing functor. +Remark 8.8.38. Note that neither of the two adjoints in the left half of the recollement +(8.8.36) are the tautological restriction functor (8.8.29) to the discrete k-shifted diagonal. +Remark 8.8.39. We can think of Mapoplax +ěk +pA, Bq|ěk`1“0 as the 8-category of degree-k chain +maps f : A‚ Ñ Bk`‚ with trivialized structure map fα Ñ βf. Note that this is not a full +subcategory of Mapoplax +k +pA, Bq. In the restriction functor +t|ěk`1 “ 0u Ă Mapoplax +ěk +pA, Bq Ñ Mapoplax +k +pA, Bq +(8.8.40) +which forgets the trivialization is neither full nor faithful. +The restriction functor to the diagonal does not, in general, have analogous adjoints. This +does happen in the special case where the differentials of the chain complexes pA‚, αq and/or +pB‚, βq have left adjoints: +114 + +Lemma 8.8.41. Consider the restriction functor +|k : Mapoplax +ěk +pA, Bq ÝÑ Mapoplax +k +pA, Bq +(8.8.42) +(1) Assume that each differential β has a left adjoint. Then this restriction functor has a +fully faithful left adjoint given by relative left Kan extension. Explicitly it it is given by +fk`1 +n +:“ βLfk +n +and +fr +n “ 0 for r ě k ` 2 +(8.8.43) +with the non-trivial vertical arrows amounting to the units fk +n Ñ ββLfk +n of the adjunc- +tion. +(2) Assume that each differential α has a left adjoint. Then this restriction functor has a +fully faithful right adjoint given by relative right Kan extension. Explicitly it it is given +by +fk`1 +n +:“ fk +n`1αL +and +fr +n “ 0 for r ě k ` 2 +(8.8.44) +with the non-trivial horizontal arrows amounting to the counits fk +n`1αLα Ñ fk +n`1 of the +adjunction. +Proof. The two statments are dual; we focus on (2). +We observe that the relevant undercategories pm, nq{Zpkq (for pm, nq : Zpěkq) have an +initial object pm, m ´ kq. Therefore the desired pointwise right q-Kan extension exists if we +can guarantee that each horizontal edge pm, m ´ kq Ñ pm, nq has a cartesian lift. In general, +the fibration +q: +ż m:Zop +n:Z +ApAn, Bmq ÝÑ Z ˆ Z +(8.8.45) +is only cartesian in the first variable, not in the second. Being cartesian in the second variable +amounts to each Apα, Bmq having a right adjoint which is guaranteed because each α: An Ñ +An´1 has a left adjoint by assumption. The explicit formulas are an immediate consequence +of this pointwise construction using βLβL “ 0 to obtain the vanishing beyond the first off- +diagonal. +Lemma 8.8.46. There is an equivalence, canonical up to shift, between +• the full subcategory +t|k “ 0u X t|ěk`2 “ 0u Ă Mapoplax +ěk +pA, Bq +(8.8.47) +of those sections which are non-zero only on the first off-diagonal and +• the 8-category Mapoplax +k`1 pA, Bq of oplax degree-pk`1q chain maps. +Explicitly it sends a section pfr +nq to a chain map with components gn :“ fk`1 +n +r´ns : ApAn, Bk`n`1q. +Remark 8.8.48. Note that the equivalence of Lemma 8.8.46 is not induced by the obvious +restriction functor +Mapoplax +ěk +pA, Bq +|k`1 +ÝÝÑ Mapoplax +k`1 pA, Bq +(8.8.49) +which, when restricted to t|k “ 0u X t|ěk`2 “ 0u only hits oplax chain maps with trivial +structure map. +115 + +Proof. According to Remark 8.8.26, the data of a section f : t|k “ 0u X t|ěk`2 “ 0u amounts +to +• 1-morphisms hn :“ fk`1 +n +: ApAn, Bk`n`1q +• and commutative squares +βhn +0 +0 +hn´1α +(8.8.50) +in ApAn, Bk`nq which amount precisely to morphisms φn : hn´1r´n ` 1sα Ñ βhnr´ns. +Thus setting gn :“ hnr´ns, this is precisely the data of an oplax degree-pk`1q map g “ +pg‚, φ‚q : Mapoplax +k`1 pA, Bq. +Proposition 8.8.51. Assume that all differentials α and β have left adjoints. +Then the +restriction functor +|k : Mapoplax +ěk +pA, Bq|ěk`2“0 ÝÑ Mapoplax +k +pA, Bq +(8.8.52) +is part of a recollement +Mapoplax +k`1 pA, Bq +Mapoplax +ěk +pA, Bq|ěk`2“0 +Mapoplax +k +pA, Bq +i +|k +j1 +j +; +(8.8.53) +with gluing functor +ρ: f‚ ÞÑ +` +fibpβLfn Ñ fn`1αLqr´ns +˘ +n . +(8.8.54) +In particular, we have the dashed equivalence of (stable) 8-categories. +Mapoplax +ěk +pA, Bq|ěk`2“0 +Mapoplax +k +pA, Bq +ш +ρ Mapoplax +k`1 pA, Bq +Mapoplax +k +pA, Bq +» +|k +p0 +(8.8.55) +Proof. Using Lemma 8.8.46 to identify the kernel of the restriction functor (8.8.52), the exis- +tence of the recollement and the induced equivalence (8.8.55) follow from the general theory +of recollements, see for instance [Lur17, A.8]. +From the explicit construction in Lemma 8.8.46 it follows that the canonical transformation +j Ñ j1 between the two adjoints is given explicitly at f : Mapoplax +k +pA, Bq by the canonical +mate +jpfqk`1 +n +“ βLfn Ñ fn`1αL “ j1pfqk`1 +n +(8.8.56) +on the first off-diagonal; it is an equivalence (fn +“ +ÝÑ fn or 0 “ +ÝÑ 0) everywhere else. The gluing +functor +Mapoplax +k +pA, Bq Ñ Kerp|kq +(8.8.57) +is given by the fiber of this transformation, therefore yields the desired formula (8.8.54) under +the identification of Lemma 8.8.46. +116 + +Definition 8.8.58. We denote by +Mapoplax +exěkpA, Bq Ă Mapoplax +ěk +pA, Bq +(8.8.59) +the full subcategory spanned by those sections pfr +nq such that all the induced squares +βfr`1 +n +βfr`2 +n´1α +fr +n +fr`1 +n´1α +˝ +(8.8.60) +in ApAn, Br`nq are bicartesian (for all r ě k) and call such sections exact. +Lemma 8.8.61. There is an equivalence of 8-categories between +• the full subcategory +Mapoplax +exěkpA, Bq|ěk`2“0 Ă Mapoplax +ěk +pA, Bq +(8.8.62) +of those sections which are exact and vanish beyong the first off-diagonal and +• the 8-category Maplax +k`1pA, Bq, of lax degree-pk`1q chain maps. +Explicitly it sends a section pfr +nq to a chain map with components gn :“ fk`1 +n +r´ns. +Proof. In Remark 8.8.26, if we restrict to squares (8.8.27) which are bicartesian, the data just +amounts (by rotating the exact triangle forward and shifting by r´ns) to objects hn “ fk`1 +n +: +ApAn, Bk`n`1q and maps βhnr´ns Ñ hn´1r´n ` 1sα in ApAn, Bk`nq. This is precisely the +data of a lax degree-k`1 chain map g with components gn :“ hnr´ns, as desired. +Remark 8.8.63. Lemma 8.8.46 and Lemma 8.8.61 explain how the 8-category Mapoplax +ěk +pA, Bq|ěk`2“0 +contains both the oplax and the lax degree-pk`1q maps A Ñ B. From the explicit construc- +tions it is immediate that these two inclusions are compatible, in the sense that there is a +commutative square +MappA‚, Bk`‚`1q +Mapoplax +k`1 pA, Bq +Maplax +k`1pA, Bq +Mapoplax +ěk +pA, Bq|ěk`2“0 +(8.8.64) +and we have +MappA‚, Bk`‚`1q “ Mapoplax +k`1 pA, Bq X Maplax +k`1pA, Bq +(8.8.65) +as full subcategories of Mapoplax +ěk +pA, Bq|ěk`2“0. +Construction 8.8.66. Let pfr +nq “ pf, hq : Mapoplax +ěk +pA, Bq be a section as in Remark 8.8.26. +If each square (8.8.27) is bicartesian, then both of the maps +fnα Ñ βhnα +and +βhnα Ñ βfn`1 +(8.8.67) +117 + +are equivalences, since their fibers/cofibers are +hn´1αα “ 0 +and +ββhn`1 “ 0, +(8.8.68) +respectively. Therefore the oplax degree-k chain map f “ fk is an actual chain map A‚ Ñ +Bk`‚. Therefore the canonical restriction functor +|k : Mapoplax +ěk +pA, Bq Ñ Mapoplax +k +pA, Bq +(8.8.69) +restricts to a functor +δ: Maplax +k`1pA, Bq » Mapoplax +exěkpA, Bq|ěk`2“0 +|k +ÝÑ MappA‚, Bk`‚q +(8.8.70) +whose kernel is precisely MappA‚, Bk`‚`1q. These differentials δ assemble to what we call the +lax mapping complex Maplax +‚ pA, Bq: +¨ ¨ ¨ +Maplax +2 pA, Bq +Maplax +1 pA, Bq +Maplax +0 pA, Bq +¨ ¨ ¨ +MappA‚, B1`‚q +MappA‚, B‚q +δ +δ +δ +δ +(8.8.71) +Unraveling, we get the explicit formula for the differential +δpg‚qn “ fibpβgn Ñ gn´1αqrns. +(8.8.72) +Remark 8.8.73. Assume that the differentials α and β have right adjoints αR and βR, +respectively. Denote by AR +‚ :“ pA´‚, αRq and BR +‚ :“ pB´‚, βRq the chain complex obtained +from A and B by passing to right adjoints of the differentials. Note that for each n P N there +is a tautological equivalence of 8-categories +Mapoplax +k +pAR, BRq +Maplax +´kpA, Bq +pf‚, fαR Ñ βRfq, +pf´‚, βf Ñ fαq +» +(8.8.74) +by noting that both sides are sections +şm:Zop +n:Z +ApAR +n, BR +mq +şn:Z +m:Zop ApAn, Bmq +Zpkq +Z ˆ Z +Zop ˆ Zop +q +q1 +– +p´1q¨ +(8.8.75) +of the same fibration. An explicit computation shows that under this equivalence, the differ- +ential +δ: Maplax +k`1pA, Bq Ñ Maplax +k pA, Bq +(8.8.76) +of the lax mapping complex (8.8.71) is identified with the gluing functor +ρ: Mapoplax +´k´1pAR, BRq Ñ Maplax +´kpAR, BRq +(8.8.77) +of Proposition 8.8.51 applied to the chain complexes AR and BR. +118 + +Once we have constructed the mapping complex, we immediately get the corresponding +notion of categorified chain homotopy. +Definition 8.8.78. Let f : A Ñ B be a chain map. A lax null-homotopy of f is a lax degree-1 +map h: Maplax +1 pA, Bq with δphq “ f. +Remark 8.8.79. Clearly, one could dualize Construction 8.8.66 and all the preceding lemmas +to obtain the oplax mapping complex and the resulting notion of an oplax null-homotopy This +is the version which we have already seen in §4.6, where this oplax mapping complex was +constructed (in the special case C “ St) via the product totalization of the canonical double +complex CpA´‚, B‚q. We shall not give a detailled proof that these two different constructions +agree; this is relatively straightforward by inspection of the terms of the complex and the +explicit formulas (8.8.72) and (4.6.2) for the differential. +In this section there appear both lax and oplax chain maps, so we are a bit more careful +to always carry the corresponding adjective with us. If the context is clear, we might drop +the adjective and just write null-homotopy or categorical null-homotopy as we did in §4.6. +Finally, we remark that the adjointability conditions for commutative squares (see Defi- +nition 4.5.1) immediately give rise to corresponding notions for chain maps (expanding Def- +inition 4.3.10). For chain maps, we distinguish to types of adjointability conditions: in the +direction of the differentials and in the direction of the chain map itself. +Definition 8.8.80. Let f : pA‚, αq Ñ pB‚, βq be a chain map. +• We say that f is left diff-adjointable / right diff-adjointable if each square in the cor- +responding diagram (8.8.2) is horizontally left/right adjointable, i.e. all differentials α +and β admit left/right adjoints and the canonical mate βLf Ñ fαL / fαR Ñ βRf is an +equivalence. +• We say that f is left/right adjointable if each square in the corresponding diagram (8.8.2) +is vertically left/right adjointable, i.e., each component fn has a left/right adjoint and +the canonical mate fLβ Ñ αfL / αfR Ñ fRβ is an equivalence. +8.9 +The oplax mapping cone construction +Let A be a finitely lax additive p8, 2q-category. The goal of this section is to construct the +oplax mapping cone ConeÐpfq of a chain map (8.8.2) in A by categorifying the usual formula +Conepfqn`1 :“ An ‘ Bn`1 +´ ´α 0 +´f β +¯ +ÝÝÝÝÝÑ An´1 ‘ Bn “: Conepfqn +(8.9.1) +for the differential. According to the philosophy outlined in §1.1, we need additional data to +specify the mapping cone complex: +• To construct the terms of the mapping cone complex ConeÐpfq :“ An´1 +lax +‘ Bn as a lax +bilimit, we need to specify 1-morphisms h: An´1 Ñ Bn or k: Bn Ñ An´1 +• We need some suitable 2-categorical data to be able to write down the ∆1 ˆ ∆1-analog +of the differential matrix (8.9.1). +119 + +We will also see that in the presence of sufficient compatible adjoints to the differentials +α, β and/or f, one can canonically construct such data using the various units/counits and +in this case we recover the fiber and cofiber of f as in §4.3. +Definition 8.9.2. We denote by +MaplhpA, Bq :“ MappA, Bq ˆMapoplax +0 +pA,Bq Mapoplax +ě0 +pA, Bq|ě2“0 +(8.9.3) +the 8-category of those sections (8.8.18) which are zero beyond the first off-diagonal and +restrict to an honest chain map (as opposed to an oplax one) on the diagonal. Such sections +are called lh-enhanced morphisms of chain complexes and are written F : pA‚, α‚q +lh +ùñ pB‚, β‚q. +The mnemonic “lh” stands for “left-horizontal” and is explained by Lemma 8.9.17, where +we construct canonical lh-enhancements in the presence of left adjoints in the horizontal +(=differential) direction. +Remark 8.9.4. Remark 8.8.26 tells us that an lh-enhanced morphism F : pA‚, α‚q +lh +ùñ pB‚, β‚q +consists of 1-morphisms +fn : An Ñ Bn +and +hn : An Ñ Bn`1 +(8.9.5) +together with (not necessarily bicartesian) commutative squares +hn´1αn +0 +fn +βn`1hn +ϵn +ηn +(8.9.6) +in ApAn, Bnq such that each composite +fn´1αn +ηn´1αn +ÝÝÝÝÝÑ βnhn´1αn +βnϵn +ÝÝÝÑ βnfn +(8.9.7) +is an equivalence (i.e., exhibits f : A Ñ B as a chain map). We say that F “ pF, h, ϵ, ηq is an +lh-enhancement of the underlying chain map f : A Ñ B. +We can also depict such an lh-enhanced morphism of chain complexes as follows +. . . +A2 +A1 +A0 +. . . +. . . +B2 +B1 +B0 +. . . +α +α +ð +ð +f +α +ð +ð +f +h +α +f +h +β +β +β +β +(8.9.8) +but note that this picture is not complete, since it does not depict the trivialization η ˝ ϵ » 0 +encoded in the square (8.9.6). +The forgetful functor +MaplhpA, Bq Ñ MappA, Bq +(8.9.9) +sends an lh-enhanced morphism F “ pf, h, ϵ, ηq to its underlying chain map by forgetting h, ϵ +and η and only remembering the maps f and the equivalences fα » βf; in the picture (8.9.8) +this just amounts to pasting the triangular 2-cells to form (commutative) squares. For each +chain map f : A‚ Ñ B‚, we write Maplh +f pA‚, B‚q for the fiber of this forgetful functor over the +object f : MappA‚, B‚q; it is the (typically not stable) 8-category of lh-enhancements of the +chain map f. +120 + +Remark 8.9.10. An lh-enhanced morphism is called exact if each square (8.9.6) is bicartesian. +We denote by +Maplh´expA, Bq :“ Mapoplax +exě0pA, Bq|ě2“0 Ă MaplhpA, Bq +(8.9.11) +the full subcategory of exact lh-enhanced morphisms. +Remark 8.9.12. Note that under the identification of Lemma 8.8.61, an exact lh-enhancement +of a chain map is precisely a lax null-homotopy in the sense of Definition 8.8.78. +The following construction of the oplax mapping cone is a tautological reformulation of +what the data of an lh-enhanced morphism entails. +Construction 8.9.13. Let F “ pf, h, ϵ, ηq: pA‚, αq +lh +ùñ pB‚, βq be an lh-enhanced morphism +of chain complexes. We define the oplax mapping cone of F to be the chain complex +ConeÐpFq : ¨ ¨ ¨ Ñ An +lax +‘h Bn`1 +δn`1 +ÝÝÝÑ An´1 +lax +‘h Bn +δn +ÝÑ An´2 +lax +‘h Bn´1 Ñ . . . +(8.9.14) +where the differential is the lax-oplax matrix +δn`1 :“ +¨ +˚ +˝ +αn +0 +fn +βn`1 +˛ +‹‚: An +oplax +‘h Bn`1 Ñ An´1 +lax +‘h Bn +(8.9.15) +induced by the commutative square (8.9.6). Using the matrix multiplication formula from +Remark 8.7.22 we compute the squared differential +δ ˝ δ » +¨ +˚ +˚ +˝ +cofpαα Ñ 0q +cofp0 Ñ 0q +cofpfα Ñ βhα Ñ βfq +cofp0 Ñ ββq +˛ +‹‹‚ +(8.9.16) +It is zero because α2 » 0, β2 » 0 and the that the composite map (8.9.7) is an equivalence. +Having constructed the mapping cone ConeÐpFq with respect to the choice of the auxiliary +lh-enhancement of the underlying chain map f, it is natural to ask whether there are universal +ways to produce such lh-enhancements. These exists as long as the differentials α and/or β +admit left adjoints: +Lemma 8.9.17. Let pA‚, αq and pB‚, βq be two chain complexes in A. Consider the forgetful +functor +p: MaplhpA‚, B‚q Ñ MappA‚, B‚q +(8.9.18) +(1) If each differential β has a left adjoint, then p admits a fully faithful left adjoint p´qβ. +(2) If each differential α has a left adjoint, then p admits a fully faithful right adjoint p´qα. +(3) Assume that both differentials α and β admit left adjoints. The canonical transformation +p´qβ Ñ p´qα is an equivalence precisely on those chain maps f : MappA‚, B‚q which +are left diff-adjointable. +121 + +Proof. The first two statements are a direct consequence of Lemma 8.8.41 (for k “ 0) by +observing that both adjoints (if they exists) take values in +MaplhpA, Bq Ă Mapoplax +ě0 +pA, Bq +(8.9.19) +when restricted to +MappA, Bq Ă Mapoplax +0 +pA, Bq. +(8.9.20) +To prove (3) fix a chain map f : MappA‚, B‚q and consider the component fβ Ñ fα. The +only place where it can possibly not be an equivalence is on the first off-diagonal. Unraveling +the pointwise formula, one observes that the value at these off-diagonal places is given by +the mates βLf Ñ fαL of the equivalences fα Ñ βf; by definition f is left diff-adjointable +precisely if these mates are all equivalences. +Remark 8.9.21. Assume that all differentials α and β admit left adjoints. Then the recolle- +ment (8.8.53) (for k “ 0) restricts to a recollement +Mapoplax +1 +pA, Bq +MaplhpA, Bq +MappA, Bq +p +(8.9.22) +and therefore to an equivalence +MaplhpA, Bq +MappA, Bq +ш +ρ Mapoplax +1 +pA, Bq +MappA, Bq +MappA, Bq +» +p0 +(8.9.23) +Pointwise over each f : MappA, Bq we thus have an equivalence +Maplh +f pA, Bq » ρpfq{ Mapoplax +1 +pA, Bq. +(8.9.24) +A glance at the explicit formula (8.8.54) shows that the gluing funtor ρ is zero precisely on +those chain maps which are left diff-adjointable; in this case the 8-category of lh-enhancements +is simply equivalent to Mapoplax +1 +pA, Bq, hence in particular stable. +The following corollary summarizes the situation over each chain map f: When the differ- +entials of the chain complexes admit adjoints, each chain map f can be canonically enhanced in +two ways yielding an initial or terminal object in the category Maplh +f pA, Bq of lh-enhancements +of f. If the chain map is left diff-adjointable, this 8-category is stable and these two canonical +lh-enhancements agree. +Corollary 8.9.25. Let f : pA‚, αq Ñ pB‚, βq be a chain map. +(1) If each differential β admits a left adjoint βL then f admits an initial lh-enhancement +fβ. +(2) Dually, if each differential α admits a left adjoint αL, then f admits a terminal lh- +enhancement fα. +122 + +(3) If the chain map f is left diff-adjointable then the two lh-enhancements fβ and fα +coincide. In this case we denote this lh-enhancement by flh. +Proof. Follows from the adjunctions of Lemma 8.9.17 viewed pointwise over f : MappA‚, B‚q. +We now identify the mapping cones constructed from the initial and terminal lh-enhancement +with those constructed in Construction 4.3.3 using the directed pushout and directed pullback. +Proposition 8.9.26. Let f : pA‚, αq Ñ pB‚, βq be a chain map. +(1) Assume that each α admits a left adjoint and let fα be its terminal lh-enhancement of +Corollary 8.9.25. Consider the oplax square +Ai +Ai´1 +Bi +Ai´1 +oplax +‘hα Bi +fi +α +ð +hα +ð +(8.9.27) +obtained by pasting ϵ with the oplax colimit cone. This square is a directed pushout, +thus yields an identification Ai´1 +ñ> +Ai Bi +» +ÝÑ Ai´1 +oplax +‘h Bi. Under this identification the +differential of ConeÐpfαq corepresents the map +pa_ +i´1α Ñ b_ +i fq ÞÑ pcofpa_ +i´1α Ñ b_ +i fqα » +ÝÑ b_ +i βfq. +(8.9.28) +(2) Dually, if each β admits a left adjoint, then the terms of the cone ConeÐpfαq are +canonically identified with Ai´1 +ñˆ +Bi´1 +Bi and the differential represents the map +pfai Ñ βbi`1q ÞÑ pfαai +» +ÝÑ β fibpfai Ñ βbi`1qqr1s +(8.9.29) +Proof. Let fα “ pf, hα, ϵα, ηαq be the terminal lh-enhancement of f. We have to show that +for each test object C : A, the functor +oplaxlim∆1pApAi´1, Cq +˝hα +ÐÝÝ ApBi, Cqq “ ApAi´1 +oplax +‘hα Bi, Cq Ñ ApAi´1, Cq +ñˆ +ApAi,Cq ApBi, Cq +(8.9.30) +is an equivalence of (stable) 8-categories. Explicitly, this functor sends a section a1 +i´1 +uÝÑ b_ +i hα +to the composite +a1 +i´1α uα +ÝÑ b_ +i hαα “ b_ +i fiαLα +b_ +i ficuα +ÝÝÝÝÝÑ b_ +i fi, +(8.9.31) +which is precisely its transpose under the adjunction p˝αq % p˝αLq. Thus the functor (8.9.30) +is an equivalence by Lemma A.1.6(1) applied to +ApAi´1, Cq ˝α +ÝÑ ApAi, Cq +˝fi +ÐÝÝ ApBi, Cq. +(8.9.32) +To compute the map corepresented by the differential, we compute for each +pa_ +i´1 +uÝÑ b_ +i q : ApAi´1 +oplax +‘hα Bi, Cq +(8.9.33) +123 + +the matrix product +pa_ +i´1 +uÝÑ b_ +i q ˝ +¨ +˚ +˝ +α +0 +fi +β +˛ +‹‚“ pcofpa_ +i´1α Ñ b_ +i fiq Ñ cofp0 Ñ b_ +i βqq, +(8.9.34) +where in the first entry we are taking the cofiber of the map u: a_ +i´1α uα +ÝÑ b_ +i hαα b_ϵα +ÝÝÝÑ b_ +i fi. +Note that in the matrix representation (8.9.34) we are omitting the application of the gluing +functor as is customary. If we put this implicit application back in, we obtain the map +cofpa_ +i´1α Ñ b_ +i fiq Ñ b_ +i βhα “ b_ +i βfi`1α! : ApAi, Cq +(8.9.35) +which yields the desired map +cofpa_ +i´1α Ñ b_ +i fiqα Ñ b_ +i βfi`1 : ApAi`1, Cq +(8.9.36) +after transposing; in , it is just the equivalence b_ +i fiα » b_ +i βfi`1 because a_ +i´1αα “ 0. +The proof of the dual statement is analogous: We apply ApC, ´q to reduce to the case +of 8-categories, where we apply Lemma A.1.6(2). Then we only have to perform the dual +matrix computation +¨ +˚ +˝ +α +0 +fi +β +˛ +‹‚˝ +¨ +˚ +˝ +ai +bi`1 +˛ +‹‚“ +¨ +˚ +˚ +˝ +cofpαai Ñ 0q +cofpfiai Ñ βbi`1q +˛ +‹‹‚“ +¨ +˚ +˝ +αai +fibpfiai Ñ βbi`1q +˛ +‹‚r1s +(8.9.37) +to obtain the desired formula. +Corollary 8.9.38. Let f : pA‚, αq Ñ pB‚, βq be a left diff-adjointable chain map. Both the +chain complexes Cofpfq and Fibpfq from Construction 4.3.3 are well defined and equivalent +up to a shift in degree, i.e. Cofpfq » Fibpfqr1s. +Remark 8.9.39. Recall from Definition 4.3.6 that the external shift of a chain complex +pA‚, αq is the chain complex +Ar1s‚ :“ pA‚´1, αr1sq, +(8.9.40) +where the terms are reindexed and the differentials are shifted internally in the stable 8- +category ApAi, Ai´1q. +Proof. The chain complex Fibpfqr1s from Construction 4.3.3 is precisely defined via the uni- +versal properties of the directed pullback to represent the map (8.9.29); note that it differs +from the formula (4.3.5) due to an internal shift of the differential introduced by the external +shifting convention. Dually, the differentials of the chain complex Cofpfq are defined via the +universal property of the directed pushout to corepresent the maps (8.9.28). Proposition 8.9.26 +shows that these chain complexes exist and can be concretely constructed as ConeÐpfαq and +ConeÐpfβq, respectively. Since we assume that the chain map f is left diff-adjointable, Corol- +lary 8.9.25 states that fα and fβ are canonically equivalent as lh-enhancements of the chain +map f. Therefore the chain complexes Cofpfq “ ConeÐpfαq and Fibpfqr1s “ ConeÐpfβq are +also equivalent. +124 + +So far we have used that one can express the directed pullback Ai´1 +ñˆ +Bi´1 +Bi and the +directed pushout Ai´1 +ñ> +Ai Bi as a lax limit/colimit of a composite involving horizontal left +adjoints. To prove Proposition 4.3.12 we need an analogous discussion using vertical right +adjoints. This change corresponts to changing the direction of the gluing map between Bn +and An´1. We start by defining the corresponding notion of enhancement. +Definition 8.9.41. An rv-enhanced morphism F : pA‚, α‚q +rv +ùñ pB‚, β‚q of chain complexes +consists of 1-morphisms +fn : An Ñ Bn +and +kn : Bn Ñ An´1 +(8.9.42) +toghether with an oplax-lax matrix of the form +δ “ +¨ +˚ +˝ +β +f +0 +α +˛ +‹‚: Bn`1 +lax +‘k An Ñ Bn +oplax +‘k An´1 +(8.9.43) +such that the composite map fα Ñ fkf Ñ βf is an equivalence (yielding the underlying +chain map f of F). The resulting chain complex pConeÐpFq‚ :“ B‚ +lax +‘k A‚´1, δq is called the +oplax mapping cone of F. +The mnemonic “rv” stands for “right-vertical” and reflects the fact that there are canonical +rv-enhancements in the presence of right adjoints in the vertical (=chain map) direction. +We shall now explain how such rv-enhanced morphisms assemble into an 8-category. For +simplicity we will restrict to those, where each fn admits a right adjoint gn :“ fnR. +Construction 8.9.44. Define +Maplax +ďkpB, Aq :“ FunZopˆZop +ˆ +Zpďkqop, +ż n:Z +m:Zop ApBn, Amq +˙ +(8.9.45) +to consist of sections defined on +Zpďkqop :“ tpm, nq | m ď n ` ku Ă Zop ˆ Zop. +(8.9.46) +125 + +Pictorially, such sections look as follows: +¨ ¨ ¨ +B2 +B1 +B0 +¨ ¨ ¨ +... +... +Ak`2 +gk +2 +Ak`1 +gk´1 +2 +gk +1 +Ak +gk´2 +2 +gk´1 +1 +gk +0 +... +... +β +β +β +β +α +α +α +α +(8.9.47) +Denote by +Maprv´LpA, Bq Ă Maplax +ď0pB, Aq +(8.9.48) +the full subcategory of those sections pgr +nq satisfying: +• The lax chain map g‚ “ g0 +‚ : B‚ Ñ A‚ on the main diagonal is left adjointable, i.e. +each gn : Bn Ñ An has a left adjoint gnL and the canonical mate gn´1Lα Ñ βgnL is an +equivalence. +• The section is zero beyond the first off-diagonal, i.e. gr +‚ “ 0 for r ď ´2. +Using the dual of Remark 8.8.26 and by passing from an adjointable lax chain map g‚ “ +g0 +‚ : B‚ Ñ A‚ to its adjoint f‚ :“ g‚L : A‚ Ñ B‚ (which is an honest chain map), it is not hard +to see that the data of such a section amounts precisely to that of an rv-enhanced morphism +whose underlying chain map f admits pointwise adjoints; the 1-morphisms kn : Bn Ñ An´1 +are the term g´1 +n +on the first off-diagonal and the matrices (8.9.43) amount precisely to the +squares +kn`1 +gn +0 +kn +. +(8.9.49) +Therefore we can view Maprv´LpA, Bq as the 8-category of those rv-enhanced morphisms, +whose underlying chain map f admits pointwise adjoints. We have the canonical forgetful +126 + +functor +Maplax +ď0pB, Aq|ď´2“0 +Maplax +0 pB, Aq +Maprv´LpA, Bq +tleft adjointable g‚u +tpointwise right adjointable f‚u +MappA, Bq +|0 +|0 +» +(8.9.50) +sending such an rv-enhanced morphism to its underlying chain map. +For each pointwise +adjointable chain map f : A Ñ B we write Maprv +f pA, Bq “ Maprv´L +f +pA, Bq for the fiber of this +dashed functor; it is the 8-category of rv-enhancements of f. +Lemma 8.9.51. The forgetful functor (8.9.50) is part of a recollement +Maplax +´1pB, Aq +Maprv´LpA, Bq +tpoinwise right adjointable f‚ : A Ñ Bu +j1 +j +, +(8.9.52) +whose gluing functor ρ computes the fiber of the canonical mate, i.e., +ρpfq “ +` +fibpαfnR Ñ fn´1Rβqrns +˘ +n . +(8.9.53) +Proof. Similarly to Lemma 8.8.31 and Corollary 8.8.35 we compute that the relative left and +right Kan extension along the diagonal Zop ãÑ Zpď0qop always exist, yielding fully faithful +left and right adjoints j and j1 to the restriction functors |0. Explicitly we have +jpgq´1 +n +“ αgn Ñ gn´1β “ j1pgq´1 +n +(8.9.54) +(the structure map of g‚) on the first off-diagonal and zero beyond it. Moreover, the kernel +of the forgetful functor is +t|0 “ 0u X t|ď´2 “ 0u Ă Maplax +ď0pB, Aq, +(8.9.55) +which, similarly to Lemma 8.8.46 we can identify with Maplax +´1pB, Aq via the assignment +g‚ +‚ ÞÑ pg´1 +n rnsqn. +(8.9.56) +The desired result follows by passing to adjoints, i.e., g‚ :“ f‚R. +Remark 8.9.57. The gluing functor for the recollement +Maplax +´1pB, Aq +Maplax +ď0pB, Aq|ď´2“0 +Maplax +0 pB, Aq +|0 +(8.9.58) +(before restricting the cokernel to the subcategory of left adjointable maps g: B Ñ A) is +nothing but the differential of the lax mapping complex Maplax +‚ pB, Aq. +127 + +As a direct consequence we get the following result, which provides the two canonical +rv-enhancements of a pointwise right adjointable chain map. +Corollary 8.9.59. Let f : pA‚, αq Ñ pB‚, βq be a pointwise right adjointable chain map. The +8-category Maprv +f pA, Bq has +(1) an initial object fα :“ jpfq with kα “ αfR and where the vertical map α Ñ kf “ αfRf +in the matrix (8.9.43) is the unit; +(2) a terminal object fβ :“ j1pfq with kβ “ fRβ and where the horizontal map β Ð fk “ +ffRβ in the matrix (8.9.43) is the counit. +(3) These two rv-enhancements coincide if and only if the chain map f is right adjointable. +In this case we denote this rv-enhancement by frv. +Analogously to Proposition 8.9.26, we can exhibit the terms of the corresponding oplax +mapping cones ConeÐpfαq and ConeÐpfβq as a directed pushout or directed pullback, re- +spectively. +Proposition 8.9.60. Let f : pA‚, αq Ñ pB‚, βq be a chain map and assume that each fi +admits a right adjoint. +(1) The oplax square +Ai +Ai´1 +Bi +Bi +lax +‘kα Ai´1 +f +α +ð +kα +ð +(8.9.61) +yields an identification Ai´1 +ñ> +Ai Bi +» +ÝÑ Bi +lax +‘kα Ai´1. Under this identification, the differ- +ential of ConeÐpfαq again corepresents the map (8.9.28). +(2) Dually, the terms of the cone ConeÐpfβq are canonically identified with Ai´1 +ñˆ +Bi´1 +Bi +and the differential again represents the map (8.9.29). +Proof. Similar to Proposition 8.9.26; omitted. +Corollary 8.9.62. The chain complexes ConeÐpfαq and ConeÐpfβq also yield a construction +for Cofpfq and Fibpfqr1s, respectively. In particular, Cofpfq and Fibpfqr1s agree when the +chain map f is right adjointable. +Corollary 8.9.63. When f is both right adjointable and left diff-adjointable, the two canonical +oplax mapping cones ConeÐpflhq and ConeÐpfrvq agree. +Remark 8.9.64. Throughout this section there was a bias in our discussion, since we im- +plicitly treated chain maps as being oplax, i.e. having directed squares of the form +Ai`1 +Ai +Bi`1 +Bi +α +fi`1 +fi +β +(8.9.65) +128 + +This was already apparent in the chosen direction for directed pushouts and directed pullbacks +in §4.1 and accounts for the two possible choices we had when it came to adjointability +conditions: having vertical right adjoints or horizontal left adjoints. We could rewrite this +whole section with the opposite conventions and obtain the lax mapping cone ConeÑpFq +associated to a chain map f with suitable enhancements. In the case where f is left adjointable +or right diff-adjointable we could again construct a canonical lax mapping cone ConeÑpFq +whose terms are identified both with Bi +ñ> +Ai Ai´1 and with Bi +ñˆ +Bi´1 +Ai´1. +8.10 +Universal property of the lax mapping cone +The main reason for introducing the mapping cone of a chain map f : pA‚, αq Ñ pB‚, βq +between chain complexes in an additive category A is that it yields an explicit model for the +cofiber of f in the in the stable 8-category KpAq of chain complexes up to chain homotopy. +In other words, it satisfies +MapKpAqpConepfq, Cq » fib pMappB, Cq Ñ MappA, Cqq +“ tpg: B Ñ C, h: gf » 0qu +(8.10.1) +naturally in C : KpAq. +Already before passing to the stable 8-category KpAq, one can see a naive version of this +universal property characterizing the mapping cone up to isomorphism in ChpAq via +ChpAqpConepfq, Cq – tpg, hq | g: B Ñ C, h: gf » 0u +(8.10.2) +naturally in C : ChpAq. In other words: maps out of Conepfq are chain maps g: B Ñ C +together with a null-homotopy of gf. +Ultimately, we are of course interested in understanding the categorified analog of the +homotopically meaningful universal property. However, this is currently out of reach since +we don’t even know what the correct analog of the stable 8-category KpAq should be and +in what sense we are supposed to view the mapping cone as a cofiber. Therefore, we now +instead describe the categorified analog of (8.10.2) in the hopes that it might lead to a better +understanding of the theory of categorified chain complexes up to homotopy. +Theorem 8.10.3. Let F : A +lh +ùñ B be an lh-enhanced morphism of chain complexes with +underlying chain map f. +(1) For each chain complex C : ChpAq there is a natural equivalence of (stable) 8-categories +between +• chain maps ConeÐpFq Ñ C and +• chain maps g: B Ñ C together with an exact lh-enhancement E of gf and a mor- +phism E Ñ gF of lh-enhancements of gf. +(2) For each chain complex C : ChpAq there is a natural equivalence of (stable) 8-categories +between +• chain maps C Ñ ConeÐpFqr´1s and +129 + +• chain maps g: C Ñ A together with an exact lh-enhancement E of fg and a mor- +phism Fg Ñ E of lh-enhancements of fg. +Before proving Theorem 8.10.3, we isolate the special case where F is the initial or terminal +lh-enhancement of f. +Corollary 8.10.4. Let f : pA‚, αq Ñ pB‚, βq be a chain map. +(1) Assume that all differentials α have left adjoints. Then for each chain complex C : ChpAq +there is an equivalence of (stable) 8-categories between +• chain maps Cofpfq Ñ C and +• chain maps g: B Ñ C together with a lax null-homotopy E of gf. +(2) Assume that all differentials β have left adjoints. Then for each chain complex C : ChpAq +there is an equivalence of (stable) 8-categories between +• chain maps C Ñ Fibpfq and +• chain maps g: C Ñ A together with a lax null-homotopy E of fg. +Proof. We prove the first statement; the second is dual. Let fα be the terminal lh-enhancement +of f. By Proposition 8.9.26 we may identify B Ñ Cofpfq with B Ñ ConeÐpfαq. Observe +further, that composition with g sends the lh-enhanced morphism fα to gpfαq » pgfqα, which +is thus a terminal object of Maplh +gfpA‚, C‚q. Therefore the claim follows from Theorem 8.10.3 +after identifying exact lh-enhancements with lax null-homotopies (see Remark 8.9.12). +Proof of Theorem 8.10.3. Fix an lh-enhanced morphism F “ pf, h, ϵ, ηq: pA‚, α‚q +lh +ùñ pB‚, β‚q +and a test chain complex pC, γq in A. We unravel the data encoded in a chain map pConeÐpFq‚, δq Ñ +pC‚, γq, using lax-oplax matrices. For each n, we have a map +Gn “ +´ +kn´1 +µn´1 +ÝÝÝÑ gn +¯ +: ConeÐpFqn “ An´1 +lax +‘ Bn Ñ Cn +(8.10.5) +and an equivalence Gnδn`1 +» +ÝÑ γn`1Gn`1, which we can expand to +pcof pkn´1αn Ñ gnfnq ÝÑ gnβn`1q “ pkn´1 ÝÑ gnq +¨ +˚ +˝ +αn +0 +fn +βn`1 +˛ +‹‚ +» +ÝÑ pγn`1kn ÝÑ γn`1gn`1q +(8.10.6) +130 + +Therefore, the map Gnδn`1 Ñ γn`1Gn`1 amounts to a cube (read back to front) +kn´1αn +0 +gnhn´1αn +0 +0 +gnfn +gnβn`1 +γn`1kn +γn`1gn`1 +µn´1αn +ζn +gnϵn +νn +gnηn +φn`1 +γn`1µn +ApAn, Cnq +ApBn`1, Cnq +´˝hn +(8.10.7) +in the contravariant Grothendieck construction of ´˝hn. The that this map is an equivalence +amounts to saying that the left and right squares of the cube are bicartesian. In particular we +can focus on the the right face and see an equivalence φn`1 : gnβn`1 +» +ÝÑ γn`1gn`1, exhibiting +g‚ : pB‚, βq Ñ pC‚, γq as a chain map. +Consider the functor +Ξ: Zop ˆ Z ˆ ∆1 Ñ St; +pm, n, ´q ÞÑ +´ +ApAn, Bmq +gn˝´ +ÝÝÝÑ ApAn, Cmq +¯ +, +(8.10.8) +which is well defined because g: B‚ Ñ C‚ is a chain map. +By direct comparison with the diagram (8.10.9) below, one verifies that all of the data +(8.10.7) can then be equivalently encoded as Zpě0qˆ∆1-sections of the mixed (contravariant, +contravariant, covariant) Grothendieck construction of Ξ that satisfy +• the restriction to Zpě0q ˆ 0 is the original lh-enhanced morphism F : A +lh +ùñ B, +• the value on each edge pn, n, 1q Ñ pn, n, 0q is cartesian, +• the restriction to Zpě0q ˆ t1u is an exact lh-enhanced morphism E : B +lh +ùñ C. +131 + +Cn`1 +gf +k +0 +Bn`1 +f +h +0 +Cn +gf +k +Bn +f +h +Cn´1 +gf +Bn´1 +f +An`1 +α +An +α +An´1 +γ +! +ν +ζ +µ +g +β +ϵ +η +γ +! +ν +ζ +µ +g +β +ϵ +! +g +η +(8.10.9) +In other words, we have exact lh-enhanced morphisms E : B +lh +ùñ C equipped with a map +E Ñ gF which induces an equivalence on the underlying chain maps. This completes the +proof. +A +Some lemmas from (2-)category theory +A.1 +About (op)lax limits of 8-categories +We collect here a few useful lemmas regarding various types of 2-categorical limits of 8- +categories or stable 8-categories. +Construction A.1.1. Let S be an 8-category and X: S Ñ Cat8 an S-indexed diagram of +8-categories. Let p: +ş +S X Ñ S be its (covariant) Grothendieck construction. Assume that for +every arrow f : s Ñ t in S, the functor Xf admits a right adjoint. In this case, the cocartesian +fibration p is also cartesian; it corresponds to the diagram XR : Sop Ñ Cat8 which is obtained +form X by passing to right adjoints. Therefore we obtain a tautological identification +laxlimS X “ tsections of pu “ oplaxlimSop XR. +(A.1.2) +Corollary A.1.3. Let f : A Ñ B be a diagram of 8-categories and assume that f has a right +adjoint fR. Then there is a natural identification +A +ш +f B “ B +Ј +fR A +(A.1.4) +given by the formula +pa, b, fa uÝÑ bq Ø pa, b, a uÝÑ fRbq +(A.1.5) +132 + +Proof. This is just the special case S “ ∆1 of the identification (A.1.2). +Lemma A.1.6. Let A +fÝÑ C +gÐÝ B be a diagram of 8-categories. +(1) Assume that f has a right adjoint fR. Then there is a natural equivalence +A +ñˆ +C B » B +Ј +fRg +A +(A.1.7) +given by the formula +pa, b, fa uÝÑ gbq Ø pa, b, a uÝÑ fRgbq. +(A.1.8) +(2) Assume that g has a left adjoint gL. Then there is a natural equivalence +A +ñˆ +C B » A +ш +gLf +B +(A.1.9) +given by the formula +pa, b, fa uÝÑ gbq Ø pa, b, gLfa uÝÑ bq. +(A.1.10) +Proof. We compute +A +ñˆ +C B “ A ˆCt0u Ct0Ñ1u ˆCt1u B » A +ш +f C ˆC B +(A.1.11) +» C +Ј +fR A ˆC B » At0Ñ1u ˆAt1u C ˆC B +(A.1.12) +» At0Ñ1u ˆAt1u B » B +Ј +fRg +A +(A.1.13) +where we have used Corollary A.1.3 in the third step and the explicit construction of the +lax/oplax limit in steps two, four and six. Chasing through the chain of identifications one +immediately obtains the desired formula. +The second statement is analogous, this time using the description +A +ñˆ +C B » A ˆC B +Ј +f C +(A.1.14) +and applying Corollary A.1.3 in the other direction. +Corollary A.1.15. Let C be an p8, 2q-category. +(1) For each arrow f : A Ñ B in C with a right adjoint fR, we have natural equivalences +A +ш +f B » B +Ј +fR A +and +B +Ð> +fR A » A +Ñ> +f B. +(A.1.16) +(2) For each diagram A +fÝÑ C +gÐÝ B in C, we have a natural equivalences +A +ш +gLf +B » A +ñˆ +C B » B +Ј +fRg +A. +(A.1.17) +assuming that g has a left adjoint or f has a right adjoint. +133 + +(3) For each diagram A +fÐÝ C +gÝÑ B in C, we have a natural equivalences +B +Ð> +fgR A » A +ñ> +C B » A +Ñ> +gfL B +(A.1.18) +assuming that g has a right adjoint or f has a left adjoint. +Each of these equivalences represents (in the case “>”) or corepresents (in the case “ˆ”) the +corresponding equivalences of Corollary A.1.3 and Lemma A.1.6. +Proof. All the relevant objects are characterized either by their represented or corepresented +functor, hence we may reduce to the case of lax limits and directed pullbacks in Cat8. This +case is established in Corollary A.1.3 and Lemma A.1.6. +A.2 +About adjoints in diagram 2-categories +Let B, C be two p8, 2q-categories. +• By FUNlaxpB, Cq and FUNoplaxpB, Cq we denote the p8, 2q-category of functors B Ñ C +and lax/oplax natural transformations η: F Ñ G between them, which assigns to each +morphism f : B Ñ B1 in B a square +FB +FB1 +GB +GB1 +Ff +ηB +ηB1 +Gf +or +FB +FB1 +GB +GB1 +Ff +ηB +ηB1 +Gf +(A.2.1) +respectively. Formally, the functors FUNlaxpB, ´q and FUNoplaxpB, ´q can be defined as +a right adjoints to the lax and oplax Gray tensor products. +• By FUNpB, Cq we denote the standard internal hom in the p8, 2q-category of p8, 2q- +categories; it can be identified with the wide, locally full subcategory of FUNlaxpB, Cq +and FUNoplaxpB, Cq containing only those 1-morphisms η, where the squares (A.2.1) +contain invertible 2-cells. +If each component ηB of a lax natural transformation η: F Ñ G has a left adjoint ηBL, +then these assemble to an oplax natural transformation ηL : F Ñ G whose oplax naturality +squares +GB +GB1 +FB +FB1 +Gf +ηBL +ηB1 L +Ff +(A.2.2) +are the canonical mates of the squares (A.2.1). +Dually, each oplax transformation η has +a canonical mate ηR (which is a lax transformation), whenever its components have right +adjoints. Finally note that each natural transformation η can be viewed both as a lax and as +an oplax transformation, thus has both a mate ηL (oplax) and ηR (lax), provided that all the +required componentwise adjoints exist. +The following result due to Haugseng characterizes the morphisms in FUNpB, Cq which +have a adjoints. +134 + +Proposition A.2.3 ([Hau21], Theorem 4.6). Let η: F Ñ G: B Ñ C be a natural transfor- +mation. +(1) As a morphism in FUNlaxpB, Cq, the transformation η has a right adjoint if and only if +each component ηB has a right adjoint in C. The right adjoint ηR is its canonical mate, +where η is viewed as an oplax transformation. +(2) As a morphism in FUNoplaxpB, Cq, the transformation η has a left adjoint if and only +if each component ηB has a left adjoint in C. The left adjoint ηL is its canonical mate, +where η is viewed as a lax transformation. +This result also explains our terminology from Definition 4.5.1. +Corollary A.2.4. Let η: F Ñ G: B Ñ C be a natural transformation. As a morphism in +FUNpB, Cq it has +(1) a right adjoint if and only if the naturality square (A.2.1) is vertically right adjointable, +(2) a left adjoint if and only if the naturality square (A.2.1) is vertically left adjointable, +In each case, the left/right adjoint is the corresponding canonical mate. +Proof. Beyond the existence of adjoints, the right/left vertical adjointability condition states +precisely that the 2-cells in the canonical mate ηR, ηL are again invertible, thus providing a +right/left adjoint in FUNpB, Cq and not just in FUNlaxpB, Cq/FUNoplaxpB, Cq. +Remark A.2.5. Let S be an 8-category and α: X Ñ Y: S Ñ C a natural transformation +of S-diagrams in C. Assume that each component αs has a left/right adjoint βs and that +all naturality squares of α are vertically left/right adjointable. 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Math., +398:Paper No. 108175, 53, 2022. +139 + diff --git a/2NE0T4oBgHgl3EQfugGB/content/tmp_files/load_file.txt b/2NE0T4oBgHgl3EQfugGB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..369775b976c975b33728bc00c526330407af54f5 --- /dev/null +++ b/2NE0T4oBgHgl3EQfugGB/content/tmp_files/load_file.txt @@ -0,0 +1,6399 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf,len=6398 +page_content='Complexes of stable 8-categories Merlin Christ∗, Tobias Dyckerhoff∗, Tashi Walde† January 9, 2023 Abstract We study complexes of stable 8-categories, referred to as categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As we demonstrate, examples of such complexes arise in a variety of subjects including represen- tation theory, algebraic geometry, symplectic geometry, and differential topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One of the key techniques we introduce is a totalization construction for categorical cubes which is particularly well-behaved in the presence of Beck-Chevalley conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a direct application we establish a categorical Koszul duality result which generalizes previously known derived Morita equivalences among higher Auslander algebras and puts them into a conceptual context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We explain how spherical categorical complexes can be interpreted as higher-dimensional perverse schobers, and introduce Calabi-Yau structures on categorical complexes to capture noncommutative orientation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A variant of homological mirror symmetry for categorical complexes is proposed and verified for CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, we develop the concept of a lax additive p8, 2q-category and propose it as a suitable framework to formulate further aspects of categorified homological algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Rules of categorification .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Examples of categorical complexes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 The categorified Dold–Kan correspondence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Categorical Koszul complexes and representation theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Fukaya-Seidel complexes in symplectic geometry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Spherical complexes and perverse schobers on Cn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 48 5.' metadata={'source': 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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 66 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Picard-Lefschetz theory and Fukaya-Seidel complexes .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 119 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10 Universal property of the lax mapping cone .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 129 A Some lemmas from (2-)category theory 132 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 About (op)lax limits of 8-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 132 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 About adjoints in diagram 2-categories .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 134 1 Introduction While originally intended as a habitat to formulate and prove homological duality results, derived categories have since evolved into invariants in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One modern perspec- tive is to treat derived categories as stable 8-categories, as introduced by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lurie [Lur17], which allows for the application of powerful tools from homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this work, we investigate complexes of stable 8-categories, referred to as categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These are sequences ¨ ¨ ¨ A2 A1 A0 ¨ ¨ ¨ d d d d of stable 8-categories Ai with exact functors d such that d2 » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the form of localization sequences, special examples of such complexes have already been crucial in the study of alge- braic K-theory and related additive or localizing invariants [Wal85, TT90, BGT13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Several developments, mainly within the vicinity of homological mirror symmetry, suggest the exis- tence of interesting more general classes of examples of categorical complexes and arouse the desire to investigate them within an effective axiomatic framework of categorified homological algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here, the primary interest lies in the stable 8-categories themselves and not just their additive invariants, so that the familiar axiomatic frameworks for homological algebra can not be applied but needs to be replaced by a suitable p8, 2q-categorical counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this article,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' using this circle of ideas as a starting point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' we construct classes of examples of categorical complexes arising in several different areas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' study various totalization constructions and their interplay with Beck–Chevalley condi- tions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' demonstrate the usefulness of these constructions by proving concrete results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' such as categorical Koszul duality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' formulate conjectures in the context of homological mirror symmetry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' discuss properties (sphericalness) and structures (Calabi–Yau) of categorical complexes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' propose a notion of lax 2-additivity for p8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2q-categories and explain why we believe this to be a reasonable axiomatic framework to study categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While the results and examples of this work seem to suggest that our approach yields a reason- able 8-bicategorical framework to study purely additive aspects of categorical complexes, we emphasize that these are merely preliminary steps: In this work, we do not offer well–behaved 3 8-bicategorical counterparts of homotopical or homological aspects, in particular, we do not introduce an 8-bicategorical version of the derived category – we aim to investigate this in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We proceed with a more detailed overview of the contents of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Rules of categorification Most examples and results of this work are based on the insight that some of the building blocks of the construction of complexes of abelian groups can be “categorified”, replacing abelian groups by stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This procedure of categorification typically follows the following set of rules: classical categorified 1) abelian group A stable 8-category A 2) element x P A object X P A 3) y ´ x conepX fÑ Y q 4) řp´1qixi totp X0 X1 X2 ¨ ¨ ¨ Xn d d d d q 5) direct sum decomposition C – A ‘ B semiorthogonal decomposition C » xA, By 6) external direct sum A ‘ B lax sum A Ø‘ F B While the first two rules should be apparent, we start commenting on rule 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is a first crucial difference between the classical and the categorified context: In order to take a “difference” between objects X, Y of a stable 8-category A, we need to be given the additional datum of a morphism f : X Ñ Y – the difference will then be the cone of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Compliance with this rule will force us to include certain 2-categorical data which becomes invisible upon passing to the Grothendieck group K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As we will see in examples, this typically results in rather natural lax variants of 1-categorical constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rule 4) is a natural generalization of Rule 3): An alternating sum over n elements will be categorified by the totalization of an n-term complex in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here we do not only need to specify the differentials of this complex, but also a coherent system of null homotopies – this is necessary to make sense of the totalization in the 8-categorical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rule 5) is almost evident after having accepted Rule 3): While in a direct sum A ‘ B, every element is uniquely the sum of elements from the components A and B, respectively, 4 in a semiorthogonal decomposition xA, By, every object is uniquely an extension of an object A P A by an object B P B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Put differently, by shifting the exact triangle of the extension, every object is uniquely the cone of a morphism Ar´1s Ñ B, thus connecting back to Rule 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Conceptually distinct to a direct sum decomposition of a given abelian group are the universal properties satisfied by the external direct sum of a pair of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When formulating a categorified analog of these universal properties, it is not sufficient to just provide a pair of stable 8-categories: As an additional datum, we need to specify a functor F : A Ñ B (similar to the additional choice of a morphism f : X Ñ Y needed in Rule 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The categorified “direct sum” is then the lax sum A Ø‘ F B of the diagram of stable 8-categories described by F (see Rule 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As explained in detail in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4, and §8, the lax sum is simultaneously a lax limit and lax colimit (lax semiadditivity), as well as an oplax limit and oplax colimit (lax additivity) of the diagram ∆1 Ñ Stk given by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lax sum admits a semiorthogonal decomposition with components A and B (which according to Rule 5 should be interpreted as a categorified direct sum decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Vice versa, a stable 8-category with a semiorthogonal decomposition can be described as a lax sum if and only if it admits a gluing functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Examples of categorical complexes We survey the main examples of categorical complexes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' chain complexes of stable 8- categories) that will be constructed in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 The categorified Dold–Kan correspondence As explained in [Dyc21], the classical Dold–Kan correspondence can be categorified along the lines of the rules explained in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This implies that every connective categorical complex arises as the totalization of a 2-simplicial stable 8-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The relevant constructions will be reviewed in some more detail in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The appearance of categorical complexes in this context can be viewed as a first evidence that one may imitate somewhat more elaborate constructions from homological algebra in an p8, 2q-categorical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Categorical Koszul complexes and representation theory The classical construction of Koszul complexes by means of totalizing tensor products of two-term complexes can be categorified resulting in a notion of categorical Koszul complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The details will be explained in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this context, we prove a Koszul duality statement (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) which amounts to a categorification of the classical self-duality of the Koszul complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a consequence, we obtain a conceptual proof of a derived Morita duality, origi- nally discovered in [Bec18], among the higher Auslander algebras as introduced in [Iya07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Complete intersection complexes and lax algebraic geometry Given a normal crossings divisor in a smooth projective variety, we consider its ˇCech nerve, formed by the smooth components of the divisor along with their intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Passing 5 to derived categories of coherent sheaves, we obtain a cubical diagram of stable 8-categories which may be totalized to yield a categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We refer to this complex as the complete intersection complex of the divisor, introduced in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These complexes naturally appear on the B-side of homological mirror symmetry and are part of a homological mirror symmetry conjecture for categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The terms of the categorical intersection complex can be described somewhat appealingly as a hybrid of “commutative data” (coherent sheaves on a smooth variety) and “non–commutative data” (lax gluing data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We feel that these types of stable 8-categories could of independent interest for a theory of “lax algebraic geometry”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this context, it is of particular relevance, that categorical complexes are capable of capturing noncommutative orientation data in the form of Calabi–Yau structures and have geometric interpretations in terms of higher–dimensional perverse schobers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This will be explained in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 and §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The construction of the complete intersection complex arises from a general totalization procedure which can be applied to any cubical diagram of stable 8-categories (see §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, we obtain a categorical complex associated to the cubical resolutions of singularities familiar from mixed Hodge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One could speculate that these categorical complexes may play an interesting role as some kind of noncommutative resolutions of singularities, but this will not be discussed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Fukaya-Seidel complexes in symplectic geometry Let X Ă CN be an n-dimensional smooth affine subvariety and let X1 Ă X be a generic hyperplane section of X cut out by the equation π1pxq “ 1 for some linear function π1 : CN Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In [Sei08], using further assumptions and auxiliary data, a left exact sequence of stable 8-categories (in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' these are modeled as A8-categories) FukpXq ãÑ FSpπ1q B ÝÑ FukpX1q (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is describes, which relates the Fukaya categories of X and X1 with the Fukaya–Seidel category FSpπ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may iterate this construction by choosing a generic hyperplane section X2 of X1, corresponding to a linear function π2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Splicing together the various left exact sequences (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) yields a categorical complex FSpπ1q ÝÑ FSpπ2q ÝÑ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' ÝÑ FukpXnq, which we call the Fukaya–Seidel complex of the family tπ‚u (see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is a consequence of classical Picard–Lefschetz theory, that the passage to Grothendieck groups yields a complex of abelian groups with homology H˚pXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This suggests that, in comparison with Fukaya categories, we may think of Fukaya–Seidel complexes as categorical invariants that attempt to capture information about the full homology as opposed to just the middle–dimensional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the given context, there is one caveat though: Currently it is not really justified to use the term “invariant” here, since it is unclear to us at the moment, in which sense the Fukaya–Seidel complex is independent of the choice of the family tπ‚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Fukaya–Seidel complexes are (conjecturally) the symplectic counterparts of complete in- tersection complexes in the context of homological mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus, just like complete intersection complexes of anticanonical normal crossing divisors, they typically carry Calabi– Yau structures and naturally fit within a higher–dimensional framework of perverse schobers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 and §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 Complexes of local systems on manifolds with corners We consider the n-simplex ∆n as an n-category, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' via the nth oriental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A functor M from ∆n into the p8, nq-category Bordn of bordisms, as defined in [Lur09b, CS19], amounts to an n-manifold M with corners and a diagram of subspaces of the boundary of M (decomposi- tions into lower–dimensional bordisms) parametrized by the barycentric subdivision of the n-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may interpret this data, augmented by the empty space, as a cubical diagram of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly to §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we may then pass to the corresponding cube of sta- ble 8-categories of Dpkq-valued local systems and totalize to obtain the categorical complex of local systems of M, introduced in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We will see that an orientation of the n-manifold M yields a Calabi–Yau structure on this complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Totalization constructions Many of the examples of categorical complexes described in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 arise as totalizations of cate- gorical multi–complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §4, we will discuss such totalization constructions systematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, we explain that there are (at least) two natural choices of totalizations, the product and coproduct totalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While, in general, these totalizations lead to different categorical complexes, we show that they do agree in the presence of suitable Beck–Chevalley conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Of particular expected relevance towards a theory categorified homological algebra is a totalization construction yielding the (op)lax cone of a categorical chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9 we investigate, in the general lax additive context (see §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6), how to categorify the usual formula d “ ´ ´dA 0 ´f dB ¯ for the differential of the mapping cone of a chain map f : A‚ Ñ B‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Mirror symmetry for categorical complexes We formulate homological mirror symmetry conjectures relating the Fukaya-Seidel complexes from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 on the symplectic side to the complete intersection complexes from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 on the algebro–geometric side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We work out some details for this homological mirror symmetry for categorical complexes in the simplest illustrative case of Hori-Vafa mirror symmetry for CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The claimed equivalence for CPn, n ě 3, is already conjectural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 Geometric structures associated with categorical complexes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Calabi-Yau structures The concept of a left Calabi–Yau structure on a functor between k-linear stable 8-categories, as introduced in [BD19], can be regarded as a noncommutative generalization of an oriented manifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In fact, one example is precisely obtained from such a manifold by considering the left Kan extension of derived local systems along the boundary inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §6, we introduce the notion of a left Calabi–Yau structure on a categorical complex, which can be thought of as a noncommutative analog of an oriented manifold with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It generalizes the left Calabi–Yau structures from [BD19] when interpreting the given functor 7 as a categorical 2-term complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The constructions of Calabi–Yau structures given in [BD19] can be generalized to provide examples of Calabi–Yau structures on categorical complexes: (1) the complete intersection complexes from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 associated with anticanonical divisors in projective varieties, see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6, (2) conjecturally, the Fukaya–Seidel complexes from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4, (3) the categorical complexes of local systems associated to n-bordisms in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, see Corol- lary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, To produce Calabi–Yau structures on categorical complexes, we also introduce a notion of a Calabi–Yau structure on categorical cubes, with the property that the coproduct totalization of a Calabi–Yau cube inherits a Calabi-Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The Calabi–Yau structures in the examples (1) and (3) above arise in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Spherical complexes and perverse schobers Spherical functors arise in several areas, in particular in homological mirror symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As proposed in [KS14], one may interpret them geometrically as categorical analogs of perverse sheaves, called perverse schobers, on the complex plane C, stratified as Czt0u Y t0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As discussed in §5, it turns out that the natural generalization of the concept of a spherical functor to categorical complexes is simply given by a spherical complex: a complex where all differentials are spherical functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Spherical n-term complexes correspond geometrically to categorified perverse sheaves on Cn stratified by the linear subspaces t0u Ă C Ă C2 Ă ¨ ¨ ¨ Ă Cn and their complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Just as for Calabi–Yau structures, many of the examples of spherical functors admit nat- ural generalizations to spherical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These spherical complexes often arise as totaliza- tions of spherical cubes satisfying a Beck–Chevalley condition (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark- ably, these cubical diagrams can also be interpreted geometrically as perverse schobers on Cn but with stratification given by the coordinate hyperplanes along with their intersections and complements (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this context, the totalization procedure can be regarded as a higher– dimensional analog of Seidel’s formation of the “directed subcategory on the vanishing cycles” as a description of the Fukaya–Seidel category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We hope to explore the geometric relevance of this in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 Lax additivity Plenty of thought has been invested in the question into what kind of categorical structure the collection of derived categories should be organized, or, as Tamarkin put it: ”What do DG- categories form?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [Tam07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Several precise answers to this question have been given within the various frameworks of enhanced triangulated categories that can be used to study derived categories, such as differential graded categories [Toe11], A8-categories [Sta63, Kel01], or stable 8-categories [Lur17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Relevant for this work, in which we take the approach via stable 8-categories, will be that the p8, 2q-category of stable 8-categories is additive in a suitable lax 2-categorical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We 8 give an informal introduction to the circle of ideas surrounding the concept of lax additivity in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 and §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 adapted specifically to the case of stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §8, we systematically develop the general theory of lax additivity and lax matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following table summarizes the main features and should be seen as a natural continuation of the table of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1: classical categorified 7) additive (8-)category A lax additive p8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2q-category A 8) hom-sets ApX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Y q have addition hom-categories ApX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Yq have colimits 9) hom-sets ApX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Y q are abelian groups hom-categories ApX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Yq are stable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='binary direct sums ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='X ‘ Y “ X ˆ Y “ X > Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='lax/oplax ∆1-bilimits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='ш ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='F Y “ X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Ð> ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='F Y “ X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Ј ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='F Y “ X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Ñ> ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='F Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='finite direct sums ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='‘k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='s“1xs “ śk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='s“1 xs “ šk ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='s“1 xs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='general lax bilimits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Àlax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='s:S Xs “ laxlims:S Xs “ laxcolims:S Xs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='matrices p m11 m12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='m21 m22 q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='lax matrices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='¨ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='˚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='˝ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='M11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='M12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='M21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='M22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='˛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='‹‚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='matrix multiplication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='pnmqus “ ř ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='t“t1pnut1 ˝ mtsq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='lax matrix multiplication ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='pNMqus “ colimγ : tÑt1pNut1 ˝ γ ˝ Mtsq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='matrix multiplication,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' reparameterized pnmqus “ ř tp´1qtpnut ˝ mtsq lax-oplax matrix multiplication pNMqus “ tottpNut ˝ Mtsq Accepting our basic premise that abelian groups are to be categorified by stable 8- categories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rule 9) requires no further comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rule 8) is a convenient intermediate step, categorifying the situation where the addition on hom-sets does not necessarily have inverses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' just like the uncategorified case, many basic lemmas are most naturally expressed in this generality leading to the notion of lax semi-additive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The direct sum of abelian groups is both a categorical product and a categorical coproduct, a universal property that is taken as the definition in general additive categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rule 11) states that the same definition can be categorified, as long as we replace finite products and coproducts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', limits and colimits indexed by finite discrete categories S “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , ku, with lax limits and colimits indexed by arbitrary 8-categories S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Apart from this change, the theory is exactly analogous: if the hom-categories have colimits (categorifying addition) then such lax limits and colimits always agree if they exits, yielding the notion of lax bilimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 9 When the hom-categories are stable, we can say even more for certain special shapes, such as S “ ∆1: in this case all four possible universal 2-categorical constructions (lax/oplax, limit/colimit) associated to an S-diagram X FÝÑ Y agree with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An extremely convenient feature of additive categories is that maps m: x1 ‘ x2 Ñ y1 ‘ y2 between direct sums can be represented as matrices of the form m “ ˆm11 : x1 Ñ y1, m12 : x2 Ñ y1 m21 : x1 Ñ y2, m22 : x2 Ñ y2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Composing such maps then just amounts to the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rule 13) shows how the usual matrix multiplication formula can be categorified, yielding an analogous theory of matrices indexed in each coordinate not by a finite set but by arbitrary 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These matrices are just a dependent version of bimodules, which by Morita theory encode functors between module categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 we have already seen how it is conceptually easier to categorify subtraction rather than addition and more generally alternating rather than ordinary sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the lax additive setting we see a similar feature, expressed in Rule 14), where a convenient “coordinate change” yields a much nicer formula for the categorified matrix multiplication when we reparameterize it to use alternating rather than ordinary sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Categorically speaking this reparameteriza- tion involves the identification of lax and oplax limits and is therefore only available for certain special indexing categories such as ∆1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7 Relations to previous work As already indicated in the very beginning, the idea to consider categorical complexes is, in principle, by no means new: short exact categorical sequences in the form of localization sequences already play a central role in algebraic K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Also the directed pullback con- struction has already been used in this context, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [LT19], where (implicitly) also the product totalization of a square is being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As the authors have told us, they have also observed a type of Beck–Chevalley phenomenon, that will be relevant for applications to K- theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A version of categorical complexes for non-stable symmetric monoidal 8-categories was introduced in [Lur09b], but the motivation in this context is quite different from ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The Calabi-Yau structures introduced in §6 can be interpreted in the framework of [ST16] but our examples have not been considered there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The work in progress [AGHJ22] studies Fukaya categories for Landau-Ginzburg models with multi-potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As indicated to us by the authors, this theory will provide a means to introduce cubical categorical diagrams in terms of partially wrapped Fukaya categories which one would expect to mirror the cubical diagrams in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Variants of this homological mirror symmetry conjecture for categorical cubes were already recorded in [Lee22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The expected relation to our formulation will be explained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The idea to use matrices to describe coordinate changes for semiorthogonal decompositions already appears in [DKS23], similar in spirit to the lax matrices in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8 Acknowledgements We are very grateful to Jeffrey Hicks, Sukjoo Lee, Nick Sheridan, and the participants of the Edinburgh Hodge Seminar in general for interesting conversations about the topic of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, we thank them for making us aware of the work in progress on Fukaya 10 categories of multi-potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Many thanks to Denis Auroux, Sheel Ganatra, Andrew Han- lon, and Maxim Jeffs for taking the time and effort to provide us with a report on their work in progress [AGHJ22] (and to Nick Sheridan for organizing the lecture series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We further thank Jacob Lurie for interesting remarks on categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We also thank Federico Barbacovi for discussions about spherical functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' thanks M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Kapranov and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Schecht- man for many inspiring discussions on perverse sheaves and schobers, in particular, this work draws substantial inspiration from their perspectives proposed in [KS14] and related subse- quent work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' thanks Claudia Scheimbauer for conversations in the context of their joint work that helped shape many of the key ideas regarding higher categorical additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' thanks the Hausdorff Research Institute for the hospitality during his stay, during which part of this work was written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2047/1 – 390685813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' acknowledge support by the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' acknowledges support of the VolkswagenStiftung through the Lichtenberg Professorship Pro- gramme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='W acknowledges support by the SFB 1085 – Higher Invariants, funded by the DFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2 Basic concepts 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Stable 8-categories As a model for “enhanced triangulated categories”, we will use stable 8-categories as intro- duced in [Lur17] which, along with [Lur09a] and [GR17], are our standard references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It will be most convenient to work with presentable stable 8-categories – the 8-category formed by these with colimit–preserving functors will be denoted by St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For applications, it is also important to consider presentable stable 8-categories linear over a given field, or more generally over a commutative ring spectrum, denoted k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Following [Lur17], we define these as modules in St over the symmetric monoidal stable 8-category Modk of k-module spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting 8-category of k-linear stable 8-categories will be denoted by Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given objects C, D P Stk, we denote by StkpC, Dq the 8-category of morphisms in Stk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', colimit–preserving k-linear functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category StkpC, Dq can be regarded as an object of Stk and thus defines an internal hom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Further, Stk admits a symmetric monoidal structure (adjoint to the internal hom) and we denote the corresponding binary tensor product by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Any compactly generated 8-category C P Stk is dualizable with respect to this monoidal structure and we denote its dual by C_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the subcategory C0 Ă C of compact objects, the 8-category C is equivalent to the 8-category IndpC0q of Ind-objects of C0, and we have C_ » StkpC, Modkq » IndpCop 0 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For a morphism F : C Ñ D in Stk between compactly generated 8-categories, we denote its dual by F _ : D_ ÝÑ C_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By the adjoint functor theorem, any functor F in Stk admits a right adjoint, denoted by F R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The adjoint F R is not necessarily a morphism in Stk, since it may not preserve colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 11 The condition that F R preserves colimits and admits a k-linear structure is equivalent to the condition that F preserves compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When we speak of a k-linear adjoint, we mean that the adjoint also lies in Stk, and the adjunction is in this case given in a 2-categorical sense in Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Complexes of stable 8-categories We define a chain complex of stable 8-categories to consist of a sequence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' d d d d of morphisms d: Ai Ñ Ai´1 in Stk satisfying d2 » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note, that the existence of an equiva- lence d2 » 0 simply amounts to the statement that d2 is a zero object in the stable 8-category StkpAi, Ai´2q and the space of zero objects is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus, in order to specify a com- plex of stable 8-categories, one only needs to provide the list of differentials, and verify the condition d2 » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, in contrast to the notion of a complex of objects in a fixed stable 8-category, no higher coherence data is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We justify this a bit more formally: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Following [Wal22], we define the category Ch :“ Z˚{ „ where Z˚ denotes the category obtained from Z by adjoining a zero object ˚ and Ch is the quotient obtained by identifying each composite d2 : i Ñ i ` 2 with the 0 morphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the unique composite i Ñ ˚ Ñ i ` 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We then define a coherent chain complex in a pointed 8-category C to be a functor Chop Ñ C that preserve zero objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Furthermore, we denote by G :“ ¨ ¨ ¨ >t´1u t´1 Ñ 0u >t0u t0 Ñ 1u >t1u t1 Ñ 2u >t2u ¨ ¨ ¨ Ă Z (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) the 1-skeleton of (the nerve of) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By definition, a map of simplicial sets G Ñ K just consists of a sequence of composable edges in the simplicial set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We regard Stk as an 8-category (by discarding noninvertible 2-morphisms) and consider further the 8-categories Fun˚pChop, Stkq of functors preserving zero objects, Funpd2qpZop, Stkq of functors such that each d2 is equivalent to the 0 functor, Funpd2qpGop, Stkq of functors such that each d2 is equivalent to the 0 functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The restriction functors Fun˚pChop, Stkq ÝÑ Funpd2qpZop, Stkq ÝÑ Funpd2qpGop, Stkq are equivalences of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 12 Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The rightmost restriction functor is an equivalence because the inclusion G Ă Z is an inner anodyne extension of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Extending a functor F : Zop Ñ Stk to Chop involves various lifting problems in the hom- spaces StkpFpiq, Fpjqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If we assume that for each k ě 2 the edge Fpdkq is equivalent to zero, then all of these lifting problems take place in the zero component of the mapping space StkpFpiq, Fpjqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Hence the statement follows from the key fact that for each A, B in Stk, the zero connected component of StkpA, Bq is just the groupoid of zero functors A Ñ B, which is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We omit the remaining details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In virtue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we may always extend a given complex in Stk presented as a sequence of differentials d satisfying d2 » 0, to a fully coherent complex in an essentially unique way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Further, we will sometimes want to turn a coherent complex concentrated in degrees n ě i ě 0 into a cubical diagram: To this end, we may define the functor q: Ppt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', nuq ÝÑ Ch, M ÞÑ $ ’ & ’ % 0 if M “ H, i if M “ t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', iu, ˚ else and consider the pullback q˚ : Fun˚pChop, Stkq ÝÑ Fun˚pPpt1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', nuqop, Stkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) For the sake of brevity, we will refer to a complex in Stk as a categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by ChpStkq the full subcategory of FunpZop, Stkq spanned by categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An categorical n-complex is a functor A˚ : Zop,n Ñ Stk, meaning collections of commuting n differentials di : Aa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',ai,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',an ÝÑ Aa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',ai´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',an , satisfying that d2 i » 0 for all 1 ď i ď n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) We denote by ChnpStkq the full subcategory of FunppZopqn, Stkq spanned by categorical n-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) A categorical n-complex A˚ is called a categorical n-cube if Aj » 0 for all j P pZopqn and j R In, where I “ r1sop “ t1 Ñ 0u Ă Zop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by CubenpStkq Ă ChnpStkq the full subcategory spanned by categorical n-cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Partially lax limits There are different types of limits of a given diagram tAiui:I in an p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' for each triangle lim Ai Aj in the limit cone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' one can require either of the following: 13 (i) strict: it commutes up to natural equivalence: lim Ai Aj (ii) lax: it commutes up to a possibly noninvertible 2-morphism as in lim Ai Aj (iii) oplax: it commutes up to a possibly noninvertible 2-morphism as in lim Ai Aj The oplax triangle may be modelled as an amalgamate of a strict and a lax triangle as in lim Ai Aj Aj id so that it suffices to distinguish between strict and lax triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One way to encode this “partial laxness” is to simply mark those edges in the diagram tAiui:I over which we require the cone to strictly commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting notion of marked limit has been developed and studied by several groups of authors, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', [AGS22] for a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Throughout this work, we will use partially lax universal constructions to describe stable 8 categories and functors among them (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §3 through §7, the focus lies on concreteness and explicit examples, so that we take a hands-on approach and use explicit models for the relevant constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The framework of 8-categories is sufficient to describe the needed universal properties, even though we often formulate them in p8, 2q-categorical terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §8 we will discuss (some of) these universal constructions from a more foundational point of view within a framework of lax additive p8, 2q-categories (also see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here, we do not make use of a concrete model for p8, 2q-categories, but rather provide a list of features that will be used (see §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Lax additivity For given abelian groups A and B, their product and coproduct are characterized by the universal cones A ˆ B B A > B B A and A, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) 14 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The fact that these constructions agree up to canonical isomorphism is typically indicated by using the terminology direct sum, and the notation A ‘ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This phenomenon is referred to as the semiadditivity of the category of abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Additivity then amounts to the extra condition of the resulting monoid structure on HompA, Bq being a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As already indicated in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, the analogous categorified universal constructions typically involve more data, invisible upon passing to K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Namely, beyond a pair of stable 8-categories A, B, we are also given an exact functor F : A Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The relevant universal constructions that we may build from this data are characterized by the universal cones depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' lax limit: A ш F B A B F oplax limit: A Ј F B A B F lax colimit: A Ð> F B A B F oplax colimit: A Ñ> F B A B F Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1: The four lax cones with base given by a functor F : A Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 (Lax Additivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the p8, 2q-category Stk of presentable stable 8-categories, the four lax universal constructions from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 exist and are canonically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As explained in detail in §8, the analogous statement holds in any lax additive p8, 2q- category and Stk is an example of such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In fact, §8 provides a more refined systematic analysis also introducing the notion of a lax semiadditive p8, 2q-category as an intermediate step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From an axiomatic perspective, the canonical equivalence of these four universal construc- tions captures the essence of what seems to make lax additive p8, 2q-categories a suitable context for categorified homological algebra (just like additive categories are used for classical homological algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While complete proofs will be deferred to §8, we explain here, how to construct a particular model for the lax limit in St and describe the four universal lax cones from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 in terms of this model (the analogous statement for Stk is somewhat more involved as we need to keep track of the Modk-module structures – we don’t discuss this here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let ΣpFq denote the 8-category of sections of the covariant Grothendieck construction of F, considered as a diagram ∆1 Ñ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Due to the simplicity of the indexing category, this can be described even more concretely as the pullback of simplicial sets ΣpFq Funp∆1, Bq A B ev0 F 15 Notationally, we simply write ΣpFq “ tpa, b, ηq | a P A, b P B, η: Fpaq Ñ bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now describe the universal lax cones that characterize ΣpFq as each of the four universal constructions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lax limit cone is evident: The functors to A and B, respectively, are simply given by projecting to the components a P A and b P B while the natural transformation which is part of the cone is given by Fpaq Ñ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that, while we specify the data of the cone on the level of objects, it is evident how to extend these formulas to actual functors and natural transformations of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lax colimit cone of ΣpFq corresponds to the functors A Ñ ΣpFq, a ÞÑ pa, Fpaq, Fpaq id Ñ Fpaqq B Ñ ΣpFq, b ÞÑ p0, b, 0 Ñ bq where the natural transformation assigns to a P A, the morphism in ΣpFq given by p 0, Fpaq, 0 Fpaq q p a, Fpaq, Fpaq Fpaq q id id id 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The oplax limit cone is given by the functors ΣpFq Ñ A, pa, b, ηq ÞÑ a, ΣpFq Ñ B, pa, b, ηq ÞÑ fibpηq, and the natural transformation which assigns to pa, b, ηq P Σ, the morphism in B given by fibpηq Ñ Fpaq which is part of the fiber square fibpηq Fpaq 0 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, the oplax colimit cone is determined by the functors A Ñ ΣpFq, a ÞÑ pa, 0, Fpaq Ñ 0q B Ñ ΣpFq, b ÞÑ p0, br1s, 0 Ñ br1sq while the natural transformation assigns to a P A, the morphism in ΣpFq given by p a, 0, Fpaq 0 q p 0, Fpaqr1s, 0 Fpaqr1s q where the square is the biCartesian square exhibiting Fpaqr1s as the suspension of Fpaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 16 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that, for F “ id: A Ñ A, we have ΣpFq “ Funp∆1, Aq so that ΣpFq is simply a presheaf category, or put differently, the category of A-valued repre- sentations of the quiver of Dynkin type A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this situation, the fact that ΣpFq, which most evidently is a lax limit of F, is in fact also a lax colimit of F amounts to the statement that any presheaf category is generated by the subcategory of representable presheaves (tensored with objects of A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As explained in more detail in §8, the equivalence (for general F) A ш F B » A Ð> F B generalizes to any so-called lax semiadditive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, it does not require stability and also holds, for example, in the p8, 2q-category PrL of presentable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In contrast to the equivalence between lax limit and colimit discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, the phenomenon that the lax and oplax limit are equivalent does not hold in a general lax semiadditive p8, 2q-category but requires lax additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For example, the p8, 2q-category St of stable presentable 8-categories has this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting equivalence A ш F B » A Ј F B (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) is exploited in [DJW21] where it is described explicitly, in terms of models for lax and oplax limits, and used to provide a framework for generalized Bernstein-Gelfand-Ponomarev reflec- tion functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This equivalence also has a natural interpretation in terms of semiorthogonal decomposi- tions, as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [DKSS21]) that a semiorthogonal decomposition of a stable 8-category C consists of a pair pX, Yq of full stable subcategories of C, such that the functor of 8-categories tx Ñ y | x : X, y : Yu ÝÑ C, given by associating to a morphism x Ñ y its cofiber, is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lax and oplax limit cones for ΣpFq naturally induce semiorthogonal decompositions with components given by the kernels of the projection maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This yields concretely: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' lax limit: ptpa, 0, Fpaq Ñ 0qu, tp0, b, 0 Ñ bquq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' oplax limit: ptp0, b, 0 Ñ bqu, tpa, b, Fpaq » Ñ bquq In terms of the terminology developed in [DKSS21], these two semiorthogonal decompositions of ΣpFq are related by mutation where the first decomposition is coCartesian and the second decomposition Cartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Within this context, the equivalence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) corresponds to the statement that the underlying category of a coCartesian (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Cartesian) semiorthogonal decomposition can be recovered from its gluing functor as a lax (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' oplax) limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When we wish to make the point that the above universal constructions are equivalent (where the equivalences are determined by the choices of universal cones in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), then we will use the notation A Ø‘ F B 17 to denote any of them, and refer to it as the lax sum of A and B along F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Whenever we would like to refer to one of the above universal lax cones, then we will use the notation for the respective universal construction from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We conclude this section with a further basic equivalence which exists in the presence of adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F : A Ñ B be a functor in Stk and suppose that F has a right adjoint F R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a canonical equivalence A ш F B » ÐÑ B Ј F R A, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) in Stk described by the formula tpa, b, Fa Ñ bqu Ø tpa, b, a Ñ F Rbqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is a special case of Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 3 Simplicial totalization and the Dold-Kan correspondence 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 The categorical cochain complex of a simplex To convey a first impression of the workings of categorical complexes, we present a class of examples which is easy to describe directly yet illustrates some of the key phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A be an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We will describe categorifications of the simplicial cochain complex C‚p∆n, Aq of an n-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have Ckp∆n, Aq “ Mapptσ: rks ãÑ rnsu, Aq with differential given by the formula pdaqpσq “ nÿ i“0 p´1qiapσ ˝ Biq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let us further focus on the 2-simplex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' where we have C‚p∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Aq : C‚p∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Aq : A – tpx012qu tX012u » A A3 – tpx01,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' x02,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' x12qu tX01 Ñ X02 Ñ X12u » xA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ay A3 – tpx0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' x2qu tX0 Ñ X1 Ñ X2u » xA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ay x012“x12´x02`x01 X012“totpX01ÑX02ÑX12q xij“xj´xi Xij“conepXiÑXjq We note the following features of the categorical complex C‚p∆2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Aq: 18 (1) To be able to define the first differential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' we have adjoined morphisms Xi Ñ Xj so that we can take their “difference” according to rule 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) To define the second differential, the objects Xij need to form a complex so that we may totalize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) The octahedral “axiom” (or rather the third isomorphism theorem) guarantees that the first differential is well defined (the cones conepXi Ñ Xjq form a complex) and further, that d2 » 0 (since the just-mentioned complex is exact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The additional lax data we needed to implement the rules of categorification from §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 actually has a natural interpretation: The simplex category ∆ can be defined as the full subcategory of the 1-category Cat of small categories spanned by the standard ordinals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a result, given ordinals rks and rns we obtain a set ∆prks, rnsq of k-simplices in ∆n described as the corresponding set of maps in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' However, the collection of categories has a natural 2-categorical structure taking in to account natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Defining ∆ to be the full 2-subcategory of the 2-category Cat instead yields morphism categories, in fact posets, ∆prks, rnsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Within this context, we can now give the general definition of the complex of the categorical simplicial cochain complex C‚ :“ C‚p∆n, Aq: (1) For 0 ď k ď n, the 8-category Ck is defined to be the full subcategory Ck Ă Funp∆prks, rnsq, Aq consisting of those diagrams X satisfying: for every non-injective map τ : rks Ñ rns, we have Xpτq » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The differential d: Ck Ñ Ck`1 is given by associating to a diagram X P Ck, the diagram dX : ∆prk ` 1s, rnsq Ñ A, σ ÞÑ totpXpdnσq Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ Xpd0σqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We comment on the defining formula for the diagram dX in (2) which needs to be made precise in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' First, the formula totpXpdnσq Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ Xpd0σqq needs to be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The expression Xpdnσq Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ Xpd0σq refers to an n ` 1-term complex in the stable 8-category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Formally, this is real- ized as a cubical diagram In Ñ A, with I “ r1sop “ t1 Ñ 0u, with zero relations encoded by certain vertices of this cube being mapped to zero objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The (cofiber) totalization of this complex is then obtained by extending with zeros to a punctured pn ` 1q-cube In`1 ztp0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0qu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a right Kan extension, and taking the colimit over In`1 ztp0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0qu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a left Kan extension to In`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Second, we need to explain how the various values dXpσq organize into an actual diagram ∆prk ` 1s, rnsq Ñ A and, further, how the association X ÞÑ dX defines a functor in Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is achieved by means of the formalism of Kan extensions as developed in [Lur09a] For most parts of this paper, we will not delve into the technical details of constructions such as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' as this would have a tendency of hiding the main ideas behind the (routine) technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Rather, we assume that the reader is familiar with the techniques alluded to and keep the treatment somewhat informal as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Typically, it is rather straightforward how to make things formally precise so that we hope that not much is lost by this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 The categorified Dold–Kan correspondence The discussion in § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 shows that interesting examples of categorical complexes arise when studying simplicial (or rather 2-simplicial) objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These ideas can be put in a more system- atic context, the categorified Dold–Kan correspondence as established in [Dyc21], which will be briefly recalled here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by Cat the 2-category of small categories, and further, by ∆ Ă Cat the full sub 2-category spanned by the standard ordinals trns|n ě 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Treating ∆ as an p8, 2q-category, we define a 2-stable 8-category to be a functor X: ∆op Ñ St of p8, 2q-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For example, this can be modelled concretely as a Set∆-enriched functor from ∆, considered as Set∆-enriched by taking the nerve of the categories of morphisms, to Set∆ taking values in stable 8-categories with exact functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We explain how to associate to X a complex of stable 8-categories by a construction which is, in a sense, dual to the one discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To this end, for n ą 0, consider the cubical diagram q: t0, 1un ÝÑ Funprn ´ 1s, rnsq, pi0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', in´1q ÞÑ 0 ` i0 ď 1 ` i1 ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' ď pn ´ 1q ` in´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that t0, 1u “ r1s as posets, we use the former notation in this section to avoid confusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 2-functoriality of X induces a cubical diagram qX : t0, 1un ÝÑ FunpXn, Xn´1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now denote by Xn :“ Xn{tdegenerate simplicesu the “Verdier quotient” by the subcategory of degenerate simplices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', the full subcategory spanned by the images of all degeneracy functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The above cube qX descends to define a cubical diagram qX : t0, 1un ÝÑ FunpXn, Xn´1q which represents a homotopy-coherent pn ` 1q-term complex formed by the face maps dn Ñ dn´1 Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 20 Passing to the totalization of this complex, we obtain a functor totpqXq: Xn Ñ Xn´1 and further, noting that this totalization maps degenerate simplices to zero objects, an induced functor d: Xn Ñ Xn´1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) We leave the proof of the following lemma as an entertaining exercise to the reader: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The differential constructed in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) satisfies d ˝ d » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting complex pX‚, dq is called the simplicial totalization of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As shown in [Dyc21], the association X ÞÑ pX‚, dq defines an equivalence which can be interpreted as a categorified variant of the classical Dold-Kan correspondence: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The simplicial totalization functor defines an equivalence of 8-categories C: St∆ ÐÑ Chě0pStq :N Note that in [Dyc21] a different description of the functor C is given, but this can be shown to be equivalent to the above, by investigating the fully faithful adjoints of the localization functor Xn Ñ Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One interesting aspect of the categorified Dold-Kan correspondence is the reverse proce- dure of constructing 2-simplicial objects from categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the classical context, this provides a bridge between homological and homotopical data, and we expect the cate- gorified Dold-Kan correspondence to play a similar role for categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4 Totalizations of categorical multicomplexes Besides the totalization of simplicial objects, another natural class of examples of categorical complexes arises by totalizing bicomplexes, or more generally multi-dimensional complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this section, we discuss these totalization constructions and establish some basic proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This will allow us to introduce and investigate several interesting classes of categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Directed pushouts and pullbacks We introduce two universal p8, 2q-categorical constructions which will be crucial in this sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) The directed pullback of a diagram B C D S G (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) 21 in Stk is an object B ñˆ D C together with a universal cone of the form B ñˆ D C B C D S G (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) It may be expressed in terms of the lax limit from §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 as B ñˆ D C » pB ш S Dq ˆD C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) or explicitly as the fiber product of simplicial sets B ñˆ D C “ B ˆD Funp∆1, Dq ˆD C “ tpb, c, ηq | b P B, c P C, η: Spbq Ñ Gpcqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Dually, the directed pushout of a diagram A B C R F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) in Stk is an object B ñ> A C together with a universal cone of the form A B C B ñ> A C F R (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) We may construct it in terms of the oplax colimit via the formula B ñ> A C “ B >A pA Ñ> R Cq or as a pushout B >A Funp∆1, Aq >A C in Stk, where the functors A Ñ Funp∆1, Aq are given by the formulas a ÞÑ p0 Ñ aq and a ÞÑ pa id Ñ aq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the presence of suitable adjoints, we may express the directed pullbacks (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' pushouts) introduced in this section in terms of the (op)lax (co)limits of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This will be most relevant for §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 where we investigate Beck–Chevalley conditions and is documented in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 22 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Suppose we are given a diagram as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (a) There is a canonical equivalence B ñˆ D C » C Ј SR˝G B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (b) Suppose further that G has a left adjoint GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a canonical equivalence B ñˆ D C » B ш GL˝S C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Suppose we are given a diagram as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (a) Then there is a canonical equivalence B ñ> A C » C Ð> F˝RR B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (b) Suppose further that F has a left adjoint F L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a canonical equivalence B ñ> A C » B Ñ> R˝F L C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Informally, the equivalences are simply described by transposition using the provided adjunctions, for example pb, c, Sb Ñ Gcq Ø pb, c, b Ñ SRGcq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) for the first statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' See Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15 for a general p8, 2q-categorical proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Commutative squares Let A B C D F R S G (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) be a commutative square in Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We introduce two means of totalizing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) to obtain a categorical complex: (1) We define the product totalization to be the sequence of functors A B ñˆ D C D d2 d1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) where d2 is the canonical functor arising from the square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), interpreted as a directed cone over (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a ÞÑ pFpaq, Rpaq, SpFpaqq » GpRpaqqq while the functor d1 is given by pb, c, η: Spbq Ñ Gpcqq ÞÑ fibpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is immediate that we have d1 ˝d2 » 0 so that we may interpret (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) as a categorical complex with D in degree 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 23 (2) We define the coproduct totalization to be the sequence of functors A B ñ> A C D d2 d1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) where the functor d2 is the cofiber of the natural transformation in the universal cone (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) while d1 is the canonical functor arising from the square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), interpreted as a directed cone under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Again, it is evident, that d1 ˝ d2 » 0: The postcomposition of the said natural transformation from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) with d1 is a natural equivalence, since the square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus, we may interpret (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) as a categorical complex as well, concentrated in degrees 2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While the two categorical totalizations of the square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) yield isomorphic complexes of abelian groups when passing to K0 (as can be deduced from the existence of semiorthogonal decompositions pB, Cq on both directed pushout and directed pullback), they are not in gen- eral equivalent as categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Nevertheless, there exists a canonical comparison morphism which will be investigated next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We construct a functor χ: B ñ> A C ÝÑ B ñˆ D C (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) in terms of the universal properties of both sides: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the functor B Ñ B ñˆ D C is given by b ÞÑ pb, 0, Spbq Ñ 0q 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the functor C Ñ B ñˆ D C is given by c ÞÑ p0, cr1s, 0 Ñ Gpcqr1sq 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the functor A Ñ Funp∆1, B ñˆ D Cq is given by a ÞÑ p Fpaq, 0, S ˝ Fpaq 0 q p 0, Rpaqr1s, 0 G ˝ Rpaqr1s q where the square is the biCartesian square exhibiting the equivalence S ˝ F » G ˝ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor χ extends to a morphism of categorical complexes A B ñ> A C D A B ñˆ D C D d2 r1s d1 χ id d2 d1 between the coproduct and product totalizations of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 24 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Direct computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now provide a criterion for when the functor χ is an equivalence so that in particular, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, the coproduct and product totalization of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) will be equivalent as categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is called vertically right adjointable, if the functors R and S have right adjoints and the resulting mate C D A B G RR SR F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) commutes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the natural transformation is an equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Suppose that the square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is vertically right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the functor χ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, the product and coproduct totalizations are canonically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One may explicitly verify that we have a commutative square B ñ> A C B ñˆ D C C Ð> F˝RR B C Ð> SR˝G B C Ј SR˝G B χr´1s » » » » where all equivalences are given by the canonical identifications that arise from the various universal cones described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The goal of the subsequent parts of §4 will be to generalize the totalization of squares to bicomplexes, and finally multicomplexes – with a special focus on cubes, as these are our main examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Categorical bicomplexes A categorical bicomplex A‚,‚ consists of the datum of a family tApi,jqupi,jqPZop,2 of objects in Stk, functors d: Api,jq Ñ Api´1,jq and δ: Api,jq Ñ Api,j´1q, equivalences dδ » δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' such that d2 » 0 and δ2 » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Up to contractible choices, we may identify a categorical bicomplex with a functor Zop,2 Ñ St (with the property d2 “ 0 and δ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a categorical bicomplex A‚,‚, we define the product totalization to be the categorical complex C‚ “ totˆpA‚,‚q given as follows: 25 (1) for n P Zop, the category Cn is the iterated directed pullback ¨ ¨ ¨ ñˆ A1,n´2 A1,n´1 ñˆ A0,n´1 A0,n ñˆ A´1,n A´1,n`1 ñˆ A´2,n`1 ¨ ¨ ¨ Thus, an object of this category consists of a sequence tai,jui`j“n together with, for every i, j, a specified morphisms dai,j Ñ δai´1,j`1 in Ai´1,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) the differential dC : Cn Ñ Cn´1 is defined as follows: for an object tai,ju of Cn, the component in Ak,l of its image under dC is given by fibpdak`1,l Ñ δak,l`1qrks The images of adjacent components of dCptai,juq under d and δ are canonically equivalent via the chosen equivalence dδ » δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The components, equipped with these equivalences then define an object of Cn´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that, in contrast to its classical counterpart, the definition of the categorical product totalization is not symmetric in each term (ignoring the differential) with respect to swapping the coordinates of the bicomplex, due to the appearance of the iterated directed pullback in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Our convention is to use the linear order of the coordinates of the bicomplex to determine the chosen direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The coproduct totalization C1 ‚ “ tot>pA‚,‚q is defined in analogy to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3): (1) for n P Zop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the category C1 n is the iterated directed pushout ¨ ¨ ¨ ñ> A2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n´1 A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n´1 ñ> A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n ñ> A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n`1 A´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n`1 ñ> A´1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='n`2 ¨ ¨ ¨ Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' via the universal property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a functor f : C1 n Ñ D corresponds to a collection of functors αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j : Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j Ñ D with i ` j “ n and natural transformations ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j fitting into diagrams of the following form: Ai`1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j Ai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j Ai`1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j´1 D d δ αi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j ηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j αi`1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='j´1 (2) We specify the composite of the differential dC1 : C1 n`1 Ñ C1 n with any functor f : C1 n Ñ D as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Specializing to f “ idC1n yields the differential dC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We describe f ˝ dC1 via the universal property of the iterated directed pushout in terms of functors βi`1,j “ fibpηi,jqri ` 1s: Ai`1,j ÝÑ D and natural transformations βi`1,j ˝ δ » βi,j`1 ˝ d (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) arising from the identities d2 » 0, δ2 » 0 and δ ˝ d » d ˝ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows from the fact that the natural transformation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is a natural equivalence, that d2 C1 » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 26 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' With some more effort, one can show that product and coproduct totaliza- tions form functors totˆ, tot> : Ch2pStkq Ñ ChpStkq , but we omit the details in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The most important example of a totalization of a bicomplex for us will be the following special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F : A‚ Ñ B‚ be a morphism of categorical complexes, depicted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' B2 B1 B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' F2 dA dA F1 F0 dB dB (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) We may interpret F as a bicomplex C‚,‚ with C0,‚ “ A‚ and C´1,‚ “ B‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We define the fiber of F as the product totalization FibpFq :“ totˆpC‚,‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly, we have FibpFqi “ Ai ñˆ Bi Bi`1, The differential d: FibpFqi Ñ FibpFqi´1 is given by pa, b, η: Fipaq Ñ dpbqq ÞÑ pdpaq, fibpηq, Fi´1dpaq » dFipaqq, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) where we note that d fibpηq » dFipaq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dually, we define the cofiber CofpFq of F as the coproduct totalization of the bicomplex (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4), with a degree shift of ´1, so that we have CofpFqi » Ai´1 ñ> Ai Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For a chain complex pA‚, dAq we define its shift Ar1s :“ CofpA Ñ 0q, which explicitly is given by Ar1si :“ Ai´1 and dAr1s i`1 :“ dA i r1s: Ai Ñ Ai´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) Dually we define Ar´1s :“ Fibp0 Ñ Aq and observe Ar1sr´1s » A » Ar´1sr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The cofiber (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' fiber) satisfy universal properties which can be formulated in terms of the notion of categorical homotopy introduced in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For example, the cofiber is the universal example of a categorical complex C‚ equipped with a morphism G: B‚ Ñ C‚ together with a categorical zero homotopy of the composite G ˝ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' See also §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We expect that this phenomenon will become relevant when trying to introduce a notion of derived category of categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In light of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9 one may anticipate that fiber and cofiber of a given morphism are equivalent up to a shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While this is not the case for a general morphism, remarkably, there is a natural class of morphisms for which the statement holds and which we introduce next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 27 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A chain map F : A‚ Ñ B‚ between categorical complexes is called right adjointable if each square Ai Ai´1 Bi Bi´1 dA Fi Fi´1 dB (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) is vertically right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12 (Beck–Chevalley).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F : A‚ Ñ B‚ be a right adjointable morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a canonical equivalence of categorical complexes χ‚ : CofpFq » ÝÑ FibpFqr1s where χi : Ai´1 ñ> Ai Bi ÝÑ Ai´1 ñˆ Bi´1 Bi, is the functor defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) corresponding to the commutative square (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Each functor χi is an equivalence by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The fact that χ‚ defines a morphism of complexes is verified by direct computation using the universal properties of the terms involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An alternative argument is also provided in Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Categorical multicomplexes Consider a categorical n-complex as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To iteratively define its totalization, we begin by introducing the partial totalization at adjacent coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let n ě 2 and A˚ : Zop,n Ñ Stk be a categorical n-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We consider A˚ as an object of FunpZop,n´2, Ch2pStkqq, valued in the bicomplexes describing chosen coordinates 1 ď i, i`1 ď n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using the functoriality of the totalization constructions of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we may partially product totalize the chosen coordinates to obtain a categorical pn ´ 1q-complex totˆ i,i`1pA˚q, defined as the image under the functor totˆ i,i`1 : FunpZop,n´2, Ch2pStkqq FunpZop,n´2,totˆq ÝÝÝÝÝÝÝÝÝÝÝÝÑ FunpZop,n´2, ChpStkqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The partial coproduct totalization tot> i,i`1pA˚q P Chn´1pStkq is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a categorical n-complex A˚, we obtain its product and coproduct totalizations by iterating the above partial totalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the moment, we fix one particular order for these partial totalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ be a categorical n-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We define its product totalization totˆpA˚q as the iterated partial totalization totˆ 1,2 ˝ totˆ 2,3 ˝ ¨ ¨ ¨ ˝ totˆ n´1,npA˚q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly, the coproduct totalization tot>pA˚q of A˚ is defined as tot> 1,2 ˝ tot> 2,3 ˝ ¨ ¨ ¨ ˝ tot> n´1,npA˚q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 28 With our conventions, both the product and coproduct totalizations of a categorical n- cube lie in degrees n to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Suppose that q : In Ñ Stk is a constant cube with value A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By an inductive argument, we obtain the following description of the product totalization of q: Cn`1 “ A, the category Cn´k consists of diagrams X : Funprks, rnsq Ñ A mapping non-injective maps rks Ñ rns to zero objects, the differential d: Cn´k Ñ Cn´k´1 is given by associating to a diagram X : Funprks, rnsq Ñ A the diagram X1 : Funprk ` 1s, rnsq Ñ A where X1pσq is the total fiber of the complex Xpσ ˝ Bnq Ñ Xpσ ˝ Bn´1q Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ Xpσ ˝ B0q in A, where X1 is formally constructed from X by means of Kan extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, we observe that the n-cells of this complex are representations in A of higher Dynkin type A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that this complex is an augmented variant of categorical cochain complex C‚p∆, Aq from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 (normalized slightly differently there, since the differentials are given by taking the total cofiber instead of the fiber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The example illustrates the general phenomenon that, even for simple cubical diagrams, the terms of the totalization turn out to be rather interesting stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is somewhat in contrast to the usual totalization of cubical diagrams of abelian groups, where the terms of the totalization are simply direct sums of the terms of the cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The coproduct totalization of q admits a dual description, we leave the analogous detailed description to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows from the Beck–Chevalley property of the cube q, that the product and coproduct totalizations are in fact equivalent as categorical complexes (repeatedly apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The totalization of q is also an example of a categorical Koszul complex, discussed in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' See Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8 for categorical Koszul duality and its implications when applied to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We expect a fully coherent associativity statement for the totalization of multicomplexes, meaning that the totalization does not depend on the order of the totaliza- tions of the adjacent coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A systematic analysis of the associativity of the totalizaton would however go beyond the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Instead, we sketch how to prove a partial result in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 below, needed for concrete applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The totalization of a cube depends on a further choice, namely the given total order of the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We leave it as an interesting problem to determine under which conditions the totalization does not depend on this order, up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 29 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ be a categorical n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there exist equivalences of categor- ical complexes totˆpA˚q » totˆ 1,2 ˝ ¨ ¨ ¨ ˝ totˆ 1,2pA˚q and tot>pA˚q » tot> 1,2 ˝ ¨ ¨ ¨ ˝ tot> 1,2pA˚q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Informally, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 states that iteratively totalizing the last two coordinates is equivalent to iteratively totalizing the first two coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, we begin with the case of categorical 3-complexes of the form I ˆZop ˆ I Ñ Stk, where I “ r1sop Ă Zop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let FunChpI ˆZop ˆ I, Stkq Ă FunpI ˆZop ˆ I, Stkq be the full subcategory spanned by categorical 3-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There exists a natural equivalence between the functors totˆ 1,2 ˝ totˆ 1,2 : FunChpI ˆZop ˆ I, Stkq Ñ ChpStkq and totˆ 1,2 ˝ totˆ 2,3 : FunChpI ˆZop ˆ I, Stkq Ñ ChpStkq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can depict a part of A˚ as follows, A1,i`1,1 A1,i`1,0 A0,i`1,1 A0,i`1,0 A1,i,1 A1,i,0 A0,i,1 A0,i,0 A1,i´1,1 A1,i´1,0 A0,i´1,1 A0,i´1,0 and relabel this part for better readability as follows: A2 B1 B4 C3 B2 C1 C4 D3 C2 D1 D4 E3 We have by definition totˆ 1,2 ˝ totˆ 1,2pA˚qi`1 “ pC1 ñˆ D3 C3q ñˆ pD1 ñ ˆ E3 D3q pC2 ñˆ D4 C4q 30 and totˆ 1,2 ˝ totˆ 2,3pA˚qi`1 “ pC1 ñˆ D1 C2q ñˆ pD3 ñ ˆ E3 D4q pC3 ñˆ D3 C4q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Unraveling the definition, we find that both of these 8-categories satisfy the following uni- versal property: a functor from X P Stk corresponds to functors Fi : X Ñ Cj in Stk for all 1 ď j ď 4, natural transformations C1 X D1 C2 α F1 F2 C1 X D3 C3 γ F1r´1s F3 C2 X D4 C4 β F2 F4 C3 X D3 C4 δ F3 F4 and a null-homotopy of δ ˝ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This null-homotopy induces a natural transformation ν from the functor X F1 ÝÑ C1 Ñ D3 to cofpδq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A natural equivalence between the two composite natural transformations C1 X D3 E3 C4 D3 ν F1 cofpδq F4 » C1 X C2 D1 E3 C4 D4 α F1 F4 F2 β » between the functors X F1 ÝÑ C1 Ñ E3 and X F4 ÝÑ C4 Ñ E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lower natural equivalence in the left diagram arises from the fact that the functor C3 Ñ D3 Ñ E3 is zero, since d2 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One checks that the differentials of both complexes are identified in terms of degreewise equivalences arising from the matching universal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' All arising suspensions can be dealt with by including suitable equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3), we can express all directed pullbacks appearing above in terms of 8-categorical limits and the above equivalences in terms of functorial equivalences between these limits, yielding the desired equivalence between totˆ 1,2 ˝ totˆ 1,2 and totˆ 1,2 ˝ totˆ 2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 31 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We only prove the statement about the product totalization, the case of the coproduct totalization is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Repeatedly applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6, we find equivalences for all 1 ă j ă n totˆ j´1,j ˝ totˆ j,j`1 ¨ ¨ ¨ ˝ totˆ j,j`1pA˚q » totˆ j´1,j ˝ totˆ j´1,j ˝ totˆ j`1,j`2 ˝ ¨ ¨ ¨ ˝ totˆ j`1,j`2pA˚q » .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' » totˆ j´1,j ˝ ¨ ¨ ¨ ˝ totˆ j´1,j ˝ totˆ j´1,jpA˚q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Combining these equivalences, we have totˆ 1,2 ˝ ¨ ¨ ¨ ˝ totˆ n´1,npA˚q » totˆ 1,2 ˝ ¨ ¨ ¨ ˝ totˆ n´3,n´2 ˝ totˆ n´2,n´1 ˝ totˆ n´2,n´1pA˚q » totˆ 1,2 ˝ ¨ ¨ ¨ ˝ totˆ n´3,n´2 ˝ totˆ n´3,n´2 ˝ totˆ n´3,n´2pA˚q » .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' » totˆ 1,2 ˝ ¨ ¨ ¨ ˝ totˆ 1,2pA˚q , as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 Beck-Chevalley conditions and spherical functors Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a commutate square in Stk: A B C D G F F 1 G1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) We say that the square is (1) horizontally right adjointable, or simply right adjointable, if G and G1 admit k-linear right adjoints GR and pG1qR and the natural transformation F ˝ GR u ˝F˝GR ùùùùùùñ pG1qR ˝ G1 ˝ F ˝ GR » ùñ pG1qR ˝ F 1 ˝ G ˝ GR pG1qR˝F 1˝cu ùùùùùùùùñ pG1qR ˝ F 1 is a natural equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) horizontally left adjointable, if G and G1 admit k-linear left adjoints GL and pG1qL and the natural transformation pG1qL ˝F 1 pG1qL˝F 1˝u ùùùùùùùñ pG1qL ˝F 1 ˝G˝GL » ùñ pG1qL ˝G1 ˝F ˝GL cu˝F˝GL ùùùùùùñ F ˝GL (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) is a natural equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) vertically left adjointable, if F and F 1 admit k-linear right adjoints F R and pF 1qR and the natural transformation G˝F R u ˝G˝F R ùùùùùùñ pF 1qR˝F 1˝G˝F R » ùñ pF 1qR˝G1˝F ˝F R pF 1qR˝G1˝cu ùùùùùùùùñ pF 1qR˝G1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) is a natural equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 32 (4) vertically right adjointable, if F and F 1 admit k-linear left adjoints F L and pF 1qL and the natural transformation pF 1qL ˝ G1 pF 1qL˝G1˝u ùùùùùùùñ pF 1qL ˝ G1 ˝ F ˝ F L » ùñ pF 1qL ˝ F 1 ˝ G ˝ F L cu˝G˝F L ùùùùùùñ G ˝ F L is a natural equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If F and F 1 admit right adjoints and G and G1 admit left adjoints, then the natural transformations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) are adjoint to another and conditions (2) and (3) from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 hence equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An analogous statement holds for conditions (1) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remarkably, all adjointability conditions are equivalent if the square is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a commutative square in Stk as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) and suppose that all functors F, F 1, G, G1 are spherical functors, see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then all four conditions of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To begin with, we note that spherical functors admit all repeated left and right adjoints, see [DKSS21, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It thus follows that conditions (2) and (3) of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 are equivalent, as are conditions (1) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It remains to show that conditions (1) and (2) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let T be the cotwist functor of G % GR and T 1 the twist functor of G1 % pG1qR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then we have GL ˝ T » GR and T 1 ˝ pG1qL » pG1qR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We find the following commutative diagram, 0 T 1 ˝ pG1qL ˝ F 1 ˝ T 0 F ˝ GR pG1qR ˝ G1 ˝ F ˝ GR pG1qR ˝ F 1 0 T 1 ˝ F ˝ GL ˝ T 0 T 1˝pG1qL˝F 1˝u1˝T ˝ u ˝F˝GR ˝ pG1qR˝F 1˝cu T 1˝cu1˝F˝GL˝T omitting some identifications in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here, u is the unit of G1 % pG1qR and cu the counit of G % GR, u1 the unit of GL % G and cu1 the counit of pG1qL % G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The fact that the upper right square and the lower left square are biCartesian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' both pullback and pushout, follows from the general properties of units and counit of spherical adjunctions, see for instance Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10 in [Chr22c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By the pasting laws for biCartesian squares, we find that the horizontal middle composite is a natural equivalence if and only if the vertical middle composite is a natural equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The equivalence of conditions (1) and (2) now follows from the fact that T and T 1 are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A categorical n-cube is called Beck-Chevalley if each rectilinear face is both horizontally and vertically right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 33 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Suppose that A˚ is a Beck-Chevalley categorical n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then each partial product totalization totˆ i,i`1 ˝ ¨ ¨ ¨ ˝ totˆ n´1,npA˚q and partial coproduct totalization tot> i,i`1 ˝ ¨ ¨ ¨ ˝ tot> n´1,npA˚q with 1 ď i ď n ´ 1 is a Beck-Chevalley pn ´ iq-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We only show that totˆ n´1,npA˚q˚ : Iˆn´2 ˆr2sop Ñ Stk is Beck-Chevalley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A similar argument applies to tot> n´1,npA˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The general case is proven analogously, by using that the a-times repeated partial totalization consists of a stacked cubes, to each of which an analogous argument as below applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let J P In´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For l “ 0, 2, we have totˆ n´1,npA˚qpJ,lq “ ApJ,l,lq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If instead l “ 1, let J1 “ pJ, 1, 0q, J2 “ pJ, 0, 1q , J0 “ pJ, 0, 0q P In .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then we have totˆ n´1,npA˚qpJ,1q » AJ1 ñˆ AJ0 AJ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a rectilinear face f : I2 Ñ In´2 ˆr2sop with fp1, 1q differing from fp0, 0q by sub- tracting 1 in the entries i and j with 1 ď i, j ď n ´ 1 and i ‰ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There are a few cases to distinguish, in which similar arguments apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We highlight two such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If i, j ‰ n ´ 1 and fp0, 0qn´1 “ 0, 2 P r2sop, then the face of totˆ n´1,npA˚q is equivalent to a corresponding face of A˚ and hence both vertically and horizontally right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If i, j ‰ n ´ 1 and fp0, 0qn´1 “ 1 P r2sop, then the face of totˆ n´1,npA˚q is of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Afp0,0q1 ñˆ Afp0,0q0 Afp0,0q2 Afp1,0q1 ñˆ Afp1,0q0 Afp1,0q2 Afp0,1q1 ñˆ Afp0,1q0 Afp0,1q2 Afp1,1q1 ñˆ Afp1,1q0 Afp1,1q2 This diagram arises from the functoriality of directed pullbacks applied to a diagram con- taining two Beck-Chevalley faces of A˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The adjointability properties now follow from the adjointability properties of these two squares and the observation that the right adjoints of the morphisms in the above diagrams are computed componentwise, see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 Mapping complexes Given complexes A‚, B‚ of abelian groups, there is an associated mapping complex MappA‚, B‚q‚ with MappA‚, B‚qn “ ź iPZ HompAi, Bn`iq 34 and differential given by the formula dptfiuqk “ dB ˝ fk ´ p´1qk´1fk´1 ˝ dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now introduce a categorical variant of this construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚, B‚ P ChpStkq be categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The oplax categorical mapping complex between A‚ and B‚ is defined as the product totalization Mapoplax ‚ pA‚, B‚q :“ totˆ StkpA´‚, B‚q (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) of the bicomplex obtained by applying Stkp´, ´q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicity: an object of Mapoplax n pA, Bq consists of a sequence tFiuiPZ of k-linear functors Fi : Ai Ñ Bn`i equipped with natural transformations φi : Fi´1 ˝ dA ñ dB ˝ Fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the differential Mapoplax n pA‚, B‚q Ñ Mapoplax n´1 pA‚, B‚q associates to this datum the se- quence tGi “ fibpφiqr´isu (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) of functors Gi : Ai Ñ Bi`n´1 together with the natural equivalences Gi´1dA “ fibpFi´2dA Ñ dBFi´1qr´i ` 1sdA (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) » dBFi´1dAr´is » dB fibpFi´1dA Ñ dBFiqr´is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) “ dBGi (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) Note that the “categorical signs”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', the powers of the suspension r1s, do not agree with the usual Koszul signs upon passing to K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since we will not use categorical mapping com- plex systematically in this work, we will not attempt to address this (purely conventional) discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More pictorially, we may interpret Mapoplax n pA‚, B‚q as the stable 8-category of diagrams in Stk of the form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 A´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Bn`2 Bn`1 Bn Bn´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' d d F2 d F1 d F0 d F´1 d d φ1 d φ1 d φ´1 d (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) which may be called an oplax morphisms from A‚ to Bn`‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The differential then associates to this oplax morphism the (strict) morphism .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 A´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Bn`1 Bn Bn´1 Bn´2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' d d G2 d G1 d G0 d G´1 d d d d d (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) with components Gi “ fibpφiqr´is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 35 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Implicit in the product totalization is a choice of ordering of the two directions in the bicomplex StkpA´‚, B‚q which corresponds to the choice of direction of the 2-cells appearing in the chain maps (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Flipping this choice yields the notion of lax chain morphisms and the corresponding lax mapping complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' See §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8 for a more systematic discussion of lax and oplax chain maps and an alternative construction of the lax/oplax mapping complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The classical Dold-Kan correspondence offers a means of turning the homological data captured by a connective chain complex C‚ into homotopical data described by the Kan complex underlying the associated Dold-Kan nerve of C‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This transformation is of particular interest when applied to mapping complexes where it leads to a model for the derived category of complexes as a topological category (and further an 8-category by passing to coherent nerves, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [Lur17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may therefore hope to gain insights as to how the derived category of categorical com- plexes should be defined by investigating the “homotopical data” captured by the categorified Dold–Kan nerve of a categorical complex, and in particular, of the categorical mapping com- plex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ and B‚ be categorical complexes and let M‚ “ τě0 Mapoplax ‚ pA‚, B‚q P ChpStq be the categorical mapping complex, truncated below and with M0 “ kerpd0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2), that the categorified Dold–Kan nerve X‚ “ NDKpM‚q is a 2-simplicial stable 8-category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a functor X‚ : ∆ ÝÑ St of p8, 2q-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We describe its low-dimensional simplices: A vertex corresponds to a morphism F : A‚ Ñ B‚ of categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An edge corresponds to the datum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a natural transformation ηp01q : F p0q ñ F p1q, of (strict) morphisms F p0q, F p1q : A‚ Ñ B‚ of categorical complexes, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' an oplax morphism Hp01q : A‚ Ñ B‚`1 which we refer to as a categorical homotopy, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' an exact triangle dHp01q F p0q 0 F p1q ηp01q A 2-simplex corresponds to the datum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a diagram F p0q F p1q F p2q (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) of natural transformations of morphisms A‚ Ñ B‚, 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a diagram (not necessarily bicartesian) Hp01q Hp02q 0 Hp12q (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) of oplax morphisms A‚ Ñ B‚`1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' an oplax morphism Hp012q : A‚ Ñ B‚`2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' an extension of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) and d(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) to a diagram dHp01q dHp02q F p0q 0 dHp12q F p1q 0 F p2q of morphisms A‚ Ñ B‚ with all squares biCartesian, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' an extension of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) to a biCartesian cube dHp012q Hp01q 0 Hp02q 0 0 0 Hp12q of oplax morphisms A‚ Ñ B‚`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As we observe from this description, the 2-simplicial object M‚ carries meaningful data, such as a reasonable notion of “categorical homotopy” between morphisms of categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' However, when analyzing the structural properties of M‚ there are many open questions that remain to be investigated in order to understand the “higher homotopical con- tent” of M‚ in analogy to its decategorified counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For example, the ordinary Dold-Kan nerve is a Kan complex (as every simplicial abelian group) so that it carries intrinsic homotopi- cal meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The categorified Dold–Kan nerve does indeed have certain categorified lifting properties, but these are much weaker so that, in particular, it is not possible to even “com- pose homotopies” in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We hope to analyze this context more systematically in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 37 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7 Koszul complexes and categorical Koszul duality Let R be a commutative ring and let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , λn be elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may then form the Koszul complex as the tensor product Kpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', λnq “ n â i“1 pR λi ÝÑ Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall the classical “self-duality” of the Koszul complex: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is a (canonical) isomorphism of complexes Kpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', λnq_ n´‚ – Kpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', λnq‚ where p´q_ “ HomRp´, Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this section, we explain how the notion of a Koszul complex, as well as the self-duality statement admit a variant for categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let k be a commutative ring spectrum and Modk the category of k-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given an object L P Modk and A P Stk, we obtain a functor A ´bkL ÝÑ A which is functorial in A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', it defines a natural transformation on the identity functor of Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We define the categorical two-term complex KpLq :“ Modk ´bkL ÝÑ Modk, concentrated in degrees 0 and 1, and further, for k-module spectra L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Ln, the categorical complex KpL1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Lnq :“ tot>pKpL1q bk KpL2q bk ¨ ¨ ¨ bk KpLnqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The main result of this section is the following theorem, which can be regarded as a categorical variant of the classical “self-duality” of the Koszul complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 (Categorical Koszul duality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Ln be dualizable k-module spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a canonical equivalence of categorical complexes KpL1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Lnq_ n´‚ » KpLn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', L1q‚ where p´q_ “ Stkp´, Modkq denotes the dual with respect to the symmetric monoidal structure on Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We prove the statement by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For n “ 1, the identification StkpModk, Modkq » Modk via the evaluation functor F ÞÑ Fpkq, extends to an equivalence between the complex KpL1q_ 1´‚ and KpL1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now assume given the equivalence KpL1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Ln´1q_ n´1´‚ » KpLn´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', L1q 38 and write A‚ :“ KpL1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Ln´1q and A1 ‚ :“ KpLn´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', L1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have KpL1, L2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Lnq “ tot>pA‚ b KpLnqq We compute tot>pA‚ b KpLnqq_ n´‚ » pA_ n´i ñˆ A_ n´i A_ n´i´1qiPZ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) » pA1 i´1 ñˆ A1 i´1 A1 iqiPZ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) » pA1 i´1 ñ> A1 i A1 iqiPZ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) » tot>pKpLnq b A1 ‚q (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) » KpLn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', L1q (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) holds since the dual of tot> is equivalent to totˆ of the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) holds by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12 applied to the morphism given by tensoring with Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This has an adjoint given by tensoring with L_ n (since we assumed the Li to be dualizable) which is again central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) holds by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We give an application of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Set Li “ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the categorical Koszul complex can be identified with the totalization of a constant cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, the categorical Koszul duality amounts to an equivalence between the category of diagrams X : Funprks, rnsq Ñ Modk mapping every non-injective map τ : rks Ñ rns to a zero object in Modk, and the category of diagrams Y : Funprn ´ ks, rnsq Ñ Modk mapping every non-injective map τ : rn ´ ks Ñ rns to a zero object in Modk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The duality described in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8 was first obtained by Beckert [Bec18] using the theory of derivators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A geometric proof based on Fukaya categories was given in [DJL21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The novelty in our proof based on categorical Koszul complexes is the inductive nature which does not feature in the previous proofs (and seems to make the argument both simpler and more conceptual).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 39 5 Spherical complexes and perverse schobers Given an adjunction F : A Ø B :G of stable 8-categories, we define the twist functor TA : A Ñ A as the cofiber of the unit u : idA Ñ FG in the stable 8-category FunpA, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the cotwist functor TB : B Ñ B as the fiber of the counit c : FG Ñ idB in the stable 8-category FunpB, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 ([AL17, DKSS21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An adjunction F % G is called spherical if both TA and TB are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A functor F : A Ñ B of stable 8-categories is called spherical if it admits a right adjoint G and the adjunction F % G is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Spherical adjunctions were originally conceived by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Anno, to describe “family versions” of the spherical objects introduced by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Seidel and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thomas in [ST01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Natural examples can be found among the various categorical structures that arise within the context of Kont- sevich’s homological mirror symmetry program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More recently, it has been proposed in [KS14] to interpret spherical functors (and, more generally, suitable diagram categories built from spherical functors) as categorified analogues of perverse sheaves (referred to as perverse schobers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is motivated by the observation that the abelian category of perverse sheaves on the complex plane C, with stratification given by the origin t0u and its complement, is classically known to be equivalent to the category of diagrams Φ Ψ, f g (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) of vector spaces Φ and Ψ with id ´fg and id ´gf invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While a fully satisfying intrinsic definition of perverse schobers still remains open in gen- eral and work in progress in two dimensions (see [DKS20, DKSS21] for partial results), in many situations one can guess ad-hoc definitions, based on diagrammatic descriptions of the respective categories of perverse sheaves such as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting notions of perverse schobers are not intrinsic and depend on auxiliary choices – however, it is still worthwhile to construct interesting examples and study applications, since these descriptions will hopefully become part of a more intrinsic theory in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In our current context of categorical complexes, another instance of such an ad-hoc notion of perverse schober arises from a diagrammatic description of perverse sheaves on Cn with respect to the stratification given by the hyperplane inclusions t0u Ă C Ă C2 Ă C3 Ă ¨ ¨ ¨ Ă Cn and their complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting notion of perverse schober in this context is a spherical complex (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3): a categorical complex of length n all of whose differentials are spherical functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 we also consider the further stratification of Cn given by the coordinate hyperplanes, their intersections and complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case, the resulting notion of per- verse schober is a Beck-Chevalley categorical n-cube, whose edges are described by spherical functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We call such a categorical cube a spherical cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Spherical adjunctions and perverse schobers on C As mentioned above, the abelian category of perverse sheaves on C with respect to the strat- ification given by t0u and its complement is equivalent to the category of diagrams Φ Ψ, f g of vector spaces Φ and Ψ with id ´fg and id ´gf invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Geometrically, the vector spaces Φ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ψ) correspond to the spaces of vanishing cycles (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' nearby cycles) associated to a given perverse sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is instructive to investigate how this linear algebraic data describes the perverse sheaf, when considering it as an object of the derived category of constructible sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To this end, we may also interpret this latter category as the category of constructible sheaves on C valued in the 8-category of cochain complexes1 of vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Such a sheaf F of cochain complexes may then be described by assembling the linear algebraic data into the diagram Ψ Φ Ψ 0 g id f öT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) where the stalk F0 of F at 0 P C is given by the cochain complex Ψ Φ g concentrated in degrees ´1 and 0, the stalk F1 of F at 1 P C is given by the cochain complex Ψ 0 concentrated in degree ´1, the restriction map res: F0 Ñ F1 corresponds to the morphism represented by the commutative square in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2), the monodromy of the stalk F1 is induced by the automorphism T “ id ´fg of Ψ, and finally, we may interpret the map f : Φ Ñ Ψ as a homotopy, as depicted in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2), between res and the composite T ˝ res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This homotopy expresses the compatibility relations arising from the C˚-family of restriction maps F0 Ñ Fp, with p P C˚, of which we only need to remember res: F0 Ñ F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 1When discussing perverse sheaves, we use cochain complexes as this is the standard convention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In most other places in this paper, we however use chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 41 It is further interesting to note that we may apply Verdier duality to describe F equivalently as a constructible cosheaf F_ valued in cochain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This cosheaf then admits the following analogous description: Φ Ψ 0 Ψ f öT id g (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) the costalk F_ 0 of F_ at 0 P C is given by the cochain complex Φ Ψ f concentrated in degrees 0 and 1, the costalk F_ 1 of F_ at 1 P C is given by the cochain complex 0 Ψ concentrated in degree 1, the corestriction map cores: F_ 1 Ñ F_ 0 corresponds to the commutative square in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3), the monodromy of the stalk F_ 1 is induced by the automorphism T “ id ´fg of Ψ, and, we may interpret the map g: Ψ Ñ Φ as a homotopy, as depicted in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) between cores and the composite cores ˝T, completing the data needed to define a constructible cosheaf valued in cochain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that the invertibility of id ´fg is in fact equivalent to the invertibility of id ´gf, ex- plaining why this condition does not appear in an apparent way in the description of F or F_.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The datum of a spherical adjunction F : A ÐÑ B :G can be interpreted in a fashion analogous to Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In other words, we may repackage the data into the diagram B A B 0 G id F öT where the exact triangles TB Ñ id Ñ FG exhibit F as a categorical homotopy, as defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the categorical context, however, it is not the case, that the invertibility of TB implies the invertibility of TA so that this perspective only partially characterizes spherical adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Nevertheless, it motivates our proposal to interpret spherical functors as 2-term categorical complexes and use this intuition to generalize to spherical complexes as explained below in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Barbacovi’s Theorem and its geometric interpretation A remarkable result by Ed Segal ([Seg18]) says that every autoequivalence on a category arises as a spherical twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, the composition of the spherical twists associated to a pair of spherical adjunctions with the same target, must again be a spherical twist (also cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [Chr22c] for an 8-categorical generalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The main result of [Bar20] provides an explicit description of a spherical adjunction that describes this composite twist in terms of the given spherical adjunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this section, we explain how Barbacovi’s construction fits into our context, provide a geometric interpretation in terms of perverse schobers, and finally prove a generalization to stable 8-categories which we will need below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To begin with, consider the complex plane C equipped the stratification given by two points tx, x1u, say x “ i and x1 “ ´i, and their complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By choosing an arc connecting the two points, we may identify the category of perverse sheaves with the category of diagrams Φ Ψ Φ1 f g1 g f1 of vector spaces such that both pf, gq and pf1, g1q satisfy the conditions from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Geometri- cally, the automorphisms T “ id ´fg and T 1 “ id ´f1g1 can be interpreted as the monodromy transformations on the space of nearby cycles Ψ corresponding to loops around the points x and x1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As mentioned above, the vector spaces Φ and Φ1 can be interpreted as the local vanishing cycles of the given perverse sheaf F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We define the space of global vanishing cycles ΣpFq as the fiber fibpRΓpC, Fq Ñ RΓptℜz ă 0u, Fqq – Φ ‘ Φ1 of the restriction map along the inclusion of the half-plane tℜz ă 0u Ă C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This vector space comes equipped with natural maps ΣpFq Ψ pf,f1q pgT 1,g1q (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) and we compute id ´pf, f1q ˆgT 1 g1 ˙ “ id ´fgT 1 ´ f1g1 “ pid ´fgqpid ´f1g1q “ TT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore, the datum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) defines a perverse sheaf on C with respect to the startification given by t0u and its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We refer to this sheaf as the amalgamate of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This amalgamate likely describes the pushforward sheaf of F under the endomorphism of C, which contracts a disc containing x and x1 to 0, but we do not verify this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 43 As we will now explain, these geometric considerations can be lifted to analogous con- structions for perverse schobers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To begin, a perverse schober on C, equipped with the above stratification and auxiliary choices, can be interpreted as a pair A B A1 F G1 G F 1 of spherical adjunctions as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a categorical lift of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), we define the functor F ñˆ B F 1 : A ñˆ B A1 Ñ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor F ñˆ B F 1 is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The suspension of the cotwist functor of the adjunction F ñˆ B F 1 % ˆ F ñˆ B F 1 ˙R is equivalent to the composite of the cotwist functors of the adjunctions F 1 % G1 and F % G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the following, we informally describe how Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 can be proven using further ideas related to perverse schobers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The full and somewhat technical proof implementing these ideas is given in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a perverse sheaf F on pC, 0q, instead of considering the vector spaces of vanishing and nearby cycles, we can equally well describe F in terms of its nearby cycles and its, as it turns out, vector space of global sections Ψ2 with support on R Ă C or even its vector space of global sections Ψn with support on any graph embedded in C with a single n-valent vertex at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting description states that the category of perverse sheaves on pC, 0q is equivalent to the category of diagrams of vector spaces fi : Φn ÐÑ Ψi :gi , 1 ď i ď n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) satisfying that figi “ idΨi, fi`1gi is invertible and fjgi “ 0 for j ‰ i, i`1, with i, j considered modulo n, see [KS16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The vector spaces Ψi, with 1 ď i ď n, are all equivalent and may be chosen for concreteness as the stalks of the n-th roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An ad-hoc categorification of this local description is described in [Chr22a], based on Waldhausen’s relative S‚-construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using these local descriptions, we can describe perverse sheaves or perverse schobers on any surface S with 0-dimensional strata P and non-empty boundary by choosing a spanning graph G Ă S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The inclusion of G into S is required to be a homotopy equivalence and each stratum p P P is required to be the image of a vertex of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A perverse sheaf or perverse schober on S can be encoded as a constructible sheaf and cosheaf on G, which restricts at each vertex of G to a diagram as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) or its categorification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The global sections of the constructible sheaf on G describe the first cohomology of the derived global sections with support on G of the perverse sheaf or perverse schober on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we consider the perverse schober F on pC, tx, x1uq defined by the pair of spherical adjunctions F % G and F 1 % G1 and describe it as a constructible sheaf on the ribbon graph G with two vertices at x, x1, depicted as follows (in blue): 44 x x1 G q p C The points p and q lie left or right of x and x1 as indicated, but are otherwise arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote the global sections of F by RΓ1pG, Fq P Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Restriction to the two points p, q on G defines a functor pevp, evqq: RΓ1pG, Fq Ñ Bˆ2 , The right adjoint is denoted pevpR, evqRq and the left adjoint is denoted pevpL, evqLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These functors turn out to be much easier to handle than the functor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is because the functors evpR, evqR, evpL, evqL are all fully faithful and, as one can compute, there are equivalences evqR » evpL ˝ T and evpR » evqL ˝ T 1, with T the cotwist functor of F % G and T 1 the cotwist functor of F 1 % G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From these facts, it also follows that pevp, evqq is a spherical functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To show the sphericalness of the functor F ñˆ B F 1, we use the fact that this is equivalent to the assertion that there exist a stable 8-category with a 4-periodic semiorthogonal decompo- sition pA ñˆ B A1, Bq with gluing functor F ñˆ B F 1, see [HLS16, DKSS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We find this 4-periodic semiorthogonal decompositions in the 8-category RΓ1pG, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here, A ñˆ B A1 Ă RΓ1pG, Fq arises as the full subcategory of global sections with support on G which vanish at p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the kernel of evp : RΓ1pG, Fq Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category B Ă RΓ1pG, Fq arises as the image of evpR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 is a special case of a more general phenomenon exhibited by perverse schobers on a stratified surface S with non-empty boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We again choose a spanning graph G of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By a G-parametrized perverse schober, we mean a constructible sheaf of stable 8-categories on G encoding a perverse schober on S, as explained above and defined in [Chr22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given an edge e of G, we denote by Fpeq the stalk of F at any point on that edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F be a G-parametrized perverse schober on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let EB be the set of external edges of G and ź ePEB eve : RΓ1pG, Fq Ñ ź ePEB Fpeq the restriction functor of global sections with support on G to stalks at these boundary edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor ś ePEB eve is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the study of partially wrapped Fukaya categories, the functor ś ePEB eve is also called the cap functor, it is adjoint to the Orlov or cup fuctor, see for instance [Syl19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We sketch in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 how the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 generalizes to a proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Spherical complexes and perverse schobers on Cn Let n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We begin with a linear algebraic description of classical perverse sheaves on Cn with respect to the stratification given by the hyperplane inclusions t0u Ă C Ă C2 Ă C3 Ă ¨ ¨ ¨ Ă Cn , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) each setting the last coordinate to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The category of perverse sheaves on Cn with respect to the stratification (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is equivalent to the category of diagrams A0 A1 A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An´1 An d δ d δ d δ (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) of vector spaces subject to the conditions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' d2 “ 0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' δ2 “ 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' for every 0 ď k ď n, the endomorphisms id ´dδ and id ´δd of Ak are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The stratification of Cn given by the coordinate hyperplanes, their intersections and complements is a refinement of the stratification (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 below, a perverse sheaf on Cn with the former stratification amounts to a certain cube of linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One readily finds that a perverse sheaf also defines a perverse sheaf on the latter stratification of Cn if and only if all entries of this cube vanish, except for a sequence of entries as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Alternatively, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 can also be directly deduced from iterated application of Beilinson’s gluing formula for categories of perverse sheaves ([Bei87]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We explain how to interpret the linear algebraic data from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 geometrically, in terms of the corresponding perverse sheaf as a constructible sheaf F valued in the 8-category of cochain complexes of vector spaces, in analogy to Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, we assemble the linear algebraic data into a diagram An An´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 öT1 An An´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 0 öT2 An An´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' öTn´1 An An´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 0 0 0 öTn An 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 0 0 0 δ d δ δ d d δ d δ d δ δ d (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) 46 where the rows of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) correspond to the stalks of Fi at the points x0, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', xn P Cn with pxiqj “ # 1 for j ď i 0 for j ą i where the part of the complex depicted by the corresponding row of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) is concen- trated in degrees ´n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , ´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus, Ai lies in degree ´i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the restriction maps resi : Fi´1 Ñ Fi, 1 ď i ď n, correspond to the commutative rectangles between the corresponding rows in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) where all vertical maps are either 0 or id, we note without proof that the monodromies of each stalk about the “previous” hy- perplane in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) are induced by the chain automorphism Ti “ id‚ ´d‚δ‚ ´ δ‚d‚ of Fi, we may interpret the maps d: A˚ Ñ A˚`1 as defining homotopies, as depicted in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5), between resi and the composite Ti ˝ resi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is interesting to note here that the invertibility of the map T “ id ´dδ´δd is equivalent to the invertibility of the maps id ´dδ and id ´δd by virtue of the formula id ´dδ ´ δd “ pid ´dδqpid ´δdq “ pid ´δdqpid ´dδq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) We leave the Verdier dual interpretation of the linear algebraic data in analogy to Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 to the reader, only noting that the roles of d and δ get swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This gives an explanation why both d and δ need to square to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 shows that this data gives a full description of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Inspired by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2, we introduce the following concept of spherical categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A categorical complex A‚ P ChpStkq is called spherical if the differential d : Ai Ñ Ai´1 is a spherical functor for all i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Spherical categorical complexes concentrated in degrees n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0 can thus be regarded as perverse schobers on Cn with respect to the stratification (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We conclude this section with some comments on the twist functors of spherical categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We fix a categorical complex A‚ P ChpStkq and i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The adjunctions di % δi and di`1 % δi`1 give rise to the unit u : idAi Ñ δidi and the counit c : di`1δi`1 Ñ idAi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can compose these two natural transformations to obtain the following two commutative diagrams in StkpAi, Aiq: di`1δi`1 idAi di`1δi`1δidi δidi cu u di`1δi`1 idAi δididi`1δi`1 δidi cu u (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) Note that di`1δi`1δidi » 0 and δididi`1δi`1 » 0 since δ2 » 0 and d2 » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 47 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let Ti be the twist of di % δi and T 1 i the cotwist of di`1 % δi`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The totalization of the left square in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) is equivalent to T 1 iTi and the totalization of the right square in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) is equivalent to TiT 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The totalization of the left diagram in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) categorifies the expression id ´dδ ´ δd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since the totalization is T 1 iTi, we find a categorification of the expression id ´T 1 iTi “ dδ ` δd, expressing that δ describes a homotopy between the chain maps T 1 ‚T‚ and idA‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly, the right diagram in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) expresses that δ is a homotopy between idA‚ and T‚T 1 ‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following lemma shows that T‚T 1 ‚ » T 1 ‚T‚, categorifying the identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Suppose that A‚ is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The two diagrams in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, the resulting equivalence of their totalizations shows that the autoequivalences Ti and T 1 i commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that composition with an exact functor defines an exact functor between sta- ble 8-categories of functors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' we find that the two commutative squares in StkpAi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Aiq can be extended to commutative diagrams where the horizontal sequences are fiber and cofiber sequences: T 1 ir´1s di`1δi`1 idAi T 1 iδidir´1s di`1δi`1δidi δidi T 1 ir´1s u cu u T 1 ir´1s di`1δi`1 idAi δidiT 1 ir´1s δididi`1δi`1 δidi u T 1 ir´1s cu u We begin by showing that the morphism T 1 ir´1su and uT 1 ir´1s in the upper diagrams are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the adjunction T 1 iδi % dipT 1 iq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Its unit is given by T 1 iupT 1 iq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that T 1 iδi » fibpdi`1δi`1δi Ñ δiq and δ2 » 0, we get T 1 iδi » δir´1s and hence also dipT 1 iq´1 » dir1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows that the unit T 1 iupT 1 iq´1 of T 1 iδi % dipT 1 iq´1 is equivalent to the unit of δir´1s % dir1s, which is equivalent to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We thus obtain T 1 ir´1su » uT 1 ir´1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since di`1δi`1δidi » δididi`1δi`1 » 0, it is now clear that the upper left squares in the above two diagrams are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Hence the entire diagrams (which are recovered as cofibers) are also equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Spherical cubes and perverse schobers on Cn We begin by recalling another classical description of the category of perverse sheaves on the stratified space Cn with strata given by the coordinate hyperplanes and their (iterated) inter- sections and complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let rr1ss be the 1-category with two objects 0, 1 and morphisms freely generated by two morphisms 1 Ñ 0 and 0 Ñ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We identify the set of objects of rr1ssn with the set Pprnsqop, by identifying J Ă rns with its characteristic function in rr1ssn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By a cubical double diagram, we mean a functor rr1ssn Ñ Vectk, which amounts to the datum of a vector space VJ assigned to each J P Pprnsqop, a pair of linear maps fi : VJYtiu Ø VJ : gi assigned to each J P Pprnsqop and i P rnszJ, satisfying fifj “ fjfi, gigj “ gjgi and gifj “ fjgi for all i, j P rns, i ‰ j, whenever these composites are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 48 With this terminology, we may then formulate the following classical description of perverse sheaves on Cn: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 ([GGM85]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The category of perverse sheaves on Cn with respect to the coordinate hyperplane stratification is equivalent to the category of cubical double diagrams rr1ssn Ñ Vectk (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) satisfying that, for all pairs of maps fi : VIYtiu Ø VI :gi, the endomorphisms gifi ´ idVIYtiu and figi ´ idVI are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As in the 1-dimensional case, we ask the pairs of linear maps pfi, giq to become pairs of adjoint functors upon categorification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remarkably, the commutativity conditions gifj “ fjgi then correspond to the Beck–Chevalley conditions already introduced and studied in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 (where the motivation for introducing them came from their effect on totalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Inspired by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, we thus introduce the following: Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A Beck-Chevalley categorical n-cube A˚ : In Ñ Stk is called spherical if every rectilinear edge is a spherical functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A natural categorification of a perverse sheaf on Cn with the above stratification might thus be a spherical categorical n-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We conclude this section, by showing that totalization takes spherical categorical cubes to spherical categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Consider a Beck-Chevalley chain map F : A‚ Ñ B‚ between spherical categorical com- plexes, satisfying that Fi : Ai Ñ Bi is a spherical functor for all i P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then FibpFq is a spherical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Consider a spherical categorical n-cube A˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the product totalization totˆpA˚q and the coproduct totalization tot>pA˚q are equivalent and spherical categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We begin by proving part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We depict F as follows: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' B2 B1 B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' dA 2 F2 F1 dA 1 F0 dB 2 dB 1 Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we obtain a spherical functor F1 ñˆ B1 dB 2 : A1 ñˆ B1 B2 ÝÑ B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor ˜dA 1 :“ dA 1 ñˆ A1 0 : A1 ñˆ B1 B2 πA1 ÝÝÑ A1 dA 1 ÝÝÑ A0 , 49 where πA1 denotes the left adjoint of the inclusion of A1, is furthermore also spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To see this, we first note that it is immediate by the fully faithfulness of pπA1qR that the cotwist functor of the adjunction ˜dA 1 % p ˜dA 1 q R is equivalent to the cotwist functor of the adjunction dA 1 % pdA 1 q R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To show that the twist is an autoequivalence, we apply [Dyc21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Clearly, the twist acts on A1 Ă A1 ñˆ B1 B2 as the twist of dA 1 % pdA 1 q R and is thus invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' On B2 Ă A1 ñˆ B1 B2, the twist clearly acts as the suspension functor, which is also invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It thus remains to show that the twist preserves Cartesian edges, which are of the form a ˚ÝÑ pdB 2 q L ˝ F1paq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By the sphericalness of dB 2 we have pdB 2 q L » T ˝ pdB 2 q R with T the twist functor of dB 2 L % dB 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The Beck-Chevalley condition and d2 “ 0 thus imply pdB 2 q L ˝ F1 ˝ pdA 1 q R ˝ dA 1 » 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) We get a cofiber sequence in A1 ñˆ B1 B2 a pdA 1 q R ˝ dA 1 paq Tpaq pdB 2 q L ˝ F1paq 0 pdB 2 q L ˝ F1paqr1s ˚ ˚ ˚ where (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) implies that the middle vertical arrow is Cartesian and the last vertical morphism is Cartesian as the cofiber of Cartesian morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This shows that the twist preserves Cartesian edge and is hence an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor A0 p ˜dA 1 q R ÝÝÝÝÑ A1 ñˆ B1 B2 F1 ñ ˆ B1 dB 2 ÝÝÝÝÝÑ B1 is clearly equivalent to F1 ˝ pdA 1 q R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows that A0 ñ> A1 ñ ˆ B1 B2 B1 » A0 ñ> A1 B1 and the Beck- Chevalley property further implies that A0 ñ> A1 B1 » A0 ñˆ B0 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We find that the differential d1 of FibpFq is equivalent to the the composite of these equivalences with the functor ˜dA 1 ñ> A1 ñˆ B1 B2 ˆ F1 ñˆ B1 dB 2 ˙ : A1 ñˆ B1 B2 ÝÑ A0 ñ> A1 ñˆ B1 B2 B1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This functor is again spherical by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The same argument applied to each degree shows that all differentials of FibpFq are spherical functors, meaning that FibpFq is a spherical categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This concludes the proof of part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For part (2), we begin by noting that the equivalence of the product and coproduct total- izations follows from repeated application of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We prove the sphericalness of the product totalization via an induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The case n “ 2 follows from part (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Fix n ě 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can consider the categorical n-cube A˚ as a morphism between two categorical 50 pn ´ 1q-cubes B˚ Ñ B1 ˚, whose totalizations are spherical categorical complexes by the in- duction assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The arising morphism β : totˆpB˚q Ñ totˆpB1 ˚q is Beck-Chevalley by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Repeated application of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 shows that β is a spherical functor in each degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The sphericalness of totˆpA˚q thus follows again from part (1) applied to the morphism of categorical complexes β, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a commutative Beck-Chevalley diagram in Stk of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A C B D F α α1 F 1 If F and F 1 are spherical functors, then the induced functor FˆÑF 1 : A ш α B ÝÑ C ш α1 D is also spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let G and G1 be the right adjoints of F and F 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let TA and TB be the twist functors of F % G and F 1 % G1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Adjoints on lax limits are determined componentwise, see Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We thus have FˆÑF 1 % GˆÑG1 and the twist functor of this adjunction can be identified with the induced functor TA ˆÑ TB : A ш α B Ñ A ш α B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This functor is invertible with inverse given by T ´1 A ˆÑ T ´1 B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An analogous argument shows that the cotwist functor FˆÑF 1 % GˆÑG1 is invertible, showing the desired sphericalness of the adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Let F : A Ø B :G and F 1 : A1 Ø B :G1 be two spherical adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote C “ A ñˆ B A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider further the lax limits A ш F B and A1 ш F 1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There are two canonical functors evB, rcof : A ш F B Ñ B, acting on objects via evBpa Ñ bq “ b and rcofpa Ñ bq “ cofpFpaq Ñ bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There are similar functors ev1 B, rcof1 : A1 ш F 1 B Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by D the limit of the following diagram in Stk: A ш F B A1 ш F 1 B B B B evB rcof ev1 B rcof1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) We further denote by B1, B2 : D Ñ B the functors contained in limit cone, going to the leftmost and rightmost copy of B and B “ pB1, B2q: D Ñ Bˆ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category D is a concrete model for the 8-category RΓ1pG, Fq mentioned in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 and the functor B describes the functor pevp, evqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The fiber of D B2 ÝÝÑ B is equivalent to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor C ãÑ D B1 ÝÝÑ B is equivalent to F ñˆ B F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 51 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The first part follows from the observation that the fiber of rcof1 : A1 ш F 1 B Ñ B is equivalent to A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The second part can be checked for instance by describing all involved functors using the universal properties of the lax limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We construct two functor A, C : Bˆ2 Ñ D, which we show in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 to be left and right adjoint to B “ pB1, B2q : D Ñ Bˆ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the Grothendieck construction p of the diagram (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) and denote by L the 8-category of sections of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category D can be identified with full subcategory of L spanned by coCartesian sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by E1 the full subcategory of L spanned by p-relative left Kan extensions of their re- striction to A ш F B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' by E2 the full subcategory of L spanned by p-relative left Kan extensions of their re- striction to the central copy of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' by E3 the full subcategory of L spanned by p-relative left Kan extensions of their re- striction to A1 ш F 1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The restriction functor E1 Ñ A ш F B is a trivial fibration by [Lur09a, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Choosing a section, we can compose to a functor H1 : B rcofR ÝÝÝÑ A ш F B Ñ E1 Ă L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly, restriction defines a trivial fibration E2 Ñ B and we obtain a functor H2 : B Ñ E2 Ă L by choosing a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor H2 is left adjoint to the restriction functor resB : L Ñ B at the central copy of B, see [Lur09a, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17], so that we obtain a counit natural transformation H2 ˝ resB Ñ idL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Precomposition with H1 yields a natural transformation η1 : H2 ˝ evB ˝rcofR » H2 ˝ resB ˝H1 Ñ H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Analogous to the above, we define a functor H3 : B pev1 BqR ÝÝÝÝÑ A1 ш F 1 B Ñ E3 Ă L and a natural transformation η2 : H2 » H2 ˝ ev1 B ˝pev1 BqR Ñ H3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The equivalence above arises from the fact that ev1 B is a reflective localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, we define the functor C1 : B Ñ L as the colimit of the diagram H2 ˝ evB ˝rcofR H1 H3 ˝ evB ˝rcofR η1 η2˝evB ˝rcofR (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) in StkpB, Lq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is straightforward to verify that C1 factors through D Ă L, and we consider C1 as a functor B Ñ D in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Exchanging the roles of F and F 1, or equivalently reflecting diagram (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) along the vertical axis, and reperforming the above construction, we obtain a functor C2 : B Ñ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote C “ pC1, C2q : Bˆ2 Ñ D .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 52 Performing the above construction of C again, but replacing all right adjoints by left adjoints, we obtain the functor A “ pA1, A2q : Bˆ2 Ñ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor A is left adjoint to B and the functor C is right adjoint to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To determine the left adjoint of B C1 ÝÑ D Ă L, we use that passing to adjoints defines an exact functor p-qR : FunRpBˆ2, Lq Ñ FunLpL, Bˆ2qop between the functor categories of right or left adjoint functors, and the description of C1 as the colimit of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The left adjoint of H1 is by [Lur09a, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17] given by the composite L Ñ A ш F B rcof ÝÝÑ B of the restriction functor to A ш F B and rcof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Composing this functor with D Ă L yields the functor B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A similar argument shows that the left adjoints of H2 ˝ evB ˝rcofR and H3 ˝ evB ˝rcofR both restrict on D, up to equivalence, to the functor E : D resB ÝÝÝÑ B rcof ˝pev1 BqL ÝÝÝÝÝÝÝÝÑ B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that restricting along D Ă L is also exact, we obtain that the left adjoint of C1 is given by B1 » B1 ˆE E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similar arguments show B2 % C2, A1 % B1 and A2 % B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From this it follows that A % B and B % C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functors A1, A2, C1, C2 correspond to evpL, evqL, evpR, evqR from §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These functors are all fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This can be checked for instance by an explicit com- putation of the derived Homs, using their pushout description in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) and the facts that H1, H2, H3 are fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The adjunctions A % B and B % C are spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We begin by checking that the twist functor of A % B and the cotwist functor of B % C are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have the following equivalences idB » evB ˝rcofR idB » ev1 B ˝prcof1qR TF%G » rcof ˝evBR TF 1%G1 » rcof1 ˝pev1 BqR where TF%G and TF 1%G1 denote the cotwist functors of the adjunctions F % G and F 1 % G1, see [Chr22a, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7] for a detailed verification of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Unraveling the construction of C, we thus find BC » idBˆ2 ‘ ˆ 0 TF 1%G1 TF%G 0 ˙ as an endofunctor of Bˆ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The second summand describes the cotwist functor T 1 of B % C and is hence an autoequivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A similar description holds for the twist functor of A % B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To deduce the sphericalness of A % B, we apply [Chr22c, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5], by which it suffices to check that the twist functor T of A % B commutes pointwise with the unit of A % B and that the essential image of A agrees with the essential image of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The former is immediate, since the unit is pointwise the inclusion of a direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Tracing through the constructions, we find equivalences A1 ˝ TF%Gpbq » C2pbq and A2 ˝ TF 1%G1 » C1pbq for all b P B, which shows the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The sphericalness of B % C follows from the sphericalness of A % B, see [Chr22c, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6], concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 53 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We show that D has a 4-periodic semiorthogonal decomposition with gluing functor F ñˆ B F 1, in the sense of [DKSS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We check that there is a semiorthogonal decomposition pImpC2q, fibpB2qq of D, where ImpC2q Ă D denotes the stable subcategory given by the essential image of C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given d P fibpB2q and C2pbq P ImpC2q, we have MapDpd, C2pbqq » MapBpB2pdq, bq » ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Further, given d P D, the unit of B2 % C2 defines a morphism ud : d Ñ C2B2pdq with B2pudq an equivalence, since C2 is fully faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This shows fibpudq P fibpB2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similar arguments show the existence of semiorthogonal decompositions pfibpB2q, ImpA2qq pImpC1q, fibpB1qq, pfibpB1q, ImpA1qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that ImpC1q “ ImpA2q and ImpC2q “ ImpA1q, we obtain the desired 4-periodic semiorthogonal decomposition of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The gluing functors of the involved semiorthogonal decompositions are hence spherical, see [DKSS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The left gluing functor of the semiorthogonal decomposition pImpC2q, fibpB2qq is defined as the composite fibpB2q ãÑ D πÝÑ ImpC2q, where π is right adjoint to the inclusion ImpC2q Ă D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Under the equivalence C2 : B » ImpC2q, π identifies with TF%G ˝ B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Under the further equivalences fibpB2q » C, the gluing functor identifies by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 with TF%G ˝ F ñˆ B F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This shows the sphericalness of F ñˆ B F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It remains to describe the cotwist functor of F ñˆ B F 1 % ˆ F ñˆ B F 1 ˙R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let ι : C Ă D denote the composite of C » fibpB2q with the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The equivalence B2C1 » TF 1%G1 induces by passing to the adjoint C2 of B2 a natural transformation η : C1 Ñ C2 ˝TF 1%G1, satisfying that fibpηq : B Ñ D factors through ι : C Ă D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote the factorization by fibpηq1 : B Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that B2 ˝ ι » 0, we find fibpηq1 » ιR ˝ ι ˝ fibpηq1 » ιR ˝ fibpηq » ´ pfibpηqL ˝ ι ¯R » ` cofpT ´1 F 1%G1 ˝ B2 Ñ B1q ˝ ι ˘R » pB1 ˝ ιqR » ˆ F ñˆ B F 1 ˙R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We thus have an equivalence ˆ F ñˆ B F 1 ˙ ˝ ˆ F ñˆ B F 1 ˙R » B1˝fibpηq » fibpB1C1 Ñ B1C2TF 1%G1q » fibpidB Ñ TF%G˝TF 1%G1q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One finds the counit of the adjunction F ñˆ B F 1 % ˆ F ñˆ B F 1 ˙R to arise as the apparent natural transformation fibpidB Ñ TF%G ˝ TF 1%G1q Ñ idB, which implies that the cotwist functor of the adjunction is equivalent to TF%G ˝ TF 1%G1r´1s, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof sketch of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let i be the number of boundary components of S and EB “ š 1ďjďi EB j the decomposition of the set EB of external edges of G into the sets of edges ending on a given boundary component of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each 1 ď j ď i, the functor ś xPEB j eve : RΓpG, Fq Ñ ś xPEB j Fpxq is spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This follows from adapting the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7 if |EB j | ě 2 or the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 if |EB j | “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The main difficulties in spelling out these adaptations is the required involved notation and we omit these details in favor of a shorter exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 54 Let now 1 ď j1, j ď i with j1 ‰ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is easy to see that ś xPEB j eve ˝ ś xPEB j1 eveR » 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Hence, the twist functors of the adjunctions ś xPEB j eve % ś xPEB j eveR and ś xPEB j1 eve % ś xPEB j1 eveR commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The twist functor of the adjunction ś xPEB eve % ś xPEB eveR is equiv- alent to the composite of the commuting twist functors of the adjunctions ś xPEB l eve % ś xPEB l eveR, 1 ď l ď i, and hence an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The cotwist functor of ś xPEB eve % ś xPEB eveR acts on Fpeq, with e P EB l an external edge, as the cotwist functor of the adjunc- tion ś xPEB l eve % ś xPEB l eveR and is hence also invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This shows the sphericalness of the functor ś xPEB eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6 Calabi-Yau complexes The goal of this section is to introduce a notion of Calabi–Yau structure on a categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, we recall the definitions of Hochschild homology and negative cyclic ho- mology of k-linear 8-categories and introduce the total Hochschild homology of a categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2, we define left Calabi–Yau structures on categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we discuss some ways in which Calabi–Yau structures arise on lax limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the final §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4, we introduce Calabi–Yau structures on categorical cubes and show that they induce Calabi–Yau structures on their totalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This will be our main technique to construct examples of categorical Calabi–Yau complexes in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Total Hochschild and negative cyclic homology Let C be a compactly generated k-linear 8-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As explained in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, the dual of C with respect to the monoidal structure on Stk can be described as C_ “ IndpCop 0 q, where C0 Ă C denotes the subcategory of compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We thus have StkpD, C_q » StkpD bk C, Modkq for any D P Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, we have StkpC, Cq » StkpC bk C_, Modkq and can define the diagonal bimodule ∆C : C bk C_ Ñ Modk as the image of the identity functor idC under this equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The k-linear Hochschild homology of C is defined as the trace HHpCq :“ p∆C ˝ φp∆Cqqpkq P Modk, where φ denotes the equivalence StkpC bk C_, Modkq » StkpModk, C bk C_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When k is the sphere spectrum, we obtain topological Hochschild homology and when k is a field, we obtain the usual Hochschild complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Hochschild homology comes equipped with an action of the circle group S1, see [HSS17], and we denote its fixed points by HHS1pCq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We call HHS1pCq the negative cyclic homology of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Hochschild homology and negative cyclic homology form functors HH, HHS1 : Stcpt k Ñ Modk on the subcategory Stcpt k Ă Stk of compactly generated 8-categories and compact objects preserving functors, see [HSS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is a canonical natural transformation HHS1 Ñ HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 55 The k-linear stable 8-category C is called smooth if it is compactly generated and ∆C admits a bimodule left dual ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C P StkpModk, Cbk C_q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case, we sometimes consider ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C as an object in StkpCbkC_, Modkq using the equivalence φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case, the k-linear Hochschild homology of C is equivalent to RHomStkpCbkC_,Modkqp∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C, ∆Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a k-linear, compact objects preserving functor F : C Ñ D, we denote HHpD, Cq “ cof HHpFq and HHS1pD, Cq “ cof HHS1pFq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Supposing that C and D are smooth, the arising morphism RHomStkpCbkC_,Modkqp∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C, ∆Cq » HHpC, Cq HHpFq ÝÝÝÝÑ HHpD, Dq » RHomStkpDbkD_,Modkqp∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D, ∆Dq admits a concrete description, as noted in [BD21, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It maps a degree m-morphism α: ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C Ñ ∆Crms to the morphism ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D Ñ ∆Drms obtained as the composite ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D F˚p∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Cq F˚p∆Cqrms ∆Drms u F˚pαq curms where F˚ is the functor StkpC bk C_, Modkq » StkpC, Cq F˝-˝G ÝÝÝÝÑ StkpD, Dq » StkpD bk D_, Modkq (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) with G the k-linear right adjoint of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Above, the natural transformation cu is the counit of F % G and u is the defined as the composite ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D Ñ ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D ˝ ∆C ˝ ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C Ñ ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D ˝ ∆D ˝ pF bk Ind fopq ˝ ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C Ñ pF bk Ind fopq ˝ ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' C » F˚p∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Cq , with Ind fop : C_ Ñ D_ obtained by restricting F to compact objects, passing to opposite categories and Ind-completing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A categorical n-complex A˚ P ChnpStkq is called smooth if it consists of smooth k-linear stable 8-categories and all differentials preserve compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChpStkq be a bounded, smooth categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The total negative cyclic homology HHS1,totpA‚q is defined as the total cofiber of the negative cyclic homology cube, obtained by applying HHS1 to the corresponding categorical cube of A‚, see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 (see [DJW19][A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2] for terminology and basic results on total fibers and cofibers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The total Hochschild homology HHtotpA‚q is defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have normalized the suspensions in the totalization such that, if A‚ “ Ar0s is concentrated in degree 0, then HHS1,totpA‚q » HHS1pAq and HHtotpA‚q » HHpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChpStkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lower truncation τěipA‚q at i P Z of A‚ is defined as the complex A‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ai`1 Ai Ai´1 Ai´2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' τěipA‚q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ai`1 Ai 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' which is identical to A‚ in degrees i or larger and vanishes in degrees less than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lower truncation forms a functor τěi : ChpStkq Ñ ChpStkq and is equipped with a canonical natural transformation idChpStkq Ñ τěi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 56 The upper truncation τ ďipA‚q at i P Z of A‚ is defines as the complex A‚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ai`2 Ai`1 Ai Ai´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' τ ďipA‚q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 0 0 Ai{Ai`1 Ai´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' x which vanishes in degrees larger than i, is in degree i given by the cofiber of Ai`1 dÝÑ Ai and in the other degrees identical to A‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Upper truncation forms a functor τ ďi : ChpStkq Ñ ChpStkq and is equipped with a canonical natural transformation idChpStkq Ñ τ ďi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that τ ďi preserves smooth categorical complexes, as smoothness of categories is pre- served under quotients along compact objects preserving functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The two-sided truncation τ ďi ějpA‚q is defined as τ ďi ˝ τějpA‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that the upper and lower truncation commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChpStkq be a bounded, smooth categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is a canonical biCartesian square in Modk: HHS1,totpτěipA‚qq HHS1,totpτ ďi`1 ěi pA‚qq HHS1pAi, Ai`1{Ai`2qris HHS1,totpτěi`1pA‚qq HHS1,totpτ ďi`1 ěi`1 pA‚qq HHS1pAi`1{Ai`2qri ` 1s ˝ » » Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This follows from the pasting law for biCartesian squares and the commutative diagram HHS1,totpτěipA‚qq HHS1pAi, Ai`1{Ai`2qris 0 HHS1,totpτěi`1pA‚qq HHS1pAi`1{Ai`2qri ` 1s HHS1pAiqri ` 1s ˝ in which the right square and the outer square are biCartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Repeatedly applying Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, we obtain that HHS1,totpA‚q is equivalent to the limit of the diagram HHS1pAi´1, Ai{Ai`1qri ´ 1s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' HHS1pAi, Ai`1{Ai`2qris HHS1pAi{Ai`1qris HHS1pAi`1, Ai`2{Ai`3qri ` 1s HHS1pAi`1{Ai`2qri ` 1s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The limit of the above diagram is equivalent to an equalizer as follows: 57 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChpStkq be a bounded, smooth categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then HHS1,totpA‚q is equivalent to the equalizer of the diagram in Modk À iPZ HHS1pAi, Ai`1{Ai`2qris À iPZ HHS1pAi{Ai`1qris , À δi À πi where δi : HHS1pAi, Ai`1{Ai`2qris Ñ HHS1pAi`1{Ai`2qri ` 1s is the fiber map and πi : HHS1pAi, Ai`1{Ai`2qris Ñ HHS1pAi{Ai`1qris is the map induced by the quotient functor Ai Ñ Ai{Ai`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The above statement also holds when replacing HHS1 with HH and HHS1,tot with HHtot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 expresses, that a Hochschild or total negative cyclic ho- mology class consists of a compatible family of relative Hochschild or relative negative cyclic homology classes of the functors Ai`1{Ai`2 Ñ Ai, shifted into the appropriate degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Left Calabi–Yau structures Let F : C Ñ D be a compact objects preserving, k-linear functor between smooth k-linear 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As explained in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, a k-linear Hochschild class σ: krns Ñ HHpD, Cq defines a diagram ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D F˚p∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Cq F˚p∆Cqrn ´ 1s ∆Crn ´ 1s together with a choice of null-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It hence gives rise to a diagram with horizontal fiber and cofiber sequences as follows: ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' D F˚p∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Cq cof fib F˚p∆Cqrn ´ 1s ∆Crn ´ 1s We call the Hochschild class σ non-degenerate if all vertical maps in the above diagram are equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) A weak left n-CY structure on the functor F consists of a non- degenerate Hochschild class σ: krns Ñ HHpD, Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) A left n-CY structure on the functor F consists of a negative cyclic class η: krns Ñ HHS1pD, Cq, whose image under HHS1pD, Cq Ñ HHpD, Cq defines a non-degenerate Hochschild class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We refer to Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 for examples of left Calabi– Yau structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 58 Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChpStkq be a bounded, smooth categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) A weak left n-Calabi–Yau structure on A‚ consists of a class σ: krns Ñ HHtotpA‚q, whose composite with the morphism from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 HHtotpA‚q ÝÑ à iPZ HHpAi, Ai`1{Ai`2qris defines a collection of weak left pn ´ iq-Calabi–Yau structures on the functors Ai`1{Ai`2 Ñ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) A left n-Calabi–Yau structure on A‚ consists of a class η: krns Ñ HHS1,totpA‚q, whose composite with the morphism from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 HHS1,totpA‚q ÝÑ à iPZ HHS1pAi, Ai`1{Ai`2qris defines a collection of left pn´iq-Calabi–Yau structures on the functors Ai`1{Ai`2 Ñ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is an apparent analogue of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 for bounded complexes of proper k-linear 8-categories, obtained by replacing left Calabi–Yau structures with right Calabi–Yau structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Calabi–Yau structures and lax limits Consider a colimit diagram in Stk A2 A B1 A1 B C x where all appearing 8-categories are smooth and all functors preserve compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Sup- pose further that the functors AˆA1 Ñ B and AˆA2 Ñ B1 carry left n-Calabi–Yau structures, which are compatible at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor A1 ˆ A2 Ñ C inherits a left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If k is a field, this is [BD19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For k an arbitrary E8-ring spectrum, a proof will appear in [Chr22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 admits a refinement, in the form of a relative Calabi–Yau structure on the directed pushout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Before stating that result in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we need to consider the following class of examples of relative left Calabi–Yau structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A be a smooth k-linear 8-category with a left pn ´ 1q-Calabi–Yau structure η: krn ´ 1s Ñ HHS1pAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the functor GA2 “ pG1, G2, G3q :“ pev0, cof, ev1q: Funp∆1, Aq ÝÑ Aˆ3 , 59 with components given by the evaluations at i P t0, 1u and the cofiber, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The notation A2 comes from regarding Funp∆1, Aq as the 8-category of A-valued representations of the A2-quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The left adjoint FA2 “ pF1, F2, F3q: Aˆ3 Ñ Funp∆1, Aq of GA2 admits a left n-Calabi–Yau structure which restricts to the given left pn ´ 1q-Calabi– Yau structure η‘3 : k‘3 Ñ HHS1pAq‘3 » HHS1pAˆ3q on Aˆ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If k is a field, this can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For A “ Modk » Dpkq with the apparent left 0-Calabi–Yau structure, this left 1-Calabi–Yau structure on FA2 is described in [BD19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For a general k-linear 8-category A, we have Funp∆1, Modkq b A » Funp∆1, Aq and Modˆ3 k bA » Aˆ3 using the symmetric monoidal structure on Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The desired left Calabi–Yau structure now arises as the image of the above pair of Calabi–Yau structures under the canonical map HHS1 1 pFunp∆1, Modkq, Modˆ3 k q ˆ HHS1 n´1pAq Ñ HHS1 n pFunp∆1, Aq, Aˆ3q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If k is not a field, a slightly more careful argument is required, this case will be treated in [Chr22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functor A ˆ A1 ˆ A2 Ñ B ñ> A B1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) inherits a left n-Calabi–Yau structure which induces the left n-Calabi–Yau structure on A1 ˆ A2 Ñ C from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 when forming the pushout along the functor A Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have the following colimit diagram in Stk: A A ˆ A Funp∆1, Aq A1 ˆ A2 B ˆ B1 B ñ> A B1 F2 pF1,F3q x Applying Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, we find that the left n-Calabi–Yau structures on FA2 and on A ˆ A ˆ A1 ˆ A2 Ñ B ˆ B1 glue to a left n-Calabi–Yau structure on the functor (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We end this section with the following further observations concerning left Calabi–Yau structures on lax limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F : A Ñ B be a k-linear, compact objects preserving functor between smooth k-linear stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Let π: A1 Ñ A be a k-linear functor which preserves compact object and admits a fully faithful right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A class η: krns ÝÑ HHS1pB, A1q 60 determines a left n-Calabi–Yau structure on F ˝ π if and only if its composite with HHS1pB, A1q ÝÑ HHS1pB, Aq determines a left n-Calabi–Yau structure on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Assume that there are k-linear functors Fi : A Ñ Bi with 1 ď i ď n, such that B » ppB1 ñ> A B2q ñ> A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='q ñ> A Bn is an iterated directed pushout and F is equivalent to the iteratively induced functor into the directed pushout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Compatible left n-Calabi–Yau structures on the functors Fi for all 1 ď i ď n canonically determine a left n-Calabi–Yau structure on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) Conversely, in the setting of (2), a left n-Calabi–Yau structure on F determines com- patible left n-Calabi–Yau structures on the functors Fi, 1 ď i ď n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' these assignments are inverse to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Part (1) follows from the observation that the functor π˚, see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), maps the diagonal bimodule ∆A1 to ∆A, by the fully faithfulness of πR Part (2) follows from repeated application of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 (with A1 “ A2 “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For part (3), we note that the projection map πi : B ↠ Bi admits a fully faithful right adjoint for each 1 ď i ď n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Again pπiq˚ maps ∆B to ∆Bi and the arising map HHS1pB, Aq ÝÑ HHS1pBi, Aq thus left n-Calabi–Yau structures to left n-Calabi–Yau structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Cubical Calabi–Yau structures Recall that I “ r1sop “ t1 Ñ 0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ : IN ÝÑ Stk be a smooth, cubical diagram of k-linear 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Applying HHS1yields a cubical diagram HHS1pA˚q: IN ÝÑ Modk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by HHS1,totpA˚q P Modk the total cofiber of HHS1pA˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To clarify the chosen grading, note that if A˚ assigns 0 to all vertices of the cube except p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0q, then we have HHS1,totpA˚q » HHS1pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For L P Modk, we refer to a map η: L Ñ HHS1,totpA˚q as an L-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that the total cofiber is the N-fold suspension of the total fiber, we may identify an L-class η with a natural transformation in FunpIN, Modkq η: Lr´Ns0 ÝÑ HHS1pA˚q where, for E P Modk, we denote by E0 the cubical diagram E0 : IN ÝÑ Modk, x ÞÑ # E for x “ p1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', 1q, 0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From this perspective, we obtain from η the following data: 61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' for every 1 ď i ď N, a morphisms in FunpIN´1, Modkq Biη: Lr´Ns0 ÝÑ HHS1pABi˚q where ABi˚ denotes the restriction of the cube A˚ to the face IN´1 Ñ IN obtained by setting the ith coordinate to 1, and hence a class Lr´1s ÝÑ HHS1,totpABi˚q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a diagram Lr´1s 0 colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q HHS1pA˚q HHS1pcolimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q A˚q HHS1pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq, and hence a relative class L ÝÑ HHS1,totpA˚q » HHS1pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q, colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q A˚q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) where above colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q denotes the colimit over the punctured cube IN ăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q “ IN ztp0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We recursively define an n-Calabi–Yau structure on cubical diagrams as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ : IN ÝÑ Stk be a smooth cubical diagram in Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A class η: krns Ñ HHS1,totpA˚q is called an n-Calabi–Yau structure on A˚, if (1) for every 1 ď i ď N, the class krn ´ 1s Ñ HHS1,totpABi˚q from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) defines a left pn ´ 1q-Calabi–Yau structure on the pN ´ 1q-cube ABi˚, (2) the class krns Ñ HHS1pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q, colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q A˚q from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) defines a left n-Calabi– Yau structure on the functor colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q A˚ ÝÑ Ap0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChpStkq be categorical complex concentrated in degrees N, N ´ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', 0 so that A‚ gives rise to a categorical N-cube A˚ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then HHS1,totpA‚q » HHS1,totpA˚q and a class η: krns Ñ HHS1,totpA‚q describes a left n-Calabi-Yau structure on A‚ if and only if it describes a left n-Calabi-Yau structure on A˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ be a smooth cubical diagram in Stk equipped with a left n-Calabi-Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then its coproduct totalization inherits a canonical left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 62 Before proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6, we collect some preparatory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We begin with an anal- ysis of the differentials in the totalization of a categorical cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For notational convenience, we will use the reparameterization of the poset IN`1 “ pr1sopqN`1 by the poset PprNsqop of subsets of the set rNs “ t0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Nu (by means of associating to a subset its character- istic function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For instance, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 1q is identified with rNs P PprNsqop, p1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0q with t1u P PprNsqop and p0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0q with H P PprNsqop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ : PprNsqop Ñ Stk be a smooth categorical N-cube with N ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When taking product and coproduct totalizations, we equip the subset of PprNsqop of subset of cardinality i with the canonical total order, where I ă J if minpIzJq ă minpJzIq, for I, J Ă rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There exist equivalences in Stk totˆpA˚qi » ñ ą totˆpA˚qi´1 AJ and tot>pA˚qi » ñ ž tot>pA˚qi`1 AJ where the iterated directed pullback and the iterated directed pushout run over all J Ă rNs with |J| “ i, with the order as specified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We only verify the formula for the product totalization, the coproduct totalization can be treated analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The proof is by induction on N using the iterative definition of the product totalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The case N “ 2 is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We proceed with the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a categorical pN ` 1q-cube A˚, we can consider it as describing a chain map f˚ : B˚ Ñ C˚ between N-cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The induction step now follows from combining the following two observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have that totˆpB˚qi » Śñ totˆpA˚qi´1 BJ and a similar statement for C˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This follows from the fact that the functor totˆpA˚qi´1 Ñ totˆpB˚qi´1 is a reflective localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Spelling out the definition of the iterated directed pullback, one finds that the order of the bracketing is irrelevant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' satisfies associativity), implying that ¨ ˝ ñ ą totˆpA˚qi´1 BJ ˛ ‚ˆñ totˆpA˚qi´1 ¨ ˝ ñ ą totˆpA˚qi´1 CJ ˛ ‚» ñ ą totˆpA˚qi´1 AJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let J Ă rNs with |J| “ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Our next goal is to find a description of the functor tot>pA˚qi`2 dÝÑ tot>pA˚qi`1 » ñ ž tot>pA˚qi`1 AJ1 ↠ AJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Fix J Ă rNs with |J| “ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 63 (1) We define the stable 8-category P J as the full subcategory PJ :“ ñ ą totˆpA˚qi´2, jPJ AJztju Ă ñ ą totˆpA˚qi´2 AJ1 , meaning the stable subcategory generated by the components AJztju of the iterated lax product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We further define kerpdqJ Ă P J as the fiber in Stk of P J Ă totˆpA˚qi´1 d ÝÑ totˆpA˚qi´2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The dual version PJ is defined as the full subcategory PJ :“ ñ ž tot>pA˚qi`2, jPrNszJ AJYtju Ă ñ ž tot>pA˚qi`2 AJ1 » tot>pA˚qi`1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The left adjoint of this inclusion is denoted by tot>pA˚qi`1 ↠ PJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We further define cokerpdqJ as the cofiber in Stk of tot>pA˚qi`2 d ÝÑ tot>pA˚qi`1 ↠ PJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let J Ă rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) The k-linear stable 8-category kerpdqJ is equivalent to the limit in Stk of the restriction of A˚ to the full subcategory PprNsqop ąJ of PprNsqop spanned by objects J1 ‰ J satisfying J1 Ă J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The k-linear 8-category cokerpdqJ is equivalent to the colimit in Stk of the restriction of A˚ to the full subcategory PprNsqop ăJ of PprNsqop spanned by objects J1 ‰ J satisfying J Ă J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Part (2) is dual to part (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' follows from passing to right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Part (1) follows from an induction over |J| and by decomposing the limit over PprNsqop ąJ via [Lur09a, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Via the universal properties of these limits and colimits, we obtain functors dJ : AJ Ñ kerpdqJ and dJ : cokerpdqJ Ñ AJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ be a categorical n-cube and let J Ă rNs with |J| “ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) There is a commutative diagram in Stk AJ kerpdqJ totˆpA˚qi totˆpA˚qi´1 totˆpA˚qi´2 dJ 0 d d with fully faithful vertical functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Furthermore, if A˚ takes values in limit preserving functors, then all functors in the above diagram also preserve limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 64 (2) There is a commutative diagram in Stk tot>pA˚qi`2 tot>pA˚qi`1 tot>pA˚qi cokerpdqJ AJ d 0 d dJ such that the vertical functors admit fully faithful right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Furthermore, if A˚ takes values in compact objects preserving functors, then all functors in the above dia- gram also preserve compact objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We only prove part (1), part (2) is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using the notation of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8, we have a commutative diagram in Stk: kerpdqJ 0 AJ P J totˆpA˚qi totˆpA˚qi´1 totˆpA˚qi´2 { d d The top right square is pullback, the dotted arrow arises via the universal property of this pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Unraveling the definition of d, one can further show that the dotted arrow is described by the functor dJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If A˚ takes values in limit preserving functors, the pullback is also equivalent to the pushout of the k-linear left adjoint diagram in Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Passing again to right adjoints, we find that all functors in the diagram preserve limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a categorical 3-cube A˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then cokerpdqtju » Atj,j1u >Ar3s Atj,j2u with j, j1, j2 P r3s pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category cokerpdqH is equivalent to the colimit in Stk of the diagram Ar3s At1,2u At1,3u At1u At2,3u At2u At3u which is also equivalent to the iterated (usual, not directed) pushout ´ At1u >At1,2u At2u ¯ >At1,3u>Ar3sAt2,3u At3u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 65 Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A‚ P ChNpStkq be a smooth cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A class η: krns Ñ HHS1,totptot>pA‚qq , which induces left pn ´ |J|q-Calabi–Yau structures on the functors dJ : cokerpdqJ Ñ AJ, for all J Ă rNs, describes a left n-Calabi–Yau structure on tot>pA‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Combine parts (1) and (2) of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is a canonical equivalence HHS1,totpA˚q » HHS1,totptot>pA˚qq, which can be obtained using the recursive definition of the coproduct totalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a left n-Calabi-Yau structure η: krns Ñ HHS1,totpA˚q of A˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Under the above equivalence, the arising top non-degeneracy condition of HHS1,totpABi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Bj˚q implies that the functor dJ : cokerpdqJ Ñ AJ with J “ ti, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , ju inherits a left pn´|J|q-Calabi–Yau structure from η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The Theorem thus follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In view of part (3) of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 also shows the ’converse’ of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6, namely that any left n-Calabi–Yau structure of tot>pA˚q canonically determines a left n-Calabi-Yau structure on A˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 7 Examples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Normal crossings divisors and categorical intersection complexes As a standing assumption in this section, we require that all schemes are separated, reduced and of finite type over a field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Morphisms between schemes are assumed to be separated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The category of such schemes is denoted Schk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let k be a field of characteristic 0, Z a smooth scheme over k and i: D Ă Z a normal crossing divisor, by which we mean a union D “ Ť 1ďiďN Di of smooth divisors Di, intersecting transversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may organize the intersections DI :“ č iPI Di, for the various subsets I Ă t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', Nu, along with their inclusions DI Ď DJ for I Ě J, into a cubical diagram D˚ of smooth schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here, we interpret the empty intersection DH as the ambient scheme Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For example, for N “ 2, 3, the resulting cube can be depicted as follows: D1 X D2 D1 D2 Z D1 X D2 X D3 D1 X D3 D2 X D3 D3 D1 X D2 D1 D2 Z 66 Consider the functor IndCoh: NpSchkq Ñ Stk, which assigns to a scheme Y its k-linear 8- category IndCohpY q of Ind-coherent sheaves and to a morphism f : Y Ñ Y 1 of schemes the k-linear functor f˚ : IndCohpY q Ñ IndCohpY 1q, see [GR17] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If f is proper, then the functor f˚ admits a right adjoint f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', which preserves colimits, thus defining a morphism in Stk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Applying IndCoh to the cube D˚, we obtain a categorical N-cube IndCohpD˚q, for example in the case N “ 2 given by: IndCohpD1 X D2q IndCohpD1q IndCohpD2q IndCohpZq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) This categorical cube is Beck-Chevalley by base change, see [GR17, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We call the coproduct totalization tot>pIndCohpD˚qq of IndCohpD˚q the categorical intersection complex of tDiu1ďiďN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the square (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1), the categorical intersection complex has 3 nontrivial terms given by IndCohpD1 X D2q ÝÑ IndCohpD1q ñ> IndCohpD1XD2q IndCohpD2q ÝÑ IndCohpZq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category IndCohpD1q ñ> IndCohpD1XD2q IndCohpD2q describes the “lax gluing” of the two schemes D1 and D2 along D1 Y D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We proceed with describing situations in which the categorical intersection complex is spherical or allows a Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Suppose that Z is a smooth, projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then IndCohpD˚q is a spherical categorical cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case, the categorical intersection complex is hence a spherical categorical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The sphericalness of the cube follows from the fact that the inclusion of any smooth divisor into a smooth projective variety induces a spherical functor when passing to Ind- coherent sheaves, see for instance [Add16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The statement about the totalization follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall the following Theorem from [BD19]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 ([BD19, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Suppose that Z is a Gorenstein scheme of dimension n and i: D Ă Z an anticanonical divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then IndCohpDq admits a canonical left pn ´ 1q-Calabi–Yau structure and the functor i˚ : IndCohpDq Ñ IndCohpZq admits a canonical compatible left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In terms of categorical complexes, this result equips the categorical 2-term complex IndCohpDq i˚ ÝÑ IndCohpZq associated to the anticanonical divisor D Ă Z with an n-dimensional Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We will now show that, if D “ Ť 1ďiďN Di is an anticanonical normal crossings divisor, 67 then the statement can be refined to provide an n-dimensional Calabi–Yau structure on the categorical intersection complex of tDiu1ďiďN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the proof, we require the sphericalness of the cube, and to this end we assume that Z is smooth projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let Z be a smooth projective scheme of dimension n and let D “ Ť 1ďiďN Di be an anticanonical normal crossings divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the categorical N-cube IndCohpD˚q is spherical and admits a canonical left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let Z be a smooth projective scheme of dimension n and let D “ Ť 1ďiďN Di be an anticanonical normal crossings divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the categorical intersection complex tot>pIndCohpD˚qq is spherical and admits a canonical left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Combine Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 and Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 provide possible answers to the question raised by Katzarkov-Kontsevich-Pantev, as to what kind of Calabi–Yau and spherical cate- gorical structures arise from an anticanonical normal crossings divisor, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='36 in [KKP08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Before proving Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, we recall some results from [BD19, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a scheme Z, we denote by RHomZp-, -q the k-linear derived Hom in IndCohpZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by ω‚ Z the dualizing complex of Z, defined as π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='pkq with π: Z Ñ ˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using the diagonal map ∆: Z Ñ Z ˆ Z, one finds a canonical morphism RHomZpOZ, ω‚ Zq Ñ RHomZˆZp∆˚OZ, ∆˚ω‚ Zq » HHpIndCohpZqq , natural in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A scheme Z is called Cohen-Macaulay of dimension d if ωX “ ω‚ Xr´ds is a coherent sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case, there exists an isomorphism Hd HHpIndCohpZqq » Hd HHS1pIndCohpZqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the setting of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4, we further have i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='ωZ » ωDr´1s and a cofiber sequence OZ Ñ i˚OD Ñ ωZr1s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) A straightforward computation shows that the morphism i˚OD Ñ ωZr1s is adjoint to an equivalence OD » ωD, see [BD19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Furthermore, the morphism i˚OD Ñ ωZr1s factors through the equivalence i˚OD » i˚ωD via the counit cu of i˚ % i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' : i˚OD » i˚ωD » i˚i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='ωZr1s cu ÝÑ ωZr1s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows that the cofiber sequence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) gives rise to a relative negative cyclic homology class describing a Calabi–Yau structure on i˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall our notation DJ :“ Ş jPJ Dj, for J Ă rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We further denote ˜DJ :“ Ť jPrNszJ DJYtju Ă DJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The inclusions DJ Ă Z is denoted fJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 68 Each component Di Ď Z is cut out by a section si : OZ Ñ OpDiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We dualize si to obtain a map s_ i : Op´Diq Ñ OZ which is part of a cofiber sequence Op´Diq OZ 0 pfiq˚ODi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The tensor product of the morphisms s_ i , 1 ď i ď n, yields a cubical diagram p: PprNsqop Ñ IndCohpZq, J ÞÑ Op´ ÿ iPJ Diq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Further, by passing to iterated cofibers, or equivalently directly tensoring the cofiber mor- phisms, we obtain a “reflected” cubical diagram q: PprNsq Ñ IndCohpZq, J ÞÑ pfJq˚ODJ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) By construction, the total fiber of the cube q is ωZ so that its total cofiber is ωZrNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Applying the functor RHomZp´, ω‚ Zq to the cube q, we obtain the cube qω : PprNsqop Ñ Modk, J ÞÑ RHomZppfJq˚ODJ, ω‚ Zq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, composing with the pushforward along the diagonal map ∆: Z Ñ Z ˆ Z, we obtain the cube Qω : Pprnsqop Ñ Modk, J ÞÑ RHomZˆZp∆˚pfJq˚ODJ, ∆˚ω‚ Zq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is a natural equivalence of cubes Qω » HHpIndCohpD˚qq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have equivalences RHomZˆZp∆˚pfJq˚ODJ, ∆˚ω‚ Zq » RHomDJˆDJp∆˚ODJ, ∆˚ω‚ DJq arising from the various adjunctions p´q˚ % p´q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' and base change associated to the pullback square DJ DJ ˆ DJ Z Z ˆ Z ∆ fJ fJˆfJ ∆ along with the equivalence f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Jω‚ Z » ω‚ DJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These can be combined with the natural equivalences RHomDJˆDJp∆˚ODJ, ∆˚ω‚ DJq » HHpIndCohpDJqq to the desired equivalences of cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The remainder of the proof justifies that these equiva- lences coherently assemble into an equivalence of cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the cases N “ 1, 2, the involved coherence issues can be deal with directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For N ą 2, we show jointly the equivalence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) and the statement that HH preserves the 69 colimit over IN ăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The fact that HH preserves the colimit in the case N “ 2 follows from the argument presented in the proof of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We ap- ply the decomposition of colimits [Lur09a, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10] and the induction assumption, to find that colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q HHpIndCohpD˚qq is equivalent to the pushout of the diagram HHpIndCohpŤ 1ďiďN´1 Di X D1qq HHpIndCohpŤ 1ďiďN´1 Diqq HHpIndCohpDt1uq , and thus given by HHpInd Cohp ˜DrNsqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Applying the induction assumption once more, we find an equivalence between the cubes Qω and HHpIndCohpD˚qq punctured at the final vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Via the universal property of the colimit, we can extend this equivalence to the entire cubes by virtue of the commutative diagram RHomZˆZp∆˚i˚O ˜DrNs, ∆˚ω‚ Zq HHpInd Cohp ˜DrNsq RHomZˆZp∆˚OZ, ∆˚ω‚ Zq HHpIndCohpZqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' » » Here i : ˜DrNs Ă Z denotes the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have constructed the equivalence of cubes (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11), concluding the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) The total Hochschild homology HHtotpIndCohpD˚qq is equivalent to the cofiber of RHomZˆZp∆˚i˚OD, ∆˚ω‚ Zq ˝∆˚u ÝÝÝÝÑ RHomZˆZp∆˚OZ, ∆˚ω‚ Zq , with u the canonical morphism OZ Ñ i˚OD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The cofiber sequence in IndCohpZq OZ Ñ i˚OD Ñ ωZ determines a total negative cyclic homology class η: krns Ñ HHS1,totpIndCohpD˚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given any categorical N-cube A˚, one has an equivalence HHtotpInd CohpA˚qq » cofpcolimJăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q HHpInd CohpAJqq Ñ HHpInd CohpAqqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10, part (1) thus follows from the equivalence colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q HHpInd CohpD˚qq » colimăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q Qω » colimJăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q RHomZˆZp∆˚pfJq˚ODJ, ∆˚ω‚ Zq » RHomZˆZp∆˚ limpIN ăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qqoppfJq˚ODJ, ∆˚ω‚ Zq » RHomZˆZp∆˚i˚OD, ∆˚ω‚ Zq » HHpInd CohpDqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 70 We proceed with part (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Pushing the given cofiber sequence forward along ∆˚, we obtain a cofiber sequence ∆˚OZ ∆˚i˚OD 0 ∆˚ωZ which determines by part (1) a class σ: krns Ñ HHtotpIndCohpD˚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This class lifts uniquely to HHS1,totpIndCohpD˚qq, as follows from the above mentioned equivalences Hn´|J| HHS1pIndCohpDJqq » Hn´|J| HHpIndCohpDJqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is an apparent analogue of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12 with the essentially same proof: Let J Ă rNs and denote by PprNsqop ěJ the subposet of PprNsqop consisting of those J1 Ă rNs satisfying that J Ă J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by IndCohpD˚ěJq the restriction of IndCohpD˚q to PprNsqop ěJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The total Hochschild homlogy HHtotpInd CohpDěJq is equivalent to the cofiber of RHomDJˆDJp∆˚piJq˚O ˜DJ, ∆˚ω‚ DJq ˝∆˚u ÝÝÝÝÑ RHomDJˆDJp∆˚ODJ, ∆˚ω‚ DJq , with iJ : ˜DJ Ă DJ the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Any commutative diagram ODJ piJq˚pO ˜DJq 0 ωDJ thus determines a total Hochschild class η: krn ´ |J|s Ñ HHS1,totpIndCohpD˚ěJqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We show that the negative cyclic class η from part (2) of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12 describes the desired left n-Calabi-Yau structure of IndCohpD˚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The recursive nondegeneracy conditions can be checked before pushing forward along the diagonal morphism ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' They amount to the following statements: (1) As in the argument for the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 recalled above, the first nondegeneracy condition amounts to the fact that the commutative diagram OZ i˚OD 0 ωZ giving rise to η is a cofiber sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 71 (2) Let J “ tj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , jlu Ă rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider pN ` 1q-cube ˜q, with qaug|PprN`1sqop ątN`1u “ q (using that PprN ` 1sqop ątN`1u » PprNsq), qaugpHq “ ω‚ Z and qaugpxq “ 0 for all other x P PprN `1sqop, exhibiting ω‚ Z as the total cofiber of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We obtain an pN ´|J|`1q-cube ˜qaug J by restricting qaug to PprN ` 1sqop ąJYtN`1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Passing to partial colimits, we obtain the commutative diagram pfJq˚ODJ pfJq˚piJq˚O ˜DJ 0 ωZr|J|s (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) with iJ : ˜DJ Ă DJ the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The diagram (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) is adjoint under pfJq˚ % f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' J to the following diagram: ODJ piJq˚O ˜DJ 0 ωDJ , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) This diagram encodes as in Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13 the induced Hochschild homology class Bj1 ¨ ¨ ¨ Bjlη: krn ´ |J|s Ñ HHS1,totpIndCohpD˚ěJqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The inductive argument below shows that the square (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) is biCartesian, yielding the non-degeneracy condition of this Hochschild class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let j P J and assume that the non-degeneracy of the class Bj1 ¨ ¨ ¨ Bjl´1η has been shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The restriction of q to PprNsqěJztjlu :“ pPprNsqop ěJztjluqop is given by the pushforward pfJztjluq˚ of a cube denoted qJztjlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The cube qJztjlu can be extended to a colimit cube qaug Jztjlu , exhibiting ω‚ DJztjlu as the total cofiber of qJztjlu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' this colimit cube is adjoint under pfJztjluq˚ % f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Jztjlu to ˜qaug Jztjlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The cube qJztjlu arises from applying the unit of the adjunction i˚ J,jl % piJ,jlq˚, with iJ,jl : DJ Ă DJztjlu, to its face PprNsqěJYtjuzPprNsqěJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We thus consider the pN´J`1q- cube qJztjlu as a morphism between this face and the opposite face, and can pass to the cofiber morphism to obtain another pN ´ J ` 1q-cube cofqJztjlu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The cube cofqJztjlu in turn arises from applying the counit of the adjunction piJ,jq˚ % i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' J,j to its face given by the cofiber pN ´ |J|q-cube, as follows from the sphericalness of piJ,jq˚ % i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' J,j, see [DKSS21, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Partially totalizing cofqJztjlu, we produce from cofqJztjlu the restriction of qaug Jztjlu to one of its faces, and thus upon passing to a partial colimit the commutative square piJ,jlq˚ODJ piJ,jlq˚piJq˚O ˜DJ 0 ωDJztjlur1s α which expresses a counit map cofpαq Ñ ωDJztjlur1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The adjoint square under piJ,jlq˚ % i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' J,jl, given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15), is thus biCartesian, concluding the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 72 Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 shows that the divisor ˜DJ Ă DJ is anticanonical for any J Ă rNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This can also be checked directly using the adjunction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note, that in the context of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, the datum that defines the Calabi–Yau structure on the categorical intersection complex tot>pIndCohpD˚qq of the normal crossings divisor D “ Ť 1ďiďN Di is identical to the datum that defines the relative Calabi–Yau structure on the functor IndCohpDq i˚ ÝÑ IndCohpZq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Namely, in both cases this is a class in the relative Hochschild homology of i˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This may be regarded as a noncommutative analog of the fact that an orientation on a manifold with corners induces a compatible system of orientations on all boundary strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We conclude the section by noting that the above construction of the categorical inter- section complex of a normal crossings divisor can be generalized to the context of cubical resolutions of schemes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [GNAPGP88]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S˚ be a In-scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', a functor In Ñ Schk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A 2-resolution of S˚ consists of a commutative diagram of In-schemes S1,1,˚ S1,0,˚ S0,1,˚ S0,0,˚ a˚ f˚ b˚ with S0,0,˚ “ S˚ and satisfying for all i P r1sn that ai and bi are closed immersions, fi is proper, S1,0,i is smooth and fi restricts to an isomorphism of schemes between S1,0,izf´1 i pS0,0,iq and S0,0,izS0,1,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider a scheme S with smooth blowup BLZpSq at a closed subvariety Z: E BLZpSq Z S { Then the above diagram defines a 2-resolution of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S be any scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there exists a 2-resolution S2 of S obtained as the pullback square of any resolution of S, see [GNAPGP88, Thm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may thus choose a 2-resolution of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We consider the first column of this 2-resolution as a I-scheme, which in turn admits a further 2-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We choose one such, denoted S3, which we consider as a I3-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We proceed in this way for all n ă N, choosing Sn`1 as a 2-resolution of the In´1-scheme obtained 73 from restricting Sn along In´1 ˆt1u ãÑ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From this, we extract an apparent IN-scheme S˚ satisfying S0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0 “ S , S1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0 “ S2 1,0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' S˚,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0 “ Sj`2 ˚,1,0 for ˚ P Ij , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' S˚,1 “ SN ˚,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S be a scheme an S˚ an IN-scheme obtained as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We call S˚ an S-augmented cubical hyperresolution if Si is smooth for all i P IN ztp0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0qu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For S “ X1 YX2 YX3 the union of three smooth subschemes with smooth intersections, a cubical hyperresolution arises from restricting the following diagram consisting of two 2-resolutions, to the ’outer’ 3-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' X1 X X2 X X3 X1 X X3 X2 X X3 X3 X1 X X2 X1 X pX2 Y X3q X1 X2 X2 Y X3 X1 Y X2 Y X3 More generally, given a scheme S, written as the union of smooth subschemes with smooth intersections, there is an apparent IN-scheme S˚ with SJ “ Ş jPJ Xj for all J P PprNsqop » IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The IN-scheme S˚ is a cubical hyperresolution of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The IN-scheme associated to a normal crossing divisor inside a scheme Z at the beginning of the section can be obtained from this hyperresolution by changing the value at the terminal vertex of the N-cube from S “ Ť jPrNs Xj to Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given an IN-scheme S˚, we obtain a functor IndCohpS˚q: IN Ñ Stk by composing with IndCoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The diagram IndCohpS˚q describes a categorical cube, which is in general neither spherical, nor does its totalization admit a Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the case where S˚ is an S-augmented cubical hyperresolution, it seems an intriguing question to explore the implications of considering the truncation τě1 tot> IndCohpS˚q as a resolution of the stable 8-category IndCohpSq in terms of smooth k-linear stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 Picard-Lefschetz theory and Fukaya-Seidel complexes We begin by explaining how classical Picard–Lefschetz theory can be used to construct certain cell complexes modelling the cohomology of affine varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 74 Let X Ă CN be an n-dimensional smooth affine subvariety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X1 Ă X be a generic hyperplane section of X in CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The pair pX, X1q induces an exact triangle S‚pXq S‚pX, X1q S‚´1pX1q B `1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) in singular homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this context, the classical Lefschetz hyperplane theorem, due to, in the given affine setup, to Andreotti-Frankel, implies that the complex S‚pX, X1q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' has homology concentrated in degree n, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' HnpX, X1q is generated by vanishing thimbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, the exact triangle (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) provides a means of computing the homology H‚pXq in terms of the group HnpX, X1q and the homology H‚pX1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may iterate this construction, choosing a sequence of generic hyperplane sections X Ą X1 Ą X2 Ą ¨ ¨ ¨ Ą Xn so that the the corresponding exact triangles of consecutive pairs combine to give a description of the singular homology H‚pXq as the homology of the complex HnpX, X1q Ñ Hn´1pX1, X2q Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ H0pXnq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) Under suitable technical assumptions and with suitable choices of symplectic structures (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [Sei08], P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Seidel showed that a categorical variant of the above discussion can be implemented to provide an effective means for computing Fukaya categories of affine varieties in terms of Lefschetz fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Namely, suppose that the hyperplane section X1 “ π´1 1 pt1uq is given as the fiber of a Lefschetz fibration π1 : X Ñ C (such as the restriction of a generic linear function on CN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then we have a “left exact” sequence of categories FukpXq ãÑ FSpπ1q B ÝÑ FukpX1q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', the category FukpXq is the kernel of the functor B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As shown in [Sei08], the category FSpπ1q admits an exceptional collection of objects given by Lagrangian vanishing thimbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Again, we may iterate this consideration to produce a categorical complex FSpπ‚q :“ FSpπ1q Ñ FSpπ2q Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ñ FukpXnq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) where FSpπkq is the Fukaya–Seidel category of a Lefschetz fibration πk : Xk´1 Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that, since each of the Fukaya–Seidel categories FSpπkq has an excep- tional collection given by Lagrangian vanishing thimbles of πk, we have, by the Lefschetz hyperplane theorem, an isomorphism of abelian groups K0pFSpπkqq – HnpXk´1, Xkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Fur- ther, the complex K0pFSpπ‚qq reproduces the complex (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that in the original reference [Sei08] all Fukaya categories are described in the frame- work of A8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may turn them into k-linear 8-categories by virtue of the coherent nerve construction REF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Further, in order to obtain presentable 8-categories, we pass to Ind- completions: Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The categorical complex IndpFSpπ‚qq is called the (Ind-completed) Fukaya– Seidel complex of the family of Lefschetz fibrations tπ‚u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As will be seen in the examples below, the categories of cycles of the complex IndpFSpπ‚qq correspond to wrapped variants of the Fukaya categories considered in [Sei08] (the Fukaya categories in [Sei08] are always generated by compact Lagrangian submanifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Mirror symmetry In the context of homological mirror symmetry, we expect equivalences between the categorical complexes from §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 and §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this section, we illustrate this “mirror symmetry for categorical complexes” in a some- what familiar context and then formulate a conjecture as to what to expect in greater gener- ality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To this end, we start recalling a well-known example of homological mirror symmetry (see [Sei01, AKO08] for details): We consider the affine hypersurface X “ txyz “ 1u Ă C3 and note that X – pC˚q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the Lefschetz fibration π1 : X Ñ C given by the restriction of the linear function π1 “ x`y`z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The general fiber of π1 is an elliptic curve with 3 punctures which degenerates to a nodal cubic over the 3 critical values of π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a regular fiber, we may take X1 :“ π1p0q “ Eztp1, p2, p3u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A depiction of the 3 vanishing spheres in X1 that correspond to the critical values can be found in [Sei01, §3B]: This example does not directly fit into the context discussed in [Sei08], but it is explained in [Sei01] how to associate to π1 a Z-graded Fukaya-Seidel category which can be described as follows: The category FSpπ1q is equivalent to the derived category of the quiver ‚ ‚ ‚ a1 a2 a3 b1 b2 b3 subject to the relations biaj “ bjai, for i ‰ j, and biai “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus, we further have FSpπ1q » CohpP2q by [Bei84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a next step, we choose the Lefschetz fibration π2 : X1 Ñ C obtained by restricting the linear function π2 :“ x ´ y on C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is a ramified covering map of degree 3, the fibers of π2 move in the pencil generated by x ´ y and w on E whose general fiber consists of 3 distinct points degenerating over the 6 nondegenerate critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Again, this setup does not directly fit into [Sei08], but the necessary modifications are explained in [Sei01].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We may take the fiber X2 :“ π´1 2 p0q “ ta, b, cu as a regular fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the 6 critical values correspond to 6 vanishing cycles (0-spheres = pairs of points) in X2: ta, bu, ta, bu, ta, cu, ta, cu, tb, cu, tb, cu 76 The directed subcategory on these can be described by the quiver ‚ ‚ ‚ ‚ ‚ ‚ a b a b a c c b c with zero relations given by the rule that the composite of composable arrows is zero iff they are labelled by different letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We omit the discussion of grading choices (which is explained in [Sei01]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Alternatively, we may describe this category as a topological Fukaya category of the Rie- mann surface Eztp1, p2, p3u with one stop at each puncture (or rather the cylindrical end corresponding to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The corresponding spine is given by the Ribbon graph Γ “ so that the formalism of [Dyc17] yields the diagram of Ribbon graphs depicted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 This latter diagram yields an equivalence between the topological Fukaya category and the colimit of the corresponding diagram of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here, ReppA2q denotes the bounded derived category of representations of the quiver A2 “ ‚ ‚, the functors of the form Cohpptq Ñ ReppA2q are induced by the inclusion of source (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' target) of the quiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The functors of the form Cohpptq Ñ CohpP1q are given by pushforward along the inclusion of the points 0 and 8, respectively, into P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Again, there are auxiliary choices to be made to determine the Z-grading, such as a trivialization of the tangent bundle (or rather its square) of Eztp1, p2, p3u, which we do not discuss here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Informally, the colimit of the diagram from Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 can thus be described by starting with three disjoint copies of CohpP1q and freely adjoining three arrows connecting the various skyscraper sheaves of the projective lines at 0 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We keep this intuition in mind by denoting the resulting category by Cohp q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) describing it as an amalgamate of commutative geometry (the projective lines) and noncom- mutative geometry (the quiver arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In terms of the constructions of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4, this means that the category is described by the directed pushout of three copies of CohpP1q along three copies of Cohpptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 77 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1: Decomposition of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' CohpP1q Cohpptq CohpP1q CohpP1q Cohpptq Cohpptq Cohpptq Cohpptq Cohpptq ReppA2q ReppA2q ReppA2q Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2: Diagram computing the topological Fukaya category of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 78 The above choices of potentials π1, π2 in total yield a Fukaya-Seidel complex of the form FSpπ1q ÝÑ FSpπ2q ÝÑ FukpX2q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) with (1) FSpπ1q » CohpP2q (2) FSpπ2q » FukpΓq » Cohp q (3) FukpX2q » Cohppt > pt > ptq On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' consider P2 with the normal crossings divisor given by three lines L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' L2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' L3 in general position: L1 L2 L3 Ă P2 The corresponding categorical cubical diagram induced by the various push-forward func- tors takes the following form: CohpHq CohpL2 X L3q CohpL1 X L3q CohpL3q CohpL1 X L2q CohpL2q CohpL1q CohpP2q Its coproduct totalization yields the categorical intersection complex Cohppt>3q Cohp q CohpP2q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) where we use the above notation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) to denote the middle term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, all terms of the complexes (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We do not verify here the expectation that the differentials in both complexes are in fact adjoint to one another (this can probably be ex- tracted with some effort from the existing literature) but rather formulate the more general conjecture: 79 Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X “ pC˚qn Ă Cn`1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there exists a family of iterated Lefschetz fibrations π‚ on X such that the corresponding Fukaya–Seidel complex FSpπ‚q is equivalent to the (right adjoint of the) categorical intersection complex of the normal crossings divisor given by n ` 1 hyperplanes in Pn in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The conjecture has its natural generality in some context of normal crossings divisors in toric Fano varieties, where Hori-Vafa mirror symmetry provides a prediction of the mirror potential , but for the sake of concreteness, we leave it as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While presenting some of this material during a talk in the Edinburgh Hodge Seminar, we learned about some work in progress on wrapped Fukaya categories of “multi-potentials” which seems to be closely related to the perspective on homological mirror symmetry via categorical complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Indeed, once the multi-potential approach is implemented, one expects that this directly yields cubical diagrams of wrapped Fukaya categories which are mirror to the cubical diagrams of coherent sheaves from which we build the categorical intersection complex via totalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This “mirror symmetry conjecture for cubes” has for example been described in the recent article [Lee22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Once this type of cubical mirror symmetry is established, our Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 then essentially reduces to a statement that our Fukaya-Seidel complexes from Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 arise as totalization of the cubical wrapped Fukaya-category diagrams that one expects to associate to a multi-potential Landau–Ginzburg model (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [AGHJ22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This statement, as well as its relevance for higher–dimensional perverse schobers (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3), seems interesting in its own right and will be investigated in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Manifolds with corners and complexes of 8-local systems We fix a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a space X, we denote by LocpXq “ FunpX, Dpkqq the stable 8-category of Dpkq-valued local systems on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a morphism f : X Ñ Y between spaces, we denote by f˚ : LocpY q Ñ LocpXq the pullback functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It admits left and right adjoints f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' and f˚, given by left Kan extension and right Kan extension, respectively, see [Lur09a, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall the following Theorem from [BD19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X be a compact oriented manifold of dimension n with boundary f : BX Ă X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the functor f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' : LocpBXq ÝÑ LocpXq admits a canonical left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The goal of this section is to show that Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 admits an extension to categorical complexes arising from oriented bordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the simplest case, an oriented n-dimensional bordisms between two closed, oriented pn ´ 1q-dimensional manifolds M, M1 consists of an oriented n-dimensional manifold N with boundary BN “ M >M1, with M and M1 carrying the induced orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If instead M and M1 are not closed and not disjoint in N, but instead overlap on their boundary, one can further ask that M and M1 again define two pn ´ 1q- dimensional bordisms between pn ´ 2q-dimensional manifolds, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This perspective can be formalized by organizing oriented bordisms up to dimension n in an p8, nq-category of oriented bordisms Bordor n , whose m-cells are m-dimensional oriented bordisms for m ď n, see 80 for instance [Lur09b, CS19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the following, we associate to a functor from the n-simplex to the p8, nq-category Bordor n a categorical n-cube whose totalization is left Calabi–Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We fix an integer n ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given an n-simplex ∆n, we denote by Sd ∆n its barycentric subdivision, considered as a poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by Mfd the 1-category with objects compact oriented manifolds of any dimension with boundary and morphisms oriented inclusions into the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A functor from the n-simplex to Bordor n amounts to a functor A˚ : Sd ∆n Ñ Mfd, mapping each m-simplex x P p∆nqm to an m-dimensional manifold Ax (possibly empty), such that the boundary of Ax is given by BAx “ m ď i“0 Adipxq for any x P p∆nqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We further ask that the intersection of Adipxq and Adjpxq is given by Adjdipxq for any j ă i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is an equivalence of posets φ: Sd ∆n » In ztp1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 1qu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the barycentric subdi- vision of ∆n is a cube with the initial vertex removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given an n-simplex A˚ in Bordor n , we hence find a categorical cube LocpA˚q: In Ñ Stk , with LocpA˚qφpxq “ LocpAφpxqq and LocpAqpp1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',1qq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We will omit the usage of the equiva- lence φ in the following, and write for instance AJ and LocpAJq for Aφ´1pJq and LocpAφ´1pJqq, with J Ă rns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ be an n-simplex in Bordor n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the categorical n-cube LocpA˚q admits a canonical left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A˚ be an n-simplex in Bordor n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the coproduct totalization tot>pLocpA˚qq admits a canonical left n-Calabi–Yau structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Combine Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Before proving Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2, we recall some details from the proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 from [BD19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Given a space X, we denote by C‚pXq P Dpkq its singular chain complex and by LX its free loop space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There are natural morphisms in Dpkq C‚pXq Ñ C‚pLXq » HHS1pLocpXqq , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) where the first morphisms arises from the inclusion of the constant loops X Ñ LX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If X is a closed n-dimensional manifold, any orientation, considered as an element of HnC‚pXq, gives via the above map rise to a left n-Calabi–Yau on LocpXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If Y is an n-dimensional manifold with boundary X, then a relative orientation of Y is a suitable n-th homology class of C‚pY, Xq :“ cofpC‚pXq Ñ C‚pY qq and its image in HHS1pLocpY q, LocpXqq gives rise to a relative Calabi–Yau structure on the functor LocpXq pXĂY q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' ÝÝÝÝÝÑ LocpY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There exists a natural map C‚pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q, BAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq ÝÑ HHS1,totpLocpA˚qq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 81 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the diagram C‚pA˚q: In Ñ Modk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The colimit of its restriction to In ăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q “ In ztp0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , 0qu is by Mayer-Vietoris equivalent to C‚pBAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the following dia- gram in Dpkq: C‚pBAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq C‚pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq C‚pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q, BA0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq colimJăp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0q HHS1,totpAJq HHS1pAp0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=',0qq HHS1,totpAq The left square commutes by the naturality of the morphisms (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Both the upper and lower sequence in this diagram are cofiber sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows that there exists a dotted arrow as indicated, making the diagram commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let x P ∆n be the top cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Any choice of orientation of Ax rel- ative to BAx determines by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 a total negative cyclic homology class η: krns Ñ HHS1,totpLocpA˚qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Its restriction to HHS1,totpLocpABi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Bj˚qq arises via an analogue of Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 for the cube ABi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='Bj˚ from an induced orientation of Adi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='djpxq relative to BAdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='djpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using that passing to local systems preserves colimits, we find for any J Ă rns a commu- tative diagram LocpBAJqq LocpŤm jRJ AJYtjuq LocpAJq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' colimIn ăJ LocpA˚q “ » The total Hochschild class of LocpA˚q thus restricts by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 to a Calabi-Yau struc- ture on the lower horizontal morphism above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This shows that η defines a left n-Calabi-Yau structure for LocpA˚q, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the pn ` mq-simplex A in Bordn`m, with Arn`ms “ Dn ˆ Dm the product of the n- and m-dimensional real unit discs and Ad0rn`ms “ pBDnq ˆ Dm “ Sn´1 ˆ Dm , Ad1rn`ms “ Dm ˆ Sm , Ad2 0rn`ms “ Sn´1 ˆ Sm´1 and Ax “ H otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The totalization tot>pLocpAqq is given by the complex LocpSn´1 ˆ Sm´1q ÝÑ LocpDn ˆ Sm´1q Ð> LocpSn´1ˆSm´1q LocpSn´1 ˆ Dmq ÝÑ LocpDn ˆ Dmq concentrated in degrees 2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Except for being left pn ` mq-Calabi–Yau, this complex is furthermore spherical as it arises from totalizing the following spherical square: LocpSn´1 ˆ Sm´1q LocpDn ˆ Sm´1q LocpSn´1 ˆ Dmq LocpDn ˆ Dmq 82 The above square arises from a pullback square of Kan fibrations and is hence Beck-Chevalley, see [Cis19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These Kan fibrations are furthermore spherical fibra- tions, meaning that their fibres are spheres, the sphericalness of the functors was thus shown in [Chr22c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We note that in general, tot>pLocpAqq is not a spherical complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A version of this example producing an n-term complex arises by starting with the product of n ` 1 discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8 Lax additivity In this section we develop the general framework of lax additive (or 2-additive) p8, 2q- categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 Additive 1-categories To explain our philosophy, let us first remind the reader of the classical story for ordinary additive categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We start by recalling the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A category A is called additive if: (1) The category A is enriched in abelian monoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', each hom-set Apx, yq has an asso- ciative, commutative addition ` with neutral element 0 such that composition Apx, yq ˆ Apy, zq Ñ Apx, zq preserves ` and 0 in each argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Each commutative monoid pApx, yq, `, 0q admits negatives, hence is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) The category A admits finite products and coproducts (including empty ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4) For each finite set of objects x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , xn P A, the natural map n ž s“1 xs ÝÑ n ź t“1 xt, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) whose components xs Ñ xt are # 1: xs Ñ xs , if s “ t 0 P Apxs, xtq , otherwise , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A category only satisfying (1), (3) and (4) is called semiadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' One typically identifies finite products and coproducts via the canonical map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) and uses the symbol ‘ (called direct sum or biproduct) for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The use of the phrase “is called additive if” implies that being additive is a property of the category A rather than extra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is justified by the that the addition on 83 the hom-sets of an additive category is uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly, it is given by the following formula: Given two maps f, g: x Ñ y, their sum is the composite x Ñ x ‘ x f‘g ÝÝÑ y ‘ y Ñ y (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) where the first map is the diagonal x Ñ x ˆ x and the last map is the codiagonal y > y Ñ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this sense, the biproduct structure ‘ determines the addition structure ` on the hom- sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The converse is also true, as explained by the following lemma: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A be a category enriched in abelian monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , xn be a finite set of objects in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Let x be an object of A equipped with a cone P “ pps : x Ñ xsqn s“1 and a cocone I “ pis : xs Ñ xqn s“1 satisfying the two equations (a) nÿ s“1 is ˝ ps “ 1 P Apx, xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) (b) pt ˝ is “ # 1 P Apxs, xtq, if s “ t 0, otherwise (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) Then P and I exhibit x as the product śn s“1 xn and as the coproduct šn s“1 xn, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Morover, the canonical comparison map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) is the identity 1: x Ñ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Assume the product x “ śn s“1 exists and let P “ pps : x Ñ xsqn s“1 be the product cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there exists a unique cocone I “ pis : xs Ñ xqn s“1 satisfying conditions (a) and (b) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) Dually, for every coproduct cocone I “ pis : xs Ñ xqn s“1 there exists a unique cone P “ pps : x Ñ xsqn s“1 satisfying (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since ‘ and ` determine each other, we have the following corollary: Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Let A be a category enriched in abelian monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If A admits finite products (equivalently, finite coproducts) then it is automatically semiadditive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Let F : A Ñ A1 be a functor between additive categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following are equivalent: (a) the functor F preserves products;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (b) the functor F preserves coproducts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (c) the functor F preserves the addition on the hom-sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' However, its proof is very instructive for categorification, so we shall explain it here: 84 Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) We show that P is a product cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the statement about I is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We need to show that for each t P A the natural map P˚ : Apt, xq Ñ n ź s“1 Apt, xsq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' f ÞÑ pps ˝ fqn s“1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using I we can produce an explicit inverse via the formula I˚ : pfsqn s“1 ÞÑ nÿ s“1 isfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) It satisfies pP˚ ˝ I˚qpfsqn s“1 “ P˚p nÿ s“1 isfsq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) “ ppu ˝ nÿ s“1 isfsqn u“1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) “ p nÿ s“1 puisfsqn u“1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) “ pfuqn u“1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) (using equation (b) in the last step) and pI˚ ˝ P˚qpfq “ I˚pps ˝ fqn s“1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) “ ÿ s“1 pispsfq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) “ p ÿ s“1 ispsq ˝ f (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17) “ 1 ˝ f “ f (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) (using equation (a) in the last step), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Moreover, equation (b) says precisely that the identity 1: x Ñ x satisfies the defining equation to be the map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) By the univeral property of the product cone P, there are unique maps is : xs Ñ x satisfying equation (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These maps then assemble into the desired cocone I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To verify equation (a) it suffices to postcompose with all the product projections pu and compute pu ˝ nÿ s“1 isps “ nÿ s“1 puisps (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19) “ pu “ 1 ˝ pu (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20) (using equation (b) in the second step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 85 There is one last aspect of additive categories which is going to be useful to categorify: matrix calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is the observation that in any category A any map f : n ž s“1 xs ÝÑ m ź t“1 yt (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21) from a coproduct to a product can be encoded through the bijection A ˜ n ž s“1 xs, m ź t“1 yt ¸ – m ź t“1 n ź s“1 Apxs, ytq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22) as an m ˆ n-matrix pftsqm,n t“1,s“1 whose entry fts is a map xs Ñ yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The special thing about semiadditive categories is that it makes sense to consider the composite h: n ž s“1 xs fÝÑ m ź t“1 yt – n ž t“1 yt gÝÑ lź u“1 zu (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='23) of two such maps by using the identification (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This composite corresponds to a matrix phusql,n u“1,s“1 P lź u“1 n ź s“1 Apxs, zuq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) It is not hard to verify that the matrix corresponding to the composite h arises from the matrices of f and g by the usual rule for matrix multiplication: hus “ m ÿ t“1 guthts (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25) From this perspecive, the identification (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) is just the identity matrix which has identities on the diagonal an zeroes everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 p8, 2q-categories In this paper, we think of p8, 2q-categories as categories enriched in the 8-category Cat8 of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For a general treatment of enriched 8-categories, we refer to the work of Gepner and Haugseng [GH15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Our goal is not to develop any p8, 2q-categorical foundations but rather to develop the theory of lax additivity while assuming that such foundations are already laid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In practice, this means that none of our arguments and constructions are performed explicitly in a model, but only using the general high-level features which any theory of p8, 2q-categories is expected to share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We treat these ingredients axiomatically: Let C be an p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It has an underlying 8-category C1, and underlying 8-groupoid C» “ pC1q».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 86 It has a hom-functor Cp´, ´q: Cop ˆ C Ñ Cat8, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) which takes values in the p8, 2q-category of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Occasionally, it is convenient to consider the hom-functor Cp´, ´q: Cop 1 ˆ C1 Ñ Cat8 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) as a functor of the underlying 8-categories, and its associated cartesian fibration Tw˚pCq “ ż ˚ Cp´, ´q Ñ C1 ˆ Cop 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) There are composition functors CpX, Y q ˆ CpY, Zq Ñ CpX, Zq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) functorial in X, Y, Z : C».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Composition is coherently associative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' this is formalized in [GH15] by encoding the p8, 2q-category C as an algebra in the monoidal 8-category pCat8, ˆq of a certain generalized nonsymmetric operad ∆op C» Ñ ∆op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More generally, the composition map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) is also natural in X : Cop 1 , Z : C1 and di- natural in Y : C1 (and not just in their groupoid cores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thinking in terms of fibrations, this means that composition can be written as the dashed functor şX Y CpX, Y q ˆpY :C1q şY Z CpY, Zq şX Z CpX, Zq pX : C1q ˆ pY : C1q ˆ pZ : C1q pX : C1q ˆ pZ : C1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) of mixed (cartesian, cocartesian) fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It makes sense to talk about adjunctions f % fR : X Ñ Y in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These are characterized by the fact that pf˝q % pfR˝q: CpT, Xq Ñ CpT, Y q and p˝fRq % p˝fq: CpX, Tq Ñ CpY, Tq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) are adjunctions of 8-categories for all T : C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the purpose of developing the theory of lax additivity we do not need the full coher- ent associativity of the composition law, but only its incoherent shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More precisely, it suffices to postcompose the enrichment with the symmetric monoidal functor pCat8, ˆq Ñ phoCat8, ˆq, and think of C as enriched in the homotopy category of 8-categories up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Throughout this section we use the notation “x : A” borrowed from homotopy type theory to say that x is a term/inhabitant/element/object of the (8-)groupoid, (8- )category, or even p8, 2q-category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When we construct an object “Fpxq : B for each x : A”, it is understood that Fpxq is supposed to be functorial in x in the appropriate sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This allows us to unambiguosly write formulas such as colimx:A Fpxq or pFpxqqx:A, which of course only make sense with the additional functoriality in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We reserve the notation x P A for the case when A is discrete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (equivalent to) a set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in this case, the question of functoriality is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 87 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 Lax limits and colimits We start by recalling the definition of a lax limits and colimits in a p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S be an 8-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' First, let X: S Ñ Cat8 be a diagram of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let laxlim X be the 8-category of sections of the (covariant) Grothendieck construction ş ˚ X Ñ S associated to functor X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Informally, objects of laxlim X consist of (1) for each object s of S, an object xs in Xs, (2) for each edge f : s Ñ t in S, a morphism xf : Xfpxsq Ñ xt in Xt, (3) for each 2-simplex s fÝÑ t gÝÑ u (with the composite gf implicit) in S a 2-simplex Xgpxtq Xgfpxsq xu xg Xgpxfq xgf (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) in Xu, (4) and so on for higher simplices of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We will denote an object of laxlims:S X as a tuple pxsqs:S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Now, let C be an arbitrary p8, 2q-category and X: S Ñ C a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A lax cone over the diagram X with vertex L is an object pφsqs:S : laxlims:S CpL, Xsq, where s ÞÑ CpL, Xsq is the S-shaped diagram in Cat8 obtained as the composite S XÝÑ C CpL,´q ÝÝÝÝÑ Cat8 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) Informally, such a cone consists of (1) for each object s : S, a structure map φs : L Ñ Xs, (2) for each arrow f : s Ñ t in S, a lax cone L Xs Xt φt φs Xf (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' a map φf : Xfφs Ñ φt in CpL, Xtq, (3) together with coherent pasting identifications, φg ˝ Xgφf » φgf for composable arrows s fÝÑ t gÝÑ u in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 88 For each other object L1 : C we have a canonical composition map CpL1, Lq ˆ laxlims:S CpL, Xsq laxlims:S CpL1, Xsq laxlims:S CpL1, Lq ˆ laxlims:S CpL, Xsq laxlims:S CpL1, Lq ˆ CpL, Xsq ´˝´ ∆ˆid » (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) which informally sends a cone pφsqs:S with vertex L and a morphism l: L1 Ñ L to the cone pφs ˝ lqs:S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dually we can define the 8-category of lax cocones on X with vertex L as laxlims:Sop CpXs, Lq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly, a such a cocone pψsqs:S has structure maps ψs : Xs Ñ L and lax triangles L Xs Xt ψs Xf ψt φf (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) over each arrow f : s Ñ t of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A cone P “ ppsqs:S : laxlims:S CpL, Xsq is called a lax limit cone if for each object L1 : C the functor ´ ˝ P : CpL1, Lq ÝÑ laxlims:S CpL1, Xsq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) is an equivalence of 8-categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in this case we call the object L a lax limit of the dia- gram X: S Ñ C and write L “ laxlims:S Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dually, we say that a cocone I “ pisqs:Sop : laxlims:Sop CpXs, Lq is a lax colimit cone if for each L1 : C the functor I ˝ ´: CpL, L1q ÝÑ laxlims:Sop CpXs, L1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) is an equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in this case we call L a lax colimit of X and write L “ laxcolims:S Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let C “ Cat8 be the p8, 2q-category of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X: S Ñ Cat8 be a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) As the notation suggests, the lax limit of X is the 8-category L :“ laxlims:S Xs :“ FunSpS, ş s:S Xsq of sections of the corresponding Grothendieck construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Indeed, naturally in L1 : Cat8 we have the equivalence Cat8pL1, Lq “ FunpL1, FunSpS, ż ˚ Xqq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) » FunSpS ˆ L1, ż ˚ Xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) » FunSpS, ż s:S FunpL1, Xsqq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) “ laxlims:SpCat8pL1, Xsqq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) 89 which is induced by composition with the canonical cone P “ pps : L “ laxlimS X Ñ Xsqs:S (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) given by evaluation of sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The lax colimit of the diagram X is the contravariant Grothendieck construction laxcolims:S Xs “ ż s:S Xs, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) exhibited by the canonical cocone I “ ˆ is : Xs Ñ ż ˚ X ˙ s:Sop (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) that includes the individual fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume now that the diagram X takes values in stable 8-categories, (3) The 8-category laxlims:S Xs “ FunSpS, ş ˚ Xq is again stable because limits and colim- its of sections are computed pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the same reason, every functor F : L1 Ñ laxlims:S X is exact if and only if each composite ps ˝ F is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It follows that the cone P exhibits the 8-category laxlims:S Xs also as a lax limit in the p8, 2q-category of stable 8-categories and exact functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (4) The 8-category ş˚ X, which is the lax colimit of X in Cat8, is typically not stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' to compute the lax colimit of X in the p8, 2q-category of stable 8-categories one therefore has to stabilize this 8-category, which is a rather tricky operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' However, we it will follow from the theory of lax matrices that this lax colimit indeed just agrees with the lax limit which can be computed in Cat8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 Lax matrices Analogously to the case of ordinary coproducts and products (which corresponds to the case where the category S is just a set), we can interpret maps from a lax colimit to a lax limit as a sort of matrices: By the defining property we have Cplaxcolims:S Xs, laxlimt:T Ytq » laxlimt:T Cplaxcolims:S, Ytq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) » laxlimt:T laxlims:Sop CpXs, Ytq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) » laxlimpt,sq:TˆSop CpXs, Ytq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) so that we can interpret a map α: laxcolims:S Xs Ñ laxlimt:T Yt as a tuple pαt,sqpt,sq:TˆSop which we think of as a matrix whose rows are indexed by T and whose columns are indexed by Sop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We define laxMatCpX, Yq :“ laxlimpt,sq:TˆSop CpXs, Ytq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) Note that this is a well defined 8-category even when laxcolim X and/or laxlim Y does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When X: t˚u Ñ C is just an object X “ X˚, we still use the notation laxMatCpX, Yq “ laxMatCpX, Yq and observe that it is precisely the 8-category of lax cones on Y with vertex X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' and analogously for lax cocones 90 Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S “ T “ ∆1 “ t0 fÝÑ 1u be the walking arrow and consider two diagrams X: S Ñ C and Y: T Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then we can compactly describe objects of laxMatCpX, Yq “ laxlim ¨ ˚ ˚ ˚ ˝ CpX0, Y0q CpX1, Y0q CpX0, Y1q CpX1, Y1q ˛ ‹‹‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) as T ˆ Sop-indexed diagrams in the Grothendieck construction, which we depict as follows: ¨ ˚ ˚ ˝ α00 α01 α10 α11 ˛ ‹‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) Explicitly unpacking this notation, such a matrix consists of: (1) four 1-morphisms α00 : X0 Ñ Y0 α01 : X1 Ñ Y0 α10 : X0 Ñ Y1 α11 : X1 Ñ Y1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) (2) four 2-morphisms Yf ˝ α00 α00 α01 ˝ Xf α10 Yf ˝ α01 α10 α11 ˝ Xf α11 αf0 α0f αf1 α1f (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) (3) assembling into a commutative square Yf ˝ α00 Yf ˝ α01 ˝ Xf α10 α11 ˝ Xf αf1 Yf˝α0f αf1˝Xf α1f (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) We now introduce the lax matrix multiplication which categorifies the classical formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The classical formula involves a finite sum of elements in some hom-set Apxs, zuq of the category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Our categorical analog of these sums will be categorical colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the remainder of this section, let C be an p8, 2q-category enriched in 8-categories with colimits, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', each hom-category CpX, Y q has all colimits and and each composition functor CpX, Y q ˆ CpY, Zq Ñ CpX, Zq preserves colimits in each variable separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 91 Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S be an 8-category, and X: S Ñ C a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Passing to the cartesian fibrations classifying the respecive hom-functors, we obtain a commutative square Tw˚pSq “ ş˚ Sp´, ´q ş˚ Cp´, ´q “ Tw˚pCq S ˆ Sop C1 ˆ Cop 1 p α q XˆXop (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) which amounts to the dashed section şps,tq CpXs, Xtq Tw˚pCq Tw˚pSq S ˆ Sop C1 ˆ Cop 1 { q p α XˆXop (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) of the pullback-fibration pX ˆ Xopq˚pqq which informally sends an arrow pf : s Ñ tq : Tw˚pSq to Xf : CpXs, Xtq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can now construct the composite functor laxlims:S CpL, Xsq ˆ laxlimt:Sop CpXt, L1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) “ FunSˆSop ˜ S ˆ Sop, ż ps,tq CpL, Xsq ˆ CpXt, L1q ¸ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) ÝÑ FunSˆSop ˜ Tw˚pSq, ż ps,tq CpL, Xsq ˆ CpXt, L1q ¸ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) ÝÑ FunSˆSop ˜ Tw˚pSq, ż ps,tq CpXs, Xtq ˆSˆSop ż ps,tq CpL, Xsq ˆ CpXt, L1q ¸ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17) “ FunSˆSop ˜ Tw˚pSq, ż s CpL, Xsq ˆS ż ps,tq CpXs, Xtq ˆSop ż t CpXt, L1q ¸ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) ÝÑ Fun ` Tw˚pSq, CpL, L1q ˘ colim ÝÝÝÑ CpL, L1q, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19) where the first arrow is pullback of sections along p: Tw˚pSq Ñ S ˆ Sop, the second arrow adds the section α in the first component of the fiber product, the third arrow is given by composition with the composition map ż s CpL, Xsq ˆS ż ps,tq CpXs, Xtq ˆSop ż t CpXt, L1q ÝÑ CpL, L1q, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20) the last arrow is just the colimit functor in the 8-category CpL, L1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' On objects, this functor takes a lax cone and a lax cocone on X, Φ : laxlims:S CpL, Xsq and Ψ : laxlimt:Sop CpXt, L1q, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21) 92 and sends them to the map Ψ ˝ Φ: L Ñ L1 defined by the formula Ψ ˝ Φ :“ colim pf:sÑtq:Tw˚pSqpΨt ˝ Xf ˝ Φsq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22) Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When S “ t1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , nu is a finite set the (cartesian) twisted arrow category Tw˚pSq Ñ S ˆ S can be canonically identified with the diagonal ∆: S Ñ S ˆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Under this identification the formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22) simplifies to pΨ ˝ Φq “ pΨ ˝S Φq “ ž s,tPS s“t Ψt ˝ id ˝Φs “ n ž s“1 Ψs ˝ Φs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) which is just the usual multiplication (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25) of the row vector Ψ with the column vector Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We assemble our categorified analog of row-column multiplication to the lax version of matrix multiplication: Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S, U be 8-categories, and X: S Ñ C and Z: U Ñ C two diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each object Y : C we consider the functor laxlims:Sop CpXs, Y q ˆ laxlimu:U CpY, Zuq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26) “ laxlimpu,sq:UˆSop CpXs, Y q ˆ CpY, Zuq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27) ÝÑ laxlimpu,sq:UˆSop CpXs, Zuq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28) induced by composition of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' On objects it takes a lax cocone and a lax cone, Ψ : laxlims:Sop CpXs, Y q and Φ : laxlimu:S CpY, Zuq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29) and sends them to the matrix Φ ˝ Ψ : laxlimpu,sq:UˆSop described by the formula pΦ ˝ Ψqus “ Φu ˝ Φs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='30) More generally, we can replace the object Y : C by a diagram Y: T Ñ C and consider the functor laxMatCpX, Yq ˆ laxMatCpY, Zq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31) “ laxlimpu,sq:UˆSop plaxlimt:T CpXs, Ytq ˆ laxlimt1:T op CpYt1, Zuqq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='32) laxlimu,sp´˝T ´q ÝÝÝÝÝÝÝÝÝÝÑ laxlimpu,sq:UˆSop CpXs, Zuq “ laxMatCpX, Zq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='33) which is given in componentwise in u, s by the composition functor from Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11 (applied to lax cones and cocones on Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly, this functor is given by the formula Φ ˝ Ψ “ Φ ˝T Ψ “ ˆ colim pf:tÑt1q:Tw˚pTqpΦut1 ˝ Yf ˝ Ψtsq ˙ pu,sq:UˆSop .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='34) This is what we call the lax matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, we can assemble all lax matrices of different shapes into a category hoLaxMatC: 93 Objects are equivalence classes of diagrams X: S Ñ C, where S is any small 8-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Morphisms from X: S Ñ C to Y: T Ñ C are equivalence classes of matrices Φ : laxMatCpX, Yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Composition is given by lax matrix multiplication of Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that the lax matrix multiplication is functorial by construction, making it in partic- ular well defined on equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To prove that hoLaxMatC is indeed a category, we will thus only need to construct the identity matrix and prove that lax matrix multiplication is associative up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In , we shall prove a slightly stronger statement: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The lax matrix multiplication of Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25 is (1) associative up to natural equivalence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' for diagrams W: R Ñ C and X: S Ñ C and Y: T Ñ C and Z: U Ñ C we have p´ ˝T ´q ˝S ´ » ´ ˝T p´ ˝S ´q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='36) as functors laxMatCpW, Xq ˆ laxMatCpX, Yq ˆ laxMatCpY, Zq ÝÑ laxMatCpW, Zq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='37) (2) unital up to natural equivalence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', for each diagram X: S Ñ C there is a matrix IX : laxMatCpX, Xq with components IX ts “ colim f:Sps,tq Xf : CpXs, Xtq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='38) such that IX ˝S ´ » id and ´ ˝S IX » id (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='39) as endofunctors of laxMatCpY, Xq and laxMatCpX, Yq, respectively (for each other dia- gram Y: U Ñ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The category hoLaxMatC is of course only the truncation of an p8, 2q- category of lax matrices, which one could construct with more effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Even Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35 only shows that lax matrix multiplication is associative up to natural equivalence, but does not exhibit any sort of coherence such as the pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We will not be needing this additional layer of coherence in this article so Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35 will suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The proof of associativity is relatively straightforward: Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35 (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For matrices F : laxlimSˆRoppWr, Xsq, G : laxlimTˆSoppXs, Ytq, H : laxlimUˆT oppYt, Zuq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='41) 94 we compute ppH ˝T Gq ˝S Fqur » colim f : sÑs1pH ˝T Gqus1XfFsr (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='42) » colim f : sÑs1pcolim g : tÑt1 Hut1YgGts1qXfFsr (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='43) “ colim f : sÑs1 colim g : tÑt1pHut1YgGts1XfFsrq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='44) “ colim pf,gq:Tw˚pSqˆTw˚pTqpHut1YgGts1XfFsrq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='45) » ¨ ¨ ¨ » pH ˝T pG ˝S Fqqur (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46) naturally in F, G, H and u : U, r : Rop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' where the third step uses that composition in C preserves colimits in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Before we can prove part (2) of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35 we need to construct the unit matrices IX : laxMatCpX, Xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the commutative square Tw˚pSq “ ş ˚ Sp´, ´q ş ˚ Cp´, ´q “ Tw˚pCq Sop ˆ S Cop 1 ˆ C1 p α q XopˆX (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='48) induced by a diagram X: S Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here the vertical maps are the cocartesian fibrations classi- fying the respective hom-functors of S and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The that the p8, 2q-category C is enriched in 8-categories with colimits means that there exists an (essentially unique) left q-Kan extension of α along p, giving rise to the dashed lift Tw˚pSq Tw˚pCq Sop ˆ S Cop 1 ˆ C1 p α q I1 XopˆX (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='49) Since the pullback of the cocartesian fibration q along XopˆX is, by definition, the cocartesian fibration ş ps,tq:SopˆS CpXs, Xtq Ñ Sop ˆS, this lift I1 corresponds to a section of this fibration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', an object IX “ I : laxlimps,tq:SopˆS CpXs, Xtq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='50) By the pointwise formula, we can explicitly compute the value of IX at ps, tq : Sop ˆ S as the colimit of the composite Tw˚pSq{ps, tq αÝÑ Tw˚pCq{pXs, Xtq Ñ CpXs, Xtq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='51) which is the functor that informally maps pf1 : s1 Ñ t1, g: s Ñ s1, h: t1 Ñ tq ÞÑ Xh ˝ Xf1 ˝ Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='52) Since the inclusion Sps, tq » Tw˚pSqps,tq ãÑ Tw˚pSq{ps, tq has a left adjoint (because q is a cocartesian fibration), it is homotopy terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' thus we can compute the components IX ts via the desired explicit formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='39) 95 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since all pointwise colimits 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='39 are taken over spaces Sps, tq (as opposed to arbitrary 8-categories), we see that for the construction of the unit matrix we could have relaxed our assumption on C and only required it to be enriched in 8-categories with groupoidal colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When S is a set this formula simplifies to Its “ colim f:Sps,tqpXfq “ # idXs, if s “ t H, if s ‰ t (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='55) which is the direct analog of formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3), with the initial object H of Cpxs, xtq taking the role of the zero object of a commutative monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Continuing Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5, we consider a diagram X: ∆1 “ t0 10 ÝÑ 1u Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The unit ∆1 ˆ p∆1qop-matrix then looks as follows: ¨ ˚ ˚ ˝ idX0 H X10 idX1 ˛ ‹‹‚: laxMatCpX, Xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='57) since the indexing space of the colimit colimf:Sps,tq Xf is either empty in the case s “ 1, t “ 0 or a singleton otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now finish the proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We only treat the case of postcomposition with IX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the other statement is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We need to show that for every diagram Y: R Ñ C, the functor IX ˝S ´: laxMatCpY, Xq Ñ laxMatCpY, Xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='58) is naturally equivalent to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Without loss of generality, we may assume that R “ t˚u so that Y “ Y˚ is a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Naturally in F : laxMatCpY, Xq and u : S we compute (in the 8-category CpY, Xsq) pIX ˝S Fqu » colim pf : sÑtq:Tw˚pSqp colim g:Spt,uq XgqXfFs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='59) “ colim pf : sÑtq:Tw˚pSq g:Spt,uq XgfFs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='60) where the shape of the second colimit is the category ż pf : sÑtq:Tw˚pSq Spt, uq “ Tw˚pSq ˆSop pS{uqop (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='61) Note that the diagram pf, gq ÞÑ XgfFs over which we are taking the colimit arises as the pullback of the diagram S{u Ñ CpY, Xuq, ph: s Ñ uq ÞÑ XhFs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='62) 96 along the functor γ : Tw˚pSq ˆSop pS{uqop Ñ S{u, pf : s Ñ t, g: t Ñ uq ÞÑ pgf : s Ñ uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='63) This functor γ has a left adjoint S{u Ñ Tw˚pSq ˆSop pS{uqop, pf : s Ñ uq ÞÑ pf : s Ñ u, idu : u Ñ uq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='64) and is therefore homotopy terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This allows us to finish the computation: pIX ˝S Fqu » colim pf : sÑtq:Tw˚pSq g:Spt,uq XgfFs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='65) » colim ph: sÑuq:S{u XhFs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='66) » XiduFu “ Fu, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='67) using in the last step that idu : u Ñ u is a terminal object of the comma category S{u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can characterize lax limit and colimits purely in terms of the matrix calculus encoded in the category hoLaxMatC: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X: S Ñ C be a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) A lax cone P : laxMatCpL, Xq is a lax limit cone if and only if it is an isomorphism in the category hoLaxMatC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) A lax cocone I : laxMatpX, Lq is a lax colimit cone if and only if it is an isomorphism in the category hoLaxMatC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We prove the statement about lax cones;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the other one is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' First assume that P is a lax limit cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This implies that the functor P ˝ ´: laxMatCpX, Lq “ laxlims:Sop CpXt, Lq Ñ laxlimpt,sq:SˆSop CpXs, Xtq “ laxMatCpX, Xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='69) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular the map P ˝ ´: hoLaxMatCpX, Lq Ñ hoLaxMatCpX, Xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='70) is a bijection which implies that P is an isomorphism in hoLaxMatC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the inverse map is the unique lax cocone I with P ˝ I “ IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Conversely, assume that P has an inverse in hoLaxMatC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', a lax cocone I : laxMatCpX, Lq satisfying P ˝ I » IX and I ˝S P » idL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then for each L1 : C we have equivalences pP ˝ ´q ˝ pI ˝S ´q “ pP ˝ pI ˝S ´qq » pP ˝ Iq ˝S ´ » IX ˝S ´ » id (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='71) and pI ˝S ´q ˝ pP ˝ ´q “ I ˝S pP ˝ ´q » pI ˝S Pq ˝ ´ » idL ˝´ “ id (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='72) as endofunctors of laxMatCpL1, Xq and CpL1, Lq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='73) respectively (using Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35), showing that P ˝ ´ is an equivalence of 8-categories, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 97 Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A diagram X: S Ñ C admits a lax limit if and only if it admits a lax colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When they exist, the unit matrix IX : laxMatCpX, Xq corresponds to an equivalence IX : laxcolimS X » ÝÑ laxlimS X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='75) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The diagram X admits a lax (co)limit if and only if it is isomorphic in hoLaxMatC to an object L: t˚u Ñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case L is both the lax limit and the lax colimit, exhibited by mutually inverse lax (co)cones I : X Ñ L and P : L Ñ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By definition, the map IX : L Ñ L corresponding to the matrix IX is determined (up to equivalence) by the property that P ˝ IX ˝ I » IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since the identity idL satisfies this property, we conclude that IX » idL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in particular this map is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The comparison map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='75) does not just depend on the objects L “ laxlim X and L1 “ laxcolim X (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='77) but on the implicit lax (co)limit cones P : laxMatCpL, Xq and I : laxMatCpX, L1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Specifically, the map IX is characterized up to equivalence by the relation P ˝ IX ˝ I » IX (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='78) or equivalently IX » pI ˝S Pq´1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='79) (since P and I are isomorphisms in hoLaxMatC and IX is the identity on X : hoLaxMatC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Having described lax (co)limits via lax matrix formulas in hoLaxMatC, we can immediately deduce that all lax (co)limits are absolute with respect to the colimit-enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let C, C1 be p8, 2q-categories enriched in 8-categories with colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F: C Ñ C1 be a functor which preserves colimits on hom-cateogories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then F preserves all lax colimits and lax limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since F preserves colimits on hom-categories, it induces a well defined functor hoLaxMatF: hoLaxMatC Ñ hoLaxMatC1, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='81) given by applying F pointwise to diagram and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since this functor necessarily sends isomorphisms to isomorphisms, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='68 implies that F sends lax (co)limit cones to lax (co)limit cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, we deduce that lax matrix multiplication corresponds to composition of maps between lax colimits/limits in the case where those lax (co)limits exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X: S Ñ C, Y: T Ñ C, Z: U Ñ C be diagrams indexed by 8- categories and admitting lax limits/colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a commutative square of 8- categories CplaxcolimS X, laxlimT Yq ˆ CplaxcolimT Y, laxlimU Zq laxMatCpX, Yq ˆ laxMatCpY, Zq CplaxcolimS X, laxlimU Zq laxMatCpX, Zq ´˝pIYq´1˝´ » ´˝T ´ » (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='83) In other words, after identifying lax colimits and lax limits via the canonical unit matrix, lax matrix multiplication corresponds precisely to function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 98 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by IX : laxMatCpX, laxcolim Xq and PX : laxMatCplaxlim X, Xq two lax (co)limits cones for the diagram X (and similarly for Y and Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The implicit identification Cplaxcolim X, laxlim Yq » ÝÑ laxMatCpX, Yq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='84) is given explicitly as PY ˝ ´ ˝ IX, and similarly for the other horizontal maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the desired commutative square is just the natural equivalence pPZ ˝ ´ ˝ IYq ˝T pPY ˝ ´ ˝ IXq » PZ ˝ ´pIY ˝T PYq ˝ ´ ˝ IX » PZp´ ˝ pIYq´1 ˝ ´q ˝ IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='85) using the equivalence (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='79) in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Continuing Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='56, we consider two maps X FÝÑ lax à ∆1 pY0 Ñ Y1q G ÝÑ Z (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='87) corresponding to matrices F “ ¨ ˚ ˚ ˚ ˝ F0 F1 ˛ ‹‹‹‚ and G “ pG0 Ð G1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='88) The cartesian twisted arrow category Tw˚p∆1q is the poset 00 01 11 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='89) The composite GF : X Ñ Z is therefore the pushout of the diagram G0F0 G1Y01F0 G1F1 G01F0 G1F01 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='90) More general ∆1 ˆp∆1qop matrices can then be multiplied in the usual row-by-column fashion since each entry pGFqus only depends on row u of G and column s of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For example, we can compute (with X “ Y and F “ I) G ˝ I “ ¨ ˚ ˝ G00 G01 G10 G11 ˛ ‹‚ ¨ ˚ ˚ ˝ idX0 H X10 idX1 ˛ ‹‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='91) » ¨ ˚ ˚ ˚ ˝ colim ` G00 Ð G01X10 “ ÝÑ G01X10 ˘ colim ` H “ ÐÝ H Ñ G01 ˘ colim ` G10 Ð G11X10 “ ÝÑ G11X10 ˘ colim ` H “ ÐÝ H Ñ G11 ˘ ˛ ‹‹‹‚» G (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='92) 99 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5 Lax additivity Classical semi-additivity of a category A manifests itself on two levels: (1) Each hom-set of A has a commutative monoid structure which allows to take sums ř sPS fs indexed by arbitary finite sets S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The category A allows direct sums À sPS xs indexed by finite sets S which are both products and coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We categorify these notions by replacing (discrete) addition ΣsPS on the hom-sets by colimits colims:S on the hom-categories and (discrete) coproducts/products š sPS » ś sPS by lax bilimits laxcolims:S » laxlims:S, which are now indexed by arbitary small 8-categories S rather than finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let C be an p8, 2q-category enriched in 8-categories with groupoidal colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X: S Ñ C a diagram indexed by an 8-category S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A lax bilimit of X consists of a lax colimit cone I : laxMatCpX, L1q and a lax limit cone P : laxMatCpL, Xq such that the canonical map IX : L1 Ñ L (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) corresponding to the unit matrix IX : laxMatCpX, Xq is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We identify L and L1 via IX and write lax à S X or lax à s:S Xs (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) for both/either of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When X: S Ñ t˚u X ÝÑ C is a constant diagram, we write S bX or XS for the constant lax bilimit Àlax s:S X “ Àlax S X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When convenient we drop the typographical distinction between a matrix F : laxMatCpX, Yq and the associated map F : Àlax X Ñ Àlax Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can now finally define the notion of lax semiadditivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An p8, 2q-category A is called (finitely) lax semiadditive if (1) it is enriched in 8-categories with (finite) colimits (with functors preserving them), (2) each diagram S Ñ A indexed by a (finite) small 8-category S admits a lax bilimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have seen in Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='74 that in the presence of sufficently many colimits in the hom-categories, every lax limit or colimit is automatically a lax bilimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus the second condition could be weakened to just require the existence of lax limits or lax colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The final step is to categorify the passage from semiadditve to additive categories which amounts to requiring the hom-monoids to be abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Following our philosophy of §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, abelian groups should be replaced by stable 8-categories leading to the following easy definition: Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A (finitely) lax semiadditive p8, 2q-category A is called (finitely) lax addi- tive if every hom-8-category ApX, Y q is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 100 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by Cop the p8, 2q-category obtained from C by reversing the directions of the 1-morphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', CoppX, Y q “ CpY, Xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If C is enriched in (stable) 8-categories with colimits, then so is Cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Moreover lax limits/colimits/bilimits in Cop correspond to lax colimits/limits/bilimits in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus an p8, 2q-category A is (finitely) lax (semi)additive if and only if Aop is (finitely) lax (semi)additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lax limits in the p8, 2q-category PrL of presentable 8-categories exist and are computed as underlying 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since PrL is enriched in 8-categories with colimits it follows that it is a lax semiadditive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The full p8, 2q-category St of presentable stable 8-categories is closed under lax colimits and enriched in stable 8-categories, thus it is lax additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The p8, 2q-category StR of presentable stable 8-categories and right adjoint functors is only finitely lax additive, since composition with a right adjoint functor is exact but does not preserve arbitrary colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The p8, 2q-category of stable 8-categories and exact functors is finitely lax additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that one can replace all presentability assumptions by just requiring the relevant 8-categories to have colimits (or limits, in the case of StR) and for the functors between them to preserve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let T, S be small 8-categories and let F : T ˆ Sop Ñ Spaces » PrLpSpaces, Spacesq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) be a matrix of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A be a lax semiadditive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each X : A, denote by F X :“ F b idX the matrix F X : T ˆ Sop FÝÑ Spaces ´bidX ÝÝÝÝÑ ApX, Xq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) where the second functor arises from the tensoring by Spaces on the 8-category ApX, Xq with colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this way, the space-valued matrix can be interpreted as a map XS Ñ XT , which we call the action of F on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly, when the hom-8-categories are pointed/stable, we can interpret in A every matrix of pointed spaces/spectra, by using the corresponding tensoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Action of matrices is compatible with matrix multiplication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', we have equivalences F X ˝ GX » pF ˝ GqX whenever F, G are composable matrices of spaces/pointed spaces/spectra and X is an object in a correspondingly enriched p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have, naturally in u : U, s : Sop: pF ˝ GqX us » ˆ colim f : tÑt1 Fut1 b Gts ˙ b idX (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) » ˆ colim f : tÑt1pFut1 b idXq ˝ pGts b idXq ˙ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) » pF X ˝ GXqus (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) using that the tensoring preserves colimits and that idX ˝ idX » idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The universal identity ∆1 ˆ p∆1qop-matrix is I “ ¨ ˚ ˚ ˝ t˚u H t˚u t˚u ˛ ‹‹‚: ∆1 ˆ p∆1qop Ñ Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17) 101 Indeed, every p8, 2q-category C whose hom-8-categories have initial objects, the unit matrix IX for the constant diagram X: ∆1 Ñ t˚u X ÝÑ C is given by IX “ IX :“ I b idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More generally, for any 8-category S the universal idenity S ˆ Sop-matrix is the just the transpose of the hom-functor I⊺ “ Sp´, ´q: Sop ˆ S Ñ Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the matrices Cof :“ ¨ ˚ ˝ S0 S0 0 S0 ˛ ‹‚: ∆1 ˆ p∆1qop Ñ Spaces˚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20) and Fib :“ ¨ ˚ ˝ 0 Sr´1s S 0 ˝ ˛ ‹‚: ∆1 ˆ p∆1qop Ñ Sp, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21) defined in pointed spaces and spectra, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If A is a pointed 8-category with colimits, then the matrix Cof acts as the cofiber map CofA “ ¨ ˚ ˝ idA idA 0 idA ˛ ‹‚: A∆1 Ñ A∆1, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22) Indeed, for every ¨ ˝ a b ˛ ‚: B Ñ A∆1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='23) we can compute the matrix product ¨ ˚ ˝ idA idA 0 idA ˛ ‹‚˝ ¨ ˝ a b ˛ ‚» ¨ ˚ ˚ ˝ colimpa Ð a Ñ bq colimp0 Ð a Ñ bq ˛ ‹‹‚» ¨ ˚ ˝ b cofpa Ñ bq ˛ ‹‚“ Cofpa Ñ bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) If A is also stable, then a similar calculation shows that the matrix Fib is inverse to Cof and acts as the fiber map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For every matrix F : T ˆ Sop Ñ ApX, Xq corresponding to an arrow XS Ñ XT in A, we denote by F⊺ the “transposed” matrix F⊺ : Sop ˆ pT opqop » T ˆ Sop FÝÑ ApX, Xq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25) describing the dual map XT op Ñ XSop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 102 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the setting of Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9 there are commutative diagrams ApX, Y Sq ApX, Y qS ApX, Y T q ApX, Y qT » ApX,F Y q F ApX,Y q » and ApS b X, Y q ApX, Y qSop ApT b X, Y q ApX, Y qT op » ApF X,Y q F ApX,Y q ⊺ » (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We do the second computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the first one is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Functorially in M : ApX, Y qT op » ApT b X, Y q and s : Sop we compute ApF X, Y qpMqs “ pM ˝ F Xqs “ colim pf : tÑt1q:Tw˚pTq Mt1 ˝ F X ts (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28) “ colim f:Tw˚pTq Mt1 ˝ pFts b idXq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29) » colim pf : tÑt1q:Tw˚pTq Fts b Mt1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='30) » colim pg : t1Ñtq:Tw˚pT opq Fts b Mt1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31) » colim pg : t1Ñtq:Tw˚pT opqpFts b idApX,Y qqpMt1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='32) “ colim pg : t1Ñtq:Tw˚pT opqppF⊺qst b idApX,Y qqpMt1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='33) “ F ApX,Y q ⊺ pMqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='34) Apart from expanding the various definitions, we have used that Mt1 ˝ ´: ApX, Xq Ñ ApX, Y q preserves colimits and hence the tensoring Fts b ´, the canonical identification Tw˚pTq » Tw˚pT opq which reverses source and target, that colimits of functors ApX, Y q Ñ ApX, Y q are computed pointwise, hence also the tensoring Fts b ´.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A be a lax semiadditive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then A is lax additive if and only if each hom-8-category ApX, Y q is pointed and the matrix Cof from Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19 acts invertibly on each X : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If this is the case, then the inverse is given by the action of Fib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assuming that all hom-8-categories ApX, Y q of the lax semiadditive p8, 2q-category A are pointed, they are stable if and only if the cofiber functor CofApX,Y q : ApX, Y q∆1 Ñ ApX, Y q∆1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='36) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26, we can identify this functor with ApX, CofY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus A is lax additive if and only if all ApX, Y q are stable, if and only if all ApX, CofY q are invertible, if and only if all CofY are invertible, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 Oplax additivity So far we have focused our discussion exclusively on lax limits and colimits, as opposed to oplax ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We could have of course passed to the 2-morphism dual everywhere (obtained 103 from an p8, 2q-category C by replacing each hom-8-category CpX, Y q by its opposite) and told an analogous story using oplax colimits/limits/bilimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This would lead to what we might call oplax (semi)additive p8, 2q-categories A, which are enriched in (stable) 8-categories with limits and allow the formation of oplax bilimits oplax à S X :“ oplaxcolimS X » oplaxlimS X (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) of any diagram X: S Ñ A indexed by a small 8-category (note the direction of the little arrow above the symbol À).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the convenience of the reader, we summarize the main formulas of this dual theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' all the constructions and proofs are dual to the ones we saw earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) For two diagrams X: S Ñ A and Y: T Ñ C, we define oplaxMatCpX, Yq :“ oplaxlimpt,sq:TˆSop CpXs, Ytq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) as the category of oplax matrices from X to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly, such matrices are sections of the contravariant Grothendieck construction ż pt,sq:TˆSop ApXs, Ytq ÝÑ T op ˆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) (2) The oplax matrix multiplication oplaxMatCpX, Yq ˆ oplaxMatCpY, Zq Ñ oplaxMatCpX, Zq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) is given by the formula pΦ ˝ Ψqus :“ lim pf : tÑt1q:Tw˚pTq Φut1YfΨts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) (3) The oplax unit matrix for a diagram X: S Ñ C is J X “ ppt, sq ÞÑ lim f:Sps,tq Xfq : oplaxMatCpX, Xq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The p8, 2q-category St is enriched in stable 8-categories and admits oplax limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' thus it is finitely oplax additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It is not oplax additive because composition of functors does not preserve arbitrary limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7 Coordinate change for ∆1-matrices We have seen that any lax semiadditive p8, 2q-category admits a nicely behaved calculus of lax matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' However, if we apply K0 componentwise to the lax matrix multiplication for lax ∆1-bilimits (see Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='56 and Example 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='86) we obtain the very unusual formula p a0 a1 q ˝ ´ b0 b1 ¯ “ a0b0 ´ a1b0 ` a1b1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) 104 or, more generally A ˝ B “ AI´1B, where I “ K0pI∆1q “ p 1 0 1 1 q is the new unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The goal of this section is to introduce a convenient “coordinate change” in the lax additive (as opposed to merely lax semiadditive) setting, which up to a sign recovers the usual matrix multiplication on K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The key ingredient is the cofiber-fiber-equivalence Cof : Funp∆1, Aq » ÐÝÑ Funp∆1, Aq :Fib (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) pfibpu1q “ b uÝÑ aq Ø pa u1 ÝÑ b1 “ cofpuqq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) for every stable 8-category A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More precisely, we make use of the following dependent version of the cofiber-fiber- equivalence which identifies the oplax limit over an arrow with the lax limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 in [DJW21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : A Ñ B be a diagram of stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a natural equivalence Cof : oplaxlim∆1pB fÐÝ Aq » ÐÑ laxlim∆1pA fÝÑ Bq :Fib (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) described by the formula pb “ fibpu1q, a, b uÝÑ faq Ø pa, b1 “ cofpuq, fa vÝÑ b1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) While not strictly necessary, it is convenient to implement this equivalence by explicit matrices using a combination of the lax and oplax matrix calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the remainder of the section, let A be a lax additive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then A is in particular enriched in 8-categories with finite limits, so that we have available the finite oplax matrix calculus (dual to the one in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) as long as we restrict to diagrams indexed by finite 8-categories S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let X: ∆1 Ñ A be a diagram, X “ pX0 FÝÑ X1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the oplax cone and cocone Fib :“ Àlax ∆1 X X0 X1 fib p0 F and Fib_ :“ Àlax ∆1 X X0 X1 F i1 fib_ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) described by the p∆1qop ˆ p∆1qop-matrix and ∆1 ˆ ∆1-matrix Fib :“ ¨ ˚ ˝ p0 fib ˛ ‹‚“ ¨ ˚ ˚ ˝ idX0 0 0 r´1sX1 ˝ ˛ ‹‹‚ and Fib_ :“ pfib_ ÝÑ i1q “ ¨ ˚ ˚ ˝ r´1sX1 0 0 idX0 ˝ ˛ ‹‹‚, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Here P “ ppsq and I “ pisq (indexed by s : ∆1) are the lax limit/colimit cone exhibiting the lax bilimit Àlax ∆1 X i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', the rows and columns of the unit matrix IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following lemma explains the name of the cones Fib and Fib_ in terms of the maps they represent/corepresent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 105 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let Y : A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) The induced map laxlims:∆1 ApY, Xsq » ÐÝ ApY, lax à ∆1 Xq Fib˝´ ÝÝÝÝÑ oplaxlims:∆1 ApY, Xsq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) is precisely the dependent fiber functor of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 for the ∆1-diagram ApY, X0q Ñ ApY, X1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) The induced map laxlims:p∆1qop ApXs, Yq » ÐÝ Ap lax à ∆1 X, Yq ´˝Fib_ ÝÝÝÝÝÑ oplaxlims:p∆1qop ApXs, Yq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) is precisely the dependent fiber functor of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 for the p∆1qop “ ∆1-diagram ApX0, Yq Ð ApX1, Yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A quick matrix computation for each x “ ¨ ˚ ˝ x0 x1 ˛ ‹‚: laxlims ApY, Xsq shows Fib ˝ x “ ¨ ˚ ˝ p0 ˝∆1 x fib ˝∆1 x ˛ ‹‚“ ¨ ˚ ˝ x0 fibpFx0 Ñ x1q ˛ ‹‚: oplaxlims ApXs, Yq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly, for each x_ “ px_ 0 ÐÝ x_ 1 q : laxlims:p∆1qop ApXs, Yq we have x_ ˝ Fib_ “ pcofpx_ 1 F Ñ x_ 0 q ÝÑ x_ 1 q : oplaxlims ApXs, Yq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As an immediate application of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 we therefore get that the oplax cone/cocones Fib and Fib_ exhibit the lax bilimit Àlax ∆1 X also as an oplax limit and colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following lemma makes a more precise statement, showing that Fib and Fib_ are inverse up to a shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The oplax cone Fib and cocone Fib_ from Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7 are, up to negative shift r´1s, mutually inverse with respect to the oplax matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In par- ticular, Fib and Fib_r1s (or Fibr1s and Fib_) exhibit the lax bilimit Àlax ∆1 X also as an oplax bilimit of the diagram X: ∆1 Ñ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An explicit computation with the oplax matrix multiplication shows: Fib ˝ Fib_ “ ¨ ˚ ˝ p0fib_ p0i1 fibfib_ fibi1 ˛ ‹‚» ¨ ˚ ˚ ˝ idX0r´1s 0 Fr´1s idX1r´1s ˛ ‹‹‚“ J Xr´1s (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) 106 and Fib_ ˝∆1 Fib » limpfib_p0 Ñ i1Fp0 Ð i1fibq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17) » lim ¨ ˚ ˝ ¨ ˚ ˝ r´1s 0 0 0 ˛ ‹‚Ñ ¨ ˚ ˝ 0 0 F 0 ˛ ‹‚Ð ¨ ˚ ˝ 0 0 0 r´1s ˛ ‹‚ ˛ ‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) » ¨ ˚ ˚ ˝ r´1s 0 Fr´1s r´1s ˛ ‹‹‚» IXr´1s, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19) where in the second computation we omit the straightforward verification that the unnamed arrows appearing in the last matrix are indeed those of IXr´1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' While there is a distinguished choice for the cofiber-fiber equivalence (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5), the Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15 provides two (but equally distinguished) ways to identify the lax bilimit Àlax ∆1 X and the oplax bilimit Àoplax ∆1 X, depending on whether we look at the represented map (using Fib and treating them as (op)lax limits) or the corepresented map (using Fib_ and treating them as (op)lax colimits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These two ways are not equivalent: they differ precisely by a suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now have several different ways to represent maps X0 lax ‘ X1 αÝÑ Y0 lax ‘ Y1, with the passage between them implemented by applying the cofiber-fiber equivalence to rows and/or columns of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' ¨ ˚ ˝ α00 α01 αv 10 αv 11 ˛ ‹‚ ´˝Fib_ ÝÝÝÝÝÑ ¨ ˚ ˚ ˝ αh 00 α01 αhv 10 αv 11 ˛ ‹‹‚ Fib˝´ ݧ§ Fib˝´ ݧ§ ¨ ˚ ˝ α00 α01 α10 α11 ˛ ‹‚ ´˝Fib_ ÝÝÝÝÝÑ ¨ ˚ ˚ ˝ αh 00 α01 αh 01 α11 ˛ ‹‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21) Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider two composable maps X βÝÑ Y0 lax ‘G Y1 αÝÑ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='23) Each of the two maps β and α can be represented by a matrix in two ways, depending on whether we treat the middle term as a lax or oplax bilimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following table shows the four corresponding possible row-column-multiplications with the standard lax multiplication in the lower left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' General 2 ˆ 2 matrices describing maps between (op)lax bilimits over ∆1 can then be multiplied in the usual row-by-column fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 107 ˝ pα0 ÐÝ α1q ` αh 0 ÝÑ α1 ˘ ¨ ˚ ˚ ˝ β0 βv 1 ˛ ‹‹‚ cofpα1βv 1 Ñ α1Gβ0 Ñ α0β0q lim ¨ ˚ ˝ αh 0β0 α1Gβ0 α1βv 1 ˛ ‹‚r1s ¨ ˚ ˝ β0 β1 ˛ ‹‚ colim ¨ ˚ ˝ α0β0 α1Gβ0 α1β1 ˛ ‹‚ cofpαh 0β0 Ñ α1Gβ0 Ñ α1β1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) Observe how the entry in top right differs from the standard oplax multiplication by a shift r1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The reason for this is that we used the canonical cofiber-fiber-equivalence (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) both horizontally and vertically, which amounts to using the identification Fib: Àlax Y Ñ Àoplax Y when discussing maps into the (op)lax bilimit but the identification Fib_ : Àoplax Y Ñ Àlax Y when discussing maps from the (op)lax bilimit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' we have seen in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15, that these two identifications are only inverse up to shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following table depicts the unit matrix with respect to each of the four multiplications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' they are just obtained from the standard lax unit matrix (lower left) by passing to horizontal and/or vertical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' lax oplax oplax ¨ ˚ ˝ id 0 0 r´1s ˝ ˛ ‹‚ ¨ ˚ ˚ ˝ r´1s 0 r´1s r´1s ˛ ‹‹‚ lax ¨ ˚ ˝ id 0 id id ˛ ‹‚ ¨ ˚ ˝ r´1s 0 0 id ˝ ˛ ‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25) From the matrix multiplication formulas of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) we can immediately see the advantage of this change of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By working with lax-oplax or oplax-lax matrices, we obtain, on K0 the formulas p a0 a1 q ´ b0 b1 ¯ “ ˘pa0b0 ´ a1b1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26) The that matrix multiplication now involves an alternating sum rather than an ordinary sum is a feature, rather than a bug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the next section we will see, for example, how we can express the differential of the mapping cone of a chain map f : pA‚, αq Ñ pB‚, βq by directly categorifying the canonical matrix δ “ ` α 0 f β ˘ without having to introduce any signs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the signs are already part of the matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Another convenient feature is that the identification between lax-oplax and oplax-lax matrices is compatible with the passage to adjoints in the following sense: Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume that G: Y0 Ñ Y1 has a right adjoint G % GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then Corol- lary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, applied to the adjunctions pG˝q % pGR˝q and p˝GRq % p˝Gq yields equivalences Ap´, Y0 lax ‘G Y1q » Ap´, Y1 oplax ‘GR Y0q and ApY1 lax ‘GR Y0, ´q » ApY0 oplax ‘G Y1, ´q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28) 108 given explicitly by passing to vertical and horizontal transposes ¨ ˚ ˝ Gy0 y1 u ˛ ‹‚Ø ¨ ˚ ˝ GRy1 y0 u ˛ ‹‚ and ´ y_ 1 vÐÝ y_ 0 GR¯ Ø ´ y_ 0 vÝÑ y_ 1 G ¯ , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29) where we have added the application of the gluing functor in the matrix to make the effect of the transposition more apparent (usually we would just write something like py_ 0 Ñ y_ 1 q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each X, Z : A, we have a commutative square ApX, Y0 lax ‘G Y1q ˆ ApY0 oplax ‘G Y1, Zq ApX, Zq ApX, Y1 oplax ‘GR Y0q ˆ ApY1 lax ‘GR Y0, Zq ApX, Zq » (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31) where the horizontal maps are the oplax-lax and lax-oplax matrix multiplication, respectively, and the left vertical map is the equivalence of Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each ´ y_ 0 vÝÑ y_ 1 ¯ : Y0 oplax ‘G Y1 Ñ Z and ¨ ˚ ˝ y0 y1 u ˛ ‹‚: X Ñ Y0 lax ‘G Y1, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='32) the two different row-column products are the cofiber in ApX, Zq of the two composite maps y_ 0 y0 vy0 ÝÝÑ y_ 1 Gy0 y_ 1 u ÝÝÑ y_ 1 y1 and y_ 0 y0 y_ 0 u ÝÝÑ y_ 0 GRy1 vy1 ÝÝÑ y_ 1 y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='33) A straightforward computation using the triangle identities for G % GR shows that these two maps are in canonically identified;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' hence so are their cofibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8 Chain complexes and chain maps Throughout this section, let A be a finitely lax additive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let Z “ pZ, ďq be the standard poset of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A chain complex in A is a functor Zop Ñ A, depicted as .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' , α α α α (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) with the conditions that each αk is a zero object in ApAn, An´1´kq for k ě 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There are various notions of chain maps, corresponding to different notions of natural transformations of diagrams Zop Ñ A in the 2-categorical context (see also §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A chain map (without further qualifier) is a commutative diagram of the form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' B2 B1 B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' α α f2 α f1 α f0 β β » β » β (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) 109 Chain complexes and chain maps in A assemble into an p8, 2q-category ChpAq, defined as a full sub-2-category of FUNpZop, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A lax chain map is a diagram of the form, commuting only up to possibly noninvertible 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' B2 B1 B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' α α f2 α f1 α f0 β β β β (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) Chain complexes and chain maps in A assemble into an p8, 2q-category ChlaxpAq, defined as a full sub-2-category of FUNlaxpZop, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dually, we define the full sub-2-category ChoplaxpAq Ă FUNoplaxpZop, Aq of chain com- plexes and oplax chain maps, which explicitly look as follows: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' B2 B1 B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' α α f2 α f1 α f0 β β β β (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) Given two chain complexes pA‚, αq and pB‚, βq, we write Maplax 0 pA, Bq Ðâ MappA, Bq ãÑ Mapoplax 0 pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) for the three corresponding 8-categories of lax chain maps, chain maps and oplax chain maps A Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More generally, we write Maplax k pA‚, B‚q :“ Maplax 0 pA‚, Bk`‚q and Mapoplax k pA‚, B‚q :“ Mapoplax 0 pA‚, Bk`‚q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) for the (stable) 8-category of lax/oplax degree-k-maps from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Abstractly, these various 8-categories are just the hom-categories in the p8, 2q-categories FUNpZop, Aq, FUNlaxpZop, Aq and FUNoplaxpZop, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For us, a more useful description/definition will be as certain sections of certain tautological fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the rest of this section, fix two chain complexes pA‚, αq and pB‚, βq and an integer k P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the functor Zop ˆ Z BˆAop ÝÝÝÝÑ A ˆ Aop Aopp´,´q ÝÝÝÝÝÝÑ St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' pm, nq ÞÑ ApAn, Bmq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) and its two mixed Grothendieck constructions q : ż m:Zop n:Z ApAn, Bmq Ñ Z ˆ Z and q1 : ż n:Z m:Zop ApAn, Bmq Ñ Zop ˆ Zop, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) (contravariant,covariant) and (covariant,contravariant), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can identify oplax and lax chain maps A Ñ B with sections of q and q1 on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' More precisely, we define Mapoplax k pA, Bq » FunZˆZ ˆ Zpkq, ż m:Zop n:Z ApAn, Bmq ˙ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) 110 and Maplax k pA, Bq » FunZopˆZop ˆ Zoppkq, ż n:Z m:Zop ApAn, Bmq ˙ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) where Zpkq ãÑ Z ˆ Z is the k-shifted diagonal n ÞÑ pn ` k, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Concretely, such a section consists of objects fn : ApAn, Bn`kq and morphisms fn Ñ fn`1 or fn`1 Ñ fn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) in the corresponding Grothendieck construction, amounting to morphisms fnα Ñ βfn`1 or βfn`1 Ñ fnα, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) in ApAn`1, Bn`kq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The full subcategories of Mapoplax 0 pA, Bq and Maplax 0 pA, Bq spanned by those sections where the corresponding maps (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) are equivalences are canonically equivalent to each other by passing to inverses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' we define this common full subcategory to be MappA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We shall not unravel the definition of FUNlax and FUNoplax and show that the mapping categories therein do indeed agree with the 8-categories constructed in Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For the purpose of this paper, the reader may take this construction as the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' It will be useful to study more general sections of the fibrations (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by Zpěkq ãÑ Z ˆ Z the full shifted triangular subposet of those pm, nq satisfying m ě k ` n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We write Mapoplax ěk pA, Bq :“ FunZˆZ ˆ Zpěkq, ż m:Zop n:Z ApAn, Bmq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) for the 8-category of shifted upper triangular sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' see Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 for a depiction of such sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each k we have the obvious restriction functors Mapoplax k pA, Bq |k ÐÝ Mapoplax ěk pA, Bq |ěk`1 ÝÝÝÑ Mapoplax ěk`1pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17) The following lemma states that we can “crop” redundant zeroes in a section f : Mapoplax ěk pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by U r k :“ tpm, nq | k ď m ´ n ď k ` ru Ă Zpěkq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20) the k-shifted diagonal strip of width r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The canonical restriction functor along U r k ãÑ Zpěkq induces an equivalence Mapoplax ěk pA, Bq|ěk`r“0 » ÝÑ FunZˆZ ˆ U r k, ż m:Zop n:Z ApAn, Bmq ˙ |ěk`r“0 , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21) where on both sides we are only considering those sections which are zero on the r-th off diagonal and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 111 ¨ ¨ ¨ A2 A1 A0 A´1 ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Bk`2 fk 2 fk`1 1 fk`2 0 fk`3 ´1 Bk`1 fk 1 fk`1 0 fk`2 ´1 Bk fk 0 fk`1 ´1 Bk´1 fk ´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' α α α α α β β β β β (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1: A section F “ pFmnqměk`n as in Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15 with with fr n “ Fr`n,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The complexes A‚ and B‚ are drawn for reference to indicate how the section spreads across the fibers of the fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' First of all, we claim that the restriction functor (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21) admits a fully faithful left ad- joint given by left q-Kan extension (q is the fibration (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The pointwise q-Kan extension formula trivializes, since for each m ě n ` k ` r the overcategory U r k{pm, nq has a terminal object given by the vertical edge pn ` k ` r, nq Ñ pm, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We thus only have to argue that there are sufficiently many cocartesian edges over these vertical edges pn ` k ` r, nq Ñ pm, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since we are, by definition, only considering sections whose value at pn ` k ` r, nq is zero, this is automatic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the resulting Kan extended diagram is zero on Zpěk`rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The result follows since by construction the essential image of this left q-Kan extension is precisely Mapoplax ěk pA, Bq|ěk`r“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Even when the r-th off-diagonal is zero, we cannot crop the diagram any further without losing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In other words, the restriction functor Mapoplax ěk pA, Bq|ěk`r“0 » ÝÑ FunZˆZ ˆ U r´1 k , ż m:Zop n:Z ApAn, Bmq ˙ , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='23) 112 is not typically an equivalence because the commutative squares βfr´1 n 0 fr´2 n fr´1 n´1α (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) at the edge of the strip U r k carry more data than just the composable arrows fr´1 n´1α Ñ fr´2 n Ñ βfr´1 n , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25) namely a trivialization of their composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the special case r “ 2, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19 says that a section f : Mapoplax ěk pA, Bq satisfying f|ěk`2 “ 0 amounts to the following data: objects fn :“ fk n : ApAn, Bk`nq, objects hn :“ fk`1 n : ApAn, Bk`n`1q, commutative squares βhn 0 fn hn´1α (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27) in ApAn, Bk`nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Restriction to the discrete k-shifted diagonal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', from the poset Zpkq ãÑ Zpěkq to its underlying discrete set) induces the dashed equivalence Mapoplax ěk pA, Bq t|ěk`1 “ 0u ś nPZ ApAn, Bk`nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' » Y (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29) when restricted to the kernel of the restriction functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Mapoplax ěk pA, Bq |ěk`1 ÝÝÝÑ Mapoplax ěk`1pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='30) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19 (with r “ 1), we may restrict our sections to the strip U 1 k Ă Zpěkq, which, as a poset, is simply isomorphic to Z via pm, nq ÞÑ m ` n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore a diagram of shape U 1 k just amounts to a sequence of objects and arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If such a diagram is zero on odd-indexed objects 2n`1 fi pn`1, nq, then all arrows are uniquely determined and the only relevant data are the values at the even-indexed objects 2n fi pn, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The restriction functor Mapoplax ěk pA, Bq Ñ Mapoplax ěk`1pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='32) 113 admits fully faithful adjoint on both sides, given by left / right q-Kan extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A section (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) lies in the essential image if and only each leftmost horizontal / bottommost vertical edge is cocartesian / cartesian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' induces an equivalence fk`1 n´1α » ÝÑ fk n / fk n » ÝÑ βfk`1 n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='33) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The pointwise left q-Kan extension formula at pn`k, nq along the inclusion Zpěk`1q ãÑ Zpěkq trivializes, since the overcategory Zpěn`k`1q{pn ` k, nq has a terminal object pn ` k, n ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the desired left q-Kan extension exists if and only if each horizontal edge pn ` k, nq Ð pn ` k, n ´ 1q admits a cocartesian lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since the fibration q: ż m:Zop n:Z ApAn, Bmq ÝÑ Z ˆ Z (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='34) is (by construction) cocartesian in the second variable, this is always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The argument for the right adjoint is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The restriction functor |ěk`1 is part of a recollement ś nPZ ApAn, Bk`nq Mapoplax ěk pA, Bq Mapoplax ěk`1pA, Bq |ěk`1 j1 j (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='36) with gluing functor f‚ ‚ ÞÑ ´ fibpfk`1 n´1α Ñ βfk`1 n q ¯ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The existence of the recollement follows from Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31 with the kernel being identified by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From the pointwise formulas for the relative Kan extensions we see that for each f‚ ‚ : Mapoplax ěk`1pA, Bq the transformation jpfq Ñ j1pfq is given on the main diagonal by the structure map jpfqk n “ fk`1 n´1α Ñ βfk`1 n “ j1pfqk n (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='37) (for n P Z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' passing to fibers yields the desired formula for the gluing functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that neither of the two adjoints in the left half of the recollement (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='36) are the tautological restriction functor (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29) to the discrete k-shifted diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can think of Mapoplax ěk pA, Bq|ěk`1“0 as the 8-category of degree-k chain maps f : A‚ Ñ Bk`‚ with trivialized structure map fα Ñ βf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that this is not a full subcategory of Mapoplax k pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the restriction functor t|ěk`1 “ 0u Ă Mapoplax ěk pA, Bq Ñ Mapoplax k pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='40) which forgets the trivialization is neither full nor faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The restriction functor to the diagonal does not, in general, have analogous adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This does happen in the special case where the differentials of the chain complexes pA‚, αq and/or pB‚, βq have left adjoints: 114 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the restriction functor |k : Mapoplax ěk pA, Bq ÝÑ Mapoplax k pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='42) (1) Assume that each differential β has a left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then this restriction functor has a fully faithful left adjoint given by relative left Kan extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly it it is given by fk`1 n :“ βLfk n and fr n “ 0 for r ě k ` 2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='43) with the non-trivial vertical arrows amounting to the units fk n Ñ ββLfk n of the adjunc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Assume that each differential α has a left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then this restriction functor has a fully faithful right adjoint given by relative right Kan extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly it it is given by fk`1 n :“ fk n`1αL and fr n “ 0 for r ě k ` 2 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='44) with the non-trivial horizontal arrows amounting to the counits fk n`1αLα Ñ fk n`1 of the adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The two statments are dual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' we focus on (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We observe that the relevant undercategories pm, nq{Zpkq (for pm, nq : Zpěkq) have an initial object pm, m ´ kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the desired pointwise right q-Kan extension exists if we can guarantee that each horizontal edge pm, m ´ kq Ñ pm, nq has a cartesian lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In general, the fibration q: ż m:Zop n:Z ApAn, Bmq ÝÑ Z ˆ Z (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='45) is only cartesian in the first variable, not in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Being cartesian in the second variable amounts to each Apα, Bmq having a right adjoint which is guaranteed because each α: An Ñ An´1 has a left adjoint by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The explicit formulas are an immediate consequence of this pointwise construction using βLβL “ 0 to obtain the vanishing beyond the first off- diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is an equivalence, canonical up to shift, between the full subcategory t|k “ 0u X t|ěk`2 “ 0u Ă Mapoplax ěk pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='47) of those sections which are non-zero only on the first off-diagonal and the 8-category Mapoplax k`1 pA, Bq of oplax degree-pk`1q chain maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly it sends a section pfr nq to a chain map with components gn :“ fk`1 n r´ns : ApAn, Bk`n`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that the equivalence of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46 is not induced by the obvious restriction functor Mapoplax ěk pA, Bq |k`1 ÝÝÑ Mapoplax k`1 pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='49) which, when restricted to t|k “ 0u X t|ěk`2 “ 0u only hits oplax chain maps with trivial structure map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 115 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' According to Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26, the data of a section f : t|k “ 0u X t|ěk`2 “ 0u amounts to 1-morphisms hn :“ fk`1 n : ApAn, Bk`n`1q and commutative squares βhn 0 0 hn´1α (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='50) in ApAn, Bk`nq which amount precisely to morphisms φn : hn´1r´n ` 1sα Ñ βhnr´ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus setting gn :“ hnr´ns, this is precisely the data of an oplax degree-pk`1q map g “ pg‚, φ‚q : Mapoplax k`1 pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume that all differentials α and β have left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the restriction functor |k : Mapoplax ěk pA, Bq|ěk`2“0 ÝÑ Mapoplax k pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='52) is part of a recollement Mapoplax k`1 pA, Bq Mapoplax ěk pA, Bq|ěk`2“0 Mapoplax k pA, Bq i |k j1 j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='53) with gluing functor ρ: f‚ ÞÑ ` fibpβLfn Ñ fn`1αLqr´ns ˘ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='54) In particular, we have the dashed equivalence of (stable) 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Mapoplax ěk pA, Bq|ěk`2“0 Mapoplax k pA, Bq ш ρ Mapoplax k`1 pA, Bq Mapoplax k pA, Bq » |k p0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='55) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46 to identify the kernel of the restriction functor (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='52), the exis- tence of the recollement and the induced equivalence (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='55) follow from the general theory of recollements, see for instance [Lur17, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From the explicit construction in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46 it follows that the canonical transformation j Ñ j1 between the two adjoints is given explicitly at f : Mapoplax k pA, Bq by the canonical mate jpfqk`1 n “ βLfn Ñ fn`1αL “ j1pfqk`1 n (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='56) on the first off-diagonal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' it is an equivalence (fn “ ÝÑ fn or 0 “ ÝÑ 0) everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The gluing functor Mapoplax k pA, Bq Ñ Kerp|kq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='57) is given by the fiber of this transformation, therefore yields the desired formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='54) under the identification of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 116 Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by Mapoplax exěkpA, Bq Ă Mapoplax ěk pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='59) the full subcategory spanned by those sections pfr nq such that all the induced squares βfr`1 n βfr`2 n´1α fr n fr`1 n´1α ˝ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='60) in ApAn, Br`nq are bicartesian (for all r ě k) and call such sections exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' There is an equivalence of 8-categories between the full subcategory Mapoplax exěkpA, Bq|ěk`2“0 Ă Mapoplax ěk pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='62) of those sections which are exact and vanish beyong the first off-diagonal and the 8-category Maplax k`1pA, Bq, of lax degree-pk`1q chain maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly it sends a section pfr nq to a chain map with components gn :“ fk`1 n r´ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26, if we restrict to squares (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27) which are bicartesian, the data just amounts (by rotating the exact triangle forward and shifting by r´ns) to objects hn “ fk`1 n : ApAn, Bk`n`1q and maps βhnr´ns Ñ hn´1r´n ` 1sα in ApAn, Bk`nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is precisely the data of a lax degree-k`1 chain map g with components gn :“ hnr´ns, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='61 explain how the 8-category Mapoplax ěk pA, Bq|ěk`2“0 contains both the oplax and the lax degree-pk`1q maps A Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' From the explicit construc- tions it is immediate that these two inclusions are compatible, in the sense that there is a commutative square MappA‚, Bk`‚`1q Mapoplax k`1 pA, Bq Maplax k`1pA, Bq Mapoplax ěk pA, Bq|ěk`2“0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='64) and we have MappA‚, Bk`‚`1q “ Mapoplax k`1 pA, Bq X Maplax k`1pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='65) as full subcategories of Mapoplax ěk pA, Bq|ěk`2“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let pfr nq “ pf, hq : Mapoplax ěk pA, Bq be a section as in Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If each square (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27) is bicartesian, then both of the maps fnα Ñ βhnα and βhnα Ñ βfn`1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='67) 117 are equivalences, since their fibers/cofibers are hn´1αα “ 0 and ββhn`1 “ 0, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='68) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the oplax degree-k chain map f “ fk is an actual chain map A‚ Ñ Bk`‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the canonical restriction functor |k : Mapoplax ěk pA, Bq Ñ Mapoplax k pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='69) restricts to a functor δ: Maplax k`1pA, Bq » Mapoplax exěkpA, Bq|ěk`2“0 |k ÝÑ MappA‚, Bk`‚q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='70) whose kernel is precisely MappA‚, Bk`‚`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These differentials δ assemble to what we call the lax mapping complex Maplax ‚ pA, Bq: ¨ ¨ ¨ Maplax 2 pA, Bq Maplax 1 pA, Bq Maplax 0 pA, Bq ¨ ¨ ¨ MappA‚, B1`‚q MappA‚, B‚q δ δ δ δ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='71) Unraveling, we get the explicit formula for the differential δpg‚qn “ fibpβgn Ñ gn´1αqrns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='72) Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume that the differentials α and β have right adjoints αR and βR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Denote by AR ‚ :“ pA´‚, αRq and BR ‚ :“ pB´‚, βRq the chain complex obtained from A and B by passing to right adjoints of the differentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that for each n P N there is a tautological equivalence of 8-categories Mapoplax k pAR, BRq Maplax ´kpA, Bq pf‚, fαR Ñ βRfq, pf´‚, βf Ñ fαq » (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='74) by noting that both sides are sections şm:Zop n:Z ApAR n, BR mq şn:Z m:Zop ApAn, Bmq Zpkq Z ˆ Z Zop ˆ Zop q q1 – p´1q¨ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='75) of the same fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An explicit computation shows that under this equivalence, the differ- ential δ: Maplax k`1pA, Bq Ñ Maplax k pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='76) of the lax mapping complex (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='71) is identified with the gluing functor ρ: Mapoplax ´k´1pAR, BRq Ñ Maplax ´kpAR, BRq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='77) of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='51 applied to the chain complexes AR and BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 118 Once we have constructed the mapping complex, we immediately get the corresponding notion of categorified chain homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : A Ñ B be a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A lax null-homotopy of f is a lax degree-1 map h: Maplax 1 pA, Bq with δphq “ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Clearly, one could dualize Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='66 and all the preceding lemmas to obtain the oplax mapping complex and the resulting notion of an oplax null-homotopy This is the version which we have already seen in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6, where this oplax mapping complex was constructed (in the special case C “ St) via the product totalization of the canonical double complex CpA´‚, B‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We shall not give a detailled proof that these two different constructions agree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' this is relatively straightforward by inspection of the terms of the complex and the explicit formulas (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='72) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) for the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this section there appear both lax and oplax chain maps, so we are a bit more careful to always carry the corresponding adjective with us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If the context is clear, we might drop the adjective and just write null-homotopy or categorical null-homotopy as we did in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally, we remark that the adjointability conditions for commutative squares (see Defi- nition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) immediately give rise to corresponding notions for chain maps (expanding Def- inition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For chain maps, we distinguish to types of adjointability conditions: in the direction of the differentials and in the direction of the chain map itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We say that f is left diff-adjointable / right diff-adjointable if each square in the cor- responding diagram (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) is horizontally left/right adjointable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' all differentials α and β admit left/right adjoints and the canonical mate βLf Ñ fαL / fαR Ñ βRf is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We say that f is left/right adjointable if each square in the corresponding diagram (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) is vertically left/right adjointable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', each component fn has a left/right adjoint and the canonical mate fLβ Ñ αfL / αfR Ñ fRβ is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9 The oplax mapping cone construction Let A be a finitely lax additive p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The goal of this section is to construct the oplax mapping cone ConeÐpfq of a chain map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) in A by categorifying the usual formula Conepfqn`1 :“ An ‘ Bn`1 ´ ´α 0 ´f β ¯ ÝÝÝÝÝÑ An´1 ‘ Bn “: Conepfqn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) for the differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' According to the philosophy outlined in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1, we need additional data to specify the mapping cone complex: To construct the terms of the mapping cone complex ConeÐpfq :“ An´1 lax ‘ Bn as a lax bilimit, we need to specify 1-morphisms h: An´1 Ñ Bn or k: Bn Ñ An´1 We need some suitable 2-categorical data to be able to write down the ∆1 ˆ ∆1-analog of the differential matrix (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 119 We will also see that in the presence of sufficient compatible adjoints to the differentials α, β and/or f, one can canonically construct such data using the various units/counits and in this case we recover the fiber and cofiber of f as in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by MaplhpA, Bq :“ MappA, Bq ˆMapoplax 0 pA,Bq Mapoplax ě0 pA, Bq|ě2“0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3) the 8-category of those sections (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) which are zero beyond the first off-diagonal and restrict to an honest chain map (as opposed to an oplax one) on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Such sections are called lh-enhanced morphisms of chain complexes and are written F : pA‚, α‚q lh ùñ pB‚, β‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The mnemonic “lh” stands for “left-horizontal” and is explained by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17, where we construct canonical lh-enhancements in the presence of left adjoints in the horizontal (=differential) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26 tells us that an lh-enhanced morphism F : pA‚, α‚q lh ùñ pB‚, β‚q consists of 1-morphisms fn : An Ñ Bn and hn : An Ñ Bn`1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) together with (not necessarily bicartesian) commutative squares hn´1αn 0 fn βn`1hn ϵn ηn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) in ApAn, Bnq such that each composite fn´1αn ηn´1αn ÝÝÝÝÝÑ βnhn´1αn βnϵn ÝÝÝÑ βnfn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) is an equivalence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', exhibits f : A Ñ B as a chain map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We say that F “ pF, h, ϵ, ηq is an lh-enhancement of the underlying chain map f : A Ñ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We can also depict such an lh-enhanced morphism of chain complexes as follows .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A2 A1 A0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' B2 B1 B0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' α α ð ð f α ð ð f h α f h β β β β (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) but note that this picture is not complete, since it does not depict the trivialization η ˝ ϵ » 0 encoded in the square (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The forgetful functor MaplhpA, Bq Ñ MappA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) sends an lh-enhanced morphism F “ pf, h, ϵ, ηq to its underlying chain map by forgetting h, ϵ and η and only remembering the maps f and the equivalences fα » βf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in the picture (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) this just amounts to pasting the triangular 2-cells to form (commutative) squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each chain map f : A‚ Ñ B‚, we write Maplh f pA‚, B‚q for the fiber of this forgetful functor over the object f : MappA‚, B‚q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' it is the (typically not stable) 8-category of lh-enhancements of the chain map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 120 Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An lh-enhanced morphism is called exact if each square (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) is bicartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We denote by Maplh´expA, Bq :“ Mapoplax exě0pA, Bq|ě2“0 Ă MaplhpA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) the full subcategory of exact lh-enhanced morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that under the identification of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='61, an exact lh-enhancement of a chain map is precisely a lax null-homotopy in the sense of Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following construction of the oplax mapping cone is a tautological reformulation of what the data of an lh-enhanced morphism entails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F “ pf, h, ϵ, ηq: pA‚, αq lh ùñ pB‚, βq be an lh-enhanced morphism of chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We define the oplax mapping cone of F to be the chain complex ConeÐpFq : ¨ ¨ ¨ Ñ An lax ‘h Bn`1 δn`1 ÝÝÝÑ An´1 lax ‘h Bn δn ÝÑ An´2 lax ‘h Bn´1 Ñ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) where the differential is the lax-oplax matrix δn`1 :“ ¨ ˚ ˝ αn 0 fn βn`1 ˛ ‹‚: An oplax ‘h Bn`1 Ñ An´1 lax ‘h Bn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15) induced by the commutative square (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using the matrix multiplication formula from Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22 we compute the squared differential δ ˝ δ » ¨ ˚ ˚ ˝ cofpαα Ñ 0q cofp0 Ñ 0q cofpfα Ñ βhα Ñ βfq cofp0 Ñ ββq ˛ ‹‹‚ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) It is zero because α2 » 0, β2 » 0 and the that the composite map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Having constructed the mapping cone ConeÐpFq with respect to the choice of the auxiliary lh-enhancement of the underlying chain map f, it is natural to ask whether there are universal ways to produce such lh-enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' These exists as long as the differentials α and/or β admit left adjoints: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let pA‚, αq and pB‚, βq be two chain complexes in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the forgetful functor p: MaplhpA‚, B‚q Ñ MappA‚, B‚q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) (1) If each differential β has a left adjoint, then p admits a fully faithful left adjoint p´qβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) If each differential α has a left adjoint, then p admits a fully faithful right adjoint p´qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) Assume that both differentials α and β admit left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The canonical transformation p´qβ Ñ p´qα is an equivalence precisely on those chain maps f : MappA‚, B‚q which are left diff-adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 121 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The first two statements are a direct consequence of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='41 (for k “ 0) by observing that both adjoints (if they exists) take values in MaplhpA, Bq Ă Mapoplax ě0 pA, Bq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='19) when restricted to MappA, Bq Ă Mapoplax 0 pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='20) To prove (3) fix a chain map f : MappA‚, B‚q and consider the component fβ Ñ fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The only place where it can possibly not be an equivalence is on the first off-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Unraveling the pointwise formula, one observes that the value at these off-diagonal places is given by the mates βLf Ñ fαL of the equivalences fα Ñ βf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' by definition f is left diff-adjointable precisely if these mates are all equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume that all differentials α and β admit left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then the recolle- ment (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='53) (for k “ 0) restricts to a recollement Mapoplax 1 pA, Bq MaplhpA, Bq MappA, Bq p (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='22) and therefore to an equivalence MaplhpA, Bq MappA, Bq ш ρ Mapoplax 1 pA, Bq MappA, Bq MappA, Bq » p0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='23) Pointwise over each f : MappA, Bq we thus have an equivalence Maplh f pA, Bq » ρpfq{ Mapoplax 1 pA, Bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='24) A glance at the explicit formula (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='54) shows that the gluing funtor ρ is zero precisely on those chain maps which are left diff-adjointable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in this case the 8-category of lh-enhancements is simply equivalent to Mapoplax 1 pA, Bq, hence in particular stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following corollary summarizes the situation over each chain map f: When the differ- entials of the chain complexes admit adjoints, each chain map f can be canonically enhanced in two ways yielding an initial or terminal object in the category Maplh f pA, Bq of lh-enhancements of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If the chain map is left diff-adjointable, this 8-category is stable and these two canonical lh-enhancements agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) If each differential β admits a left adjoint βL then f admits an initial lh-enhancement fβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Dually, if each differential α admits a left adjoint αL, then f admits a terminal lh- enhancement fα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 122 (3) If the chain map f is left diff-adjointable then the two lh-enhancements fβ and fα coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case we denote this lh-enhancement by flh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Follows from the adjunctions of Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17 viewed pointwise over f : MappA‚, B‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We now identify the mapping cones constructed from the initial and terminal lh-enhancement with those constructed in Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 using the directed pushout and directed pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Assume that each α admits a left adjoint and let fα be its terminal lh-enhancement of Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the oplax square Ai Ai´1 Bi Ai´1 oplax ‘hα Bi fi α ð hα ð (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='27) obtained by pasting ϵ with the oplax colimit cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This square is a directed pushout, thus yields an identification Ai´1 ñ> Ai Bi » ÝÑ Ai´1 oplax ‘h Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Under this identification the differential of ConeÐpfαq corepresents the map pa_ i´1α Ñ b_ i fq ÞÑ pcofpa_ i´1α Ñ b_ i fqα » ÝÑ b_ i βfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28) (2) Dually, if each β admits a left adjoint, then the terms of the cone ConeÐpfαq are canonically identified with Ai´1 ñˆ Bi´1 Bi and the differential represents the map pfai Ñ βbi`1q ÞÑ pfαai » ÝÑ β fibpfai Ñ βbi`1qqr1s (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let fα “ pf, hα, ϵα, ηαq be the terminal lh-enhancement of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have to show that for each test object C : A, the functor oplaxlim∆1pApAi´1, Cq ˝hα ÐÝÝ ApBi, Cqq “ ApAi´1 oplax ‘hα Bi, Cq Ñ ApAi´1, Cq ñˆ ApAi,Cq ApBi, Cq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='30) is an equivalence of (stable) 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly, this functor sends a section a1 i´1 uÝÑ b_ i hα to the composite a1 i´1α uα ÝÑ b_ i hαα “ b_ i fiαLα b_ i ficuα ÝÝÝÝÝÑ b_ i fi, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31) which is precisely its transpose under the adjunction p˝αq % p˝αLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Thus the functor (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='30) is an equivalence by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6(1) applied to ApAi´1, Cq ˝α ÝÑ ApAi, Cq ˝fi ÐÝÝ ApBi, Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='32) To compute the map corepresented by the differential, we compute for each pa_ i´1 uÝÑ b_ i q : ApAi´1 oplax ‘hα Bi, Cq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='33) 123 the matrix product pa_ i´1 uÝÑ b_ i q ˝ ¨ ˚ ˝ α 0 fi β ˛ ‹‚“ pcofpa_ i´1α Ñ b_ i fiq Ñ cofp0 Ñ b_ i βqq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='34) where in the first entry we are taking the cofiber of the map u: a_ i´1α uα ÝÑ b_ i hαα b_ϵα ÝÝÝÑ b_ i fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Note that in the matrix representation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='34) we are omitting the application of the gluing functor as is customary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If we put this implicit application back in, we obtain the map cofpa_ i´1α Ñ b_ i fiq Ñ b_ i βhα “ b_ i βfi`1α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' : ApAi, Cq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35) which yields the desired map cofpa_ i´1α Ñ b_ i fiqα Ñ b_ i βfi`1 : ApAi`1, Cq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='36) after transposing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in , it is just the equivalence b_ i fiα » b_ i βfi`1 because a_ i´1αα “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The proof of the dual statement is analogous: We apply ApC, ´q to reduce to the case of 8-categories, where we apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then we only have to perform the dual matrix computation ¨ ˚ ˝ α 0 fi β ˛ ‹‚˝ ¨ ˚ ˝ ai bi`1 ˛ ‹‚“ ¨ ˚ ˚ ˝ cofpαai Ñ 0q cofpfiai Ñ βbi`1q ˛ ‹‹‚“ ¨ ˚ ˝ αai fibpfiai Ñ βbi`1q ˛ ‹‚r1s (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='37) to obtain the desired formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a left diff-adjointable chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Both the chain complexes Cofpfq and Fibpfq from Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 are well defined and equivalent up to a shift in degree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Cofpfq » Fibpfqr1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Recall from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6 that the external shift of a chain complex pA‚, αq is the chain complex Ar1s‚ :“ pA‚´1, αr1sq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='40) where the terms are reindexed and the differentials are shifted internally in the stable 8- category ApAi, Ai´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The chain complex Fibpfqr1s from Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 is precisely defined via the uni- versal properties of the directed pullback to represent the map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' note that it differs from the formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) due to an internal shift of the differential introduced by the external shifting convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dually, the differentials of the chain complex Cofpfq are defined via the universal property of the directed pushout to corepresent the maps (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26 shows that these chain complexes exist and can be concretely constructed as ConeÐpfαq and ConeÐpfβq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Since we assume that the chain map f is left diff-adjointable, Corol- lary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='25 states that fα and fβ are canonically equivalent as lh-enhancements of the chain map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the chain complexes Cofpfq “ ConeÐpfαq and Fibpfqr1s “ ConeÐpfβq are also equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 124 So far we have used that one can express the directed pullback Ai´1 ñˆ Bi´1 Bi and the directed pushout Ai´1 ñ> Ai Bi as a lax limit/colimit of a composite involving horizontal left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' To prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12 we need an analogous discussion using vertical right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This change corresponts to changing the direction of the gluing map between Bn and An´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We start by defining the corresponding notion of enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' An rv-enhanced morphism F : pA‚, α‚q rv ùñ pB‚, β‚q of chain complexes consists of 1-morphisms fn : An Ñ Bn and kn : Bn Ñ An´1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='42) toghether with an oplax-lax matrix of the form δ “ ¨ ˚ ˝ β f 0 α ˛ ‹‚: Bn`1 lax ‘k An Ñ Bn oplax ‘k An´1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='43) such that the composite map fα Ñ fkf Ñ βf is an equivalence (yielding the underlying chain map f of F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The resulting chain complex pConeÐpFq‚ :“ B‚ lax ‘k A‚´1, δq is called the oplax mapping cone of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The mnemonic “rv” stands for “right-vertical” and reflects the fact that there are canonical rv-enhancements in the presence of right adjoints in the vertical (=chain map) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We shall now explain how such rv-enhanced morphisms assemble into an 8-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For simplicity we will restrict to those, where each fn admits a right adjoint gn :“ fnR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Define Maplax ďkpB, Aq :“ FunZopˆZop ˆ Zpďkqop, ż n:Z m:Zop ApBn, Amq ˙ (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='45) to consist of sections defined on Zpďkqop :“ tpm, nq | m ď n ` ku Ă Zop ˆ Zop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46) 125 Pictorially, such sections look as follows: ¨ ¨ ¨ B2 B1 B0 ¨ ¨ ¨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ak`2 gk 2 Ak`1 gk´1 2 gk 1 Ak gk´2 2 gk´1 1 gk 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' β β β β α α α α (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='47) Denote by Maprv´LpA, Bq Ă Maplax ď0pB, Aq (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='48) the full subcategory of those sections pgr nq satisfying: The lax chain map g‚ “ g0 ‚ : B‚ Ñ A‚ on the main diagonal is left adjointable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' each gn : Bn Ñ An has a left adjoint gnL and the canonical mate gn´1Lα Ñ βgnL is an equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The section is zero beyond the first off-diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' gr ‚ “ 0 for r ď ´2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Using the dual of Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26 and by passing from an adjointable lax chain map g‚ “ g0 ‚ : B‚ Ñ A‚ to its adjoint f‚ :“ g‚L : A‚ Ñ B‚ (which is an honest chain map), it is not hard to see that the data of such a section amounts precisely to that of an rv-enhanced morphism whose underlying chain map f admits pointwise adjoints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the 1-morphisms kn : Bn Ñ An´1 are the term g´1 n on the first off-diagonal and the matrices (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='43) amount precisely to the squares kn`1 gn 0 kn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='49) Therefore we can view Maprv´LpA, Bq as the 8-category of those rv-enhanced morphisms, whose underlying chain map f admits pointwise adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We have the canonical forgetful 126 functor Maplax ď0pB, Aq|ď´2“0 Maplax 0 pB, Aq Maprv´LpA, Bq tleft adjointable g‚u tpointwise right adjointable f‚u MappA, Bq |0 |0 » (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='50) sending such an rv-enhanced morphism to its underlying chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each pointwise adjointable chain map f : A Ñ B we write Maprv f pA, Bq “ Maprv´L f pA, Bq for the fiber of this dashed functor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' it is the 8-category of rv-enhancements of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The forgetful functor (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='50) is part of a recollement Maplax ´1pB, Aq Maprv´LpA, Bq tpoinwise right adjointable f‚ : A Ñ Bu j1 j , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='52) whose gluing functor ρ computes the fiber of the canonical mate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', ρpfq “ ` fibpαfnR Ñ fn´1Rβqrns ˘ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='53) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similarly to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='31 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='35 we compute that the relative left and right Kan extension along the diagonal Zop ãÑ Zpď0qop always exist, yielding fully faithful left and right adjoints j and j1 to the restriction functors |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Explicitly we have jpgq´1 n “ αgn Ñ gn´1β “ j1pgq´1 n (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='54) (the structure map of g‚) on the first off-diagonal and zero beyond it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Moreover, the kernel of the forgetful functor is t|0 “ 0u X t|ď´2 “ 0u Ă Maplax ď0pB, Aq, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='55) which, similarly to Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='46 we can identify with Maplax ´1pB, Aq via the assignment g‚ ‚ ÞÑ pg´1 n rnsqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='56) The desired result follows by passing to adjoints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', g‚ :“ f‚R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The gluing functor for the recollement Maplax ´1pB, Aq Maplax ď0pB, Aq|ď´2“0 Maplax 0 pB, Aq |0 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='58) (before restricting the cokernel to the subcategory of left adjointable maps g: B Ñ A) is nothing but the differential of the lax mapping complex Maplax ‚ pB, Aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 127 As a direct consequence we get the following result, which provides the two canonical rv-enhancements of a pointwise right adjointable chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a pointwise right adjointable chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The 8-category Maprv f pA, Bq has (1) an initial object fα :“ jpfq with kα “ αfR and where the vertical map α Ñ kf “ αfRf in the matrix (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='43) is the unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) a terminal object fβ :“ j1pfq with kβ “ fRβ and where the horizontal map β Ð fk “ ffRβ in the matrix (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='43) is the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (3) These two rv-enhancements coincide if and only if the chain map f is right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case we denote this rv-enhancement by frv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Analogously to Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26, we can exhibit the terms of the corresponding oplax mapping cones ConeÐpfαq and ConeÐpfβq as a directed pushout or directed pullback, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a chain map and assume that each fi admits a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) The oplax square Ai Ai´1 Bi Bi lax ‘kα Ai´1 f α ð kα ð (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='61) yields an identification Ai´1 ñ> Ai Bi » ÝÑ Bi lax ‘kα Ai´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Under this identification, the differ- ential of ConeÐpfαq again corepresents the map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Dually, the terms of the cone ConeÐpfβq are canonically identified with Ai´1 ñˆ Bi´1 Bi and the differential again represents the map (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Similar to Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The chain complexes ConeÐpfαq and ConeÐpfβq also yield a construction for Cofpfq and Fibpfqr1s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular, Cofpfq and Fibpfqr1s agree when the chain map f is right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' When f is both right adjointable and left diff-adjointable, the two canonical oplax mapping cones ConeÐpflhq and ConeÐpfrvq agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Throughout this section there was a bias in our discussion, since we im- plicitly treated chain maps as being oplax, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' having directed squares of the form Ai`1 Ai Bi`1 Bi α fi`1 fi β (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='65) 128 This was already apparent in the chosen direction for directed pushouts and directed pullbacks in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 and accounts for the two possible choices we had when it came to adjointability conditions: having vertical right adjoints or horizontal left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We could rewrite this whole section with the opposite conventions and obtain the lax mapping cone ConeÑpFq associated to a chain map f with suitable enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In the case where f is left adjointable or right diff-adjointable we could again construct a canonical lax mapping cone ConeÑpFq whose terms are identified both with Bi ñ> Ai Ai´1 and with Bi ñˆ Bi´1 Ai´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10 Universal property of the lax mapping cone The main reason for introducing the mapping cone of a chain map f : pA‚, αq Ñ pB‚, βq between chain complexes in an additive category A is that it yields an explicit model for the cofiber of f in the in the stable 8-category KpAq of chain complexes up to chain homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In other words, it satisfies MapKpAqpConepfq, Cq » fib pMappB, Cq Ñ MappA, Cqq “ tpg: B Ñ C, h: gf » 0qu (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) naturally in C : KpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Already before passing to the stable 8-category KpAq, one can see a naive version of this universal property characterizing the mapping cone up to isomorphism in ChpAq via ChpAqpConepfq, Cq – tpg, hq | g: B Ñ C, h: gf » 0u (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) naturally in C : ChpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In other words: maps out of Conepfq are chain maps g: B Ñ C together with a null-homotopy of gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Ultimately, we are of course interested in understanding the categorified analog of the homotopically meaningful universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' However, this is currently out of reach since we don’t even know what the correct analog of the stable 8-category KpAq should be and in what sense we are supposed to view the mapping cone as a cofiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore, we now instead describe the categorified analog of (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) in the hopes that it might lead to a better understanding of the theory of categorified chain complexes up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let F : A lh ùñ B be an lh-enhanced morphism of chain complexes with underlying chain map f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) For each chain complex C : ChpAq there is a natural equivalence of (stable) 8-categories between chain maps ConeÐpFq Ñ C and chain maps g: B Ñ C together with an exact lh-enhancement E of gf and a mor- phism E Ñ gF of lh-enhancements of gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) For each chain complex C : ChpAq there is a natural equivalence of (stable) 8-categories between chain maps C Ñ ConeÐpFqr´1s and 129 chain maps g: C Ñ A together with an exact lh-enhancement E of fg and a mor- phism Fg Ñ E of lh-enhancements of fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Before proving Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3, we isolate the special case where F is the initial or terminal lh-enhancement of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : pA‚, αq Ñ pB‚, βq be a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Assume that all differentials α have left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then for each chain complex C : ChpAq there is an equivalence of (stable) 8-categories between chain maps Cofpfq Ñ C and chain maps g: B Ñ C together with a lax null-homotopy E of gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) Assume that all differentials β have left adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then for each chain complex C : ChpAq there is an equivalence of (stable) 8-categories between chain maps C Ñ Fibpfq and chain maps g: C Ñ A together with a lax null-homotopy E of fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We prove the first statement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' the second is dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let fα be the terminal lh-enhancement of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='26 we may identify B Ñ Cofpfq with B Ñ ConeÐpfαq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Observe further, that composition with g sends the lh-enhanced morphism fα to gpfαq » pgfqα, which is thus a terminal object of Maplh gfpA‚, C‚q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore the claim follows from Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 after identifying exact lh-enhancements with lax null-homotopies (see Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Fix an lh-enhanced morphism F “ pf, h, ϵ, ηq: pA‚, α‚q lh ùñ pB‚, β‚q and a test chain complex pC, γq in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We unravel the data encoded in a chain map pConeÐpFq‚, δq Ñ pC‚, γq, using lax-oplax matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' For each n, we have a map Gn “ ´ kn´1 µn´1 ÝÝÝÑ gn ¯ : ConeÐpFqn “ An´1 lax ‘ Bn Ñ Cn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) and an equivalence Gnδn`1 » ÝÑ γn`1Gn`1, which we can expand to pcof pkn´1αn Ñ gnfnq ÝÑ gnβn`1q “ pkn´1 ÝÑ gnq ¨ ˚ ˝ αn 0 fn βn`1 ˛ ‹‚ » ÝÑ pγn`1kn ÝÑ γn`1gn`1q (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) 130 Therefore, the map Gnδn`1 Ñ γn`1Gn`1 amounts to a cube (read back to front) kn´1αn 0 gnhn´1αn 0 0 gnfn gnβn`1 γn`1kn γn`1gn`1 µn´1αn ζn gnϵn νn gnηn φn`1 γn`1µn ApAn, Cnq ApBn`1, Cnq ´˝hn (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) in the contravariant Grothendieck construction of ´˝hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The that this map is an equivalence amounts to saying that the left and right squares of the cube are bicartesian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In particular we can focus on the the right face and see an equivalence φn`1 : gnβn`1 » ÝÑ γn`1gn`1, exhibiting g‚ : pB‚, βq Ñ pC‚, γq as a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Consider the functor Ξ: Zop ˆ Z ˆ ∆1 Ñ St;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' pm, n, ´q ÞÑ ´ ApAn, Bmq gn˝´ ÝÝÝÑ ApAn, Cmq ¯ , (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) which is well defined because g: B‚ Ñ C‚ is a chain map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By direct comparison with the diagram (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) below, one verifies that all of the data (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) can then be equivalently encoded as Zpě0qˆ∆1-sections of the mixed (contravariant, contravariant, covariant) Grothendieck construction of Ξ that satisfy the restriction to Zpě0q ˆ 0 is the original lh-enhanced morphism F : A lh ùñ B, the value on each edge pn, n, 1q Ñ pn, n, 0q is cartesian, the restriction to Zpě0q ˆ t1u is an exact lh-enhanced morphism E : B lh ùñ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 131 Cn`1 gf k 0 Bn`1 f h 0 Cn gf k Bn f h Cn´1 gf Bn´1 f An`1 α An α An´1 γ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' ν ζ µ g β ϵ η γ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' ν ζ µ g β ϵ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' g η (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) In other words, we have exact lh-enhanced morphisms E : B lh ùñ C equipped with a map E Ñ gF which induces an equivalence on the underlying chain maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A Some lemmas from (2-)category theory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1 About (op)lax limits of 8-categories We collect here a few useful lemmas regarding various types of 2-categorical limits of 8- categories or stable 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Construction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S be an 8-category and X: S Ñ Cat8 an S-indexed diagram of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let p: ş S X Ñ S be its (covariant) Grothendieck construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume that for every arrow f : s Ñ t in S, the functor Xf admits a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' In this case, the cocartesian fibration p is also cartesian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' it corresponds to the diagram XR : Sop Ñ Cat8 which is obtained form X by passing to right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Therefore we obtain a tautological identification laxlimS X “ tsections of pu “ oplaxlimSop XR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let f : A Ñ B be a diagram of 8-categories and assume that f has a right adjoint fR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a natural identification A ш f B “ B Ј fR A (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4) given by the formula pa, b, fa uÝÑ bq Ø pa, b, a uÝÑ fRbq (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5) 132 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This is just the special case S “ ∆1 of the identification (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let A fÝÑ C gÐÝ B be a diagram of 8-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) Assume that f has a right adjoint fR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a natural equivalence A ñˆ C B » B Ј fRg A (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) given by the formula pa, b, fa uÝÑ gbq Ø pa, b, a uÝÑ fRgbq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) (2) Assume that g has a left adjoint gL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Then there is a natural equivalence A ñˆ C B » A ш gLf B (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='9) given by the formula pa, b, fa uÝÑ gbq Ø pa, b, gLfa uÝÑ bq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' We compute A ñˆ C B “ A ˆCt0u Ct0Ñ1u ˆCt1u B » A ш f C ˆC B (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='11) » C Ј fR A ˆC B » At0Ñ1u ˆAt1u C ˆC B (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='12) » At0Ñ1u ˆAt1u B » B Ј fRg A (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='13) where we have used Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 in the third step and the explicit construction of the lax/oplax limit in steps two, four and six.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Chasing through the chain of identifications one immediately obtains the desired formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The second statement is analogous, this time using the description A ñˆ C B » A ˆC B Ј f C (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='14) and applying Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 in the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let C be an p8, 2q-category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) For each arrow f : A Ñ B in C with a right adjoint fR, we have natural equivalences A ш f B » B Ј fR A and B Ð> fR A » A Ñ> f B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='16) (2) For each diagram A fÝÑ C gÐÝ B in C, we have a natural equivalences A ш gLf B » A ñˆ C B » B Ј fRg A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='17) assuming that g has a left adjoint or f has a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 133 (3) For each diagram A fÐÝ C gÝÑ B in C, we have a natural equivalences B Ð> fgR A » A ñ> C B » A Ñ> gfL B (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='18) assuming that g has a right adjoint or f has a left adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Each of these equivalences represents (in the case “>”) or corepresents (in the case “ˆ”) the corresponding equivalences of Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' All the relevant objects are characterized either by their represented or corepresented functor, hence we may reduce to the case of lax limits and directed pullbacks in Cat8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This case is established in Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2 About adjoints in diagram 2-categories Let B, C be two p8, 2q-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By FUNlaxpB, Cq and FUNoplaxpB, Cq we denote the p8, 2q-category of functors B Ñ C and lax/oplax natural transformations η: F Ñ G between them, which assigns to each morphism f : B Ñ B1 in B a square FB FB1 GB GB1 Ff ηB ηB1 Gf or FB FB1 GB GB1 Ff ηB ηB1 Gf (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Formally, the functors FUNlaxpB, ´q and FUNoplaxpB, ´q can be defined as a right adjoints to the lax and oplax Gray tensor products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' By FUNpB, Cq we denote the standard internal hom in the p8, 2q-category of p8, 2q- categories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' it can be identified with the wide, locally full subcategory of FUNlaxpB, Cq and FUNoplaxpB, Cq containing only those 1-morphisms η, where the squares (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) contain invertible 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' If each component ηB of a lax natural transformation η: F Ñ G has a left adjoint ηBL, then these assemble to an oplax natural transformation ηL : F Ñ G whose oplax naturality squares GB GB1 FB FB1 Gf ηBL ηB1 L Ff (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2) are the canonical mates of the squares (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dually, each oplax transformation η has a canonical mate ηR (which is a lax transformation), whenever its components have right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Finally note that each natural transformation η can be viewed both as a lax and as an oplax transformation, thus has both a mate ηL (oplax) and ηR (lax), provided that all the required componentwise adjoints exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The following result due to Haugseng characterizes the morphisms in FUNpB, Cq which have a adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 134 Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='3 ([Hau21], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let η: F Ñ G: B Ñ C be a natural transfor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (1) As a morphism in FUNlaxpB, Cq, the transformation η has a right adjoint if and only if each component ηB has a right adjoint in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The right adjoint ηR is its canonical mate, where η is viewed as an oplax transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' (2) As a morphism in FUNoplaxpB, Cq, the transformation η has a left adjoint if and only if each component ηB has a left adjoint in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' The left adjoint ηL is its canonical mate, where η is viewed as a lax transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' This result also explains our terminology from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let η: F Ñ G: B Ñ C be a natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' As a morphism in FUNpB, Cq it has (1) a right adjoint if and only if the naturality square (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is vertically right adjointable, (2) a left adjoint if and only if the naturality square (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='1) is vertically left adjointable, In each case, the left/right adjoint is the corresponding canonical mate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Beyond the existence of adjoints, the right/left vertical adjointability condition states precisely that the 2-cells in the canonical mate ηR, ηL are again invertible, thus providing a right/left adjoint in FUNpB, Cq and not just in FUNlaxpB, Cq/FUNoplaxpB, Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Let S be an 8-category and α: X Ñ Y: S Ñ C a natural transformation of S-diagrams in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assume that each component αs has a left/right adjoint βs and that all naturality squares of α are vertically left/right adjointable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='4 tells us that in this case the components βs assemble to a natural transformation β : Y Ñ X which is in the diagram category FUNpS, Cq a left/right adjoint to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Assuming that C has lax limits or colimits of shape S, we can apply the 2-functors laxcolim: FUNpS, Cq Ñ C and laxlim: FUNpS, Cq Ñ C (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='6) to get corresponding adjunctions laxcolims αs : laxcolim X Ø laxcolim Y : laxcolims βs (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='7) and laxlims αs : laxlim X Ø laxlim Y : laxlims βs (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content='8) which in the lax semiadditive case are identified with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' in prepara- tion, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [Chr22c] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Christ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Spherical monadic adjunctions of stable infinity categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Math.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Sigma, 9:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' e10, 49, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 136 [DJW19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Dyckerhoff, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Jasso, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Walde.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' [Wal22] Tashi Walde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Homotopy coherent theorems of Dold-Kan type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=', 398:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 108175, 53, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} +page_content=' 139' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NE0T4oBgHgl3EQfugGB/content/2301.02606v1.pdf'} diff --git a/59AyT4oBgHgl3EQfpfhu/content/tmp_files/2301.00526v1.pdf.txt b/59AyT4oBgHgl3EQfpfhu/content/tmp_files/2301.00526v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc133dabbcaa6bdb33a40784b49f75656061bfbd --- /dev/null +++ b/59AyT4oBgHgl3EQfpfhu/content/tmp_files/2301.00526v1.pdf.txt @@ -0,0 +1,1227 @@ + +1 + Unusual acceleration and size effects in grain boundary migration with shear coupling + +Liang Yanga, Xinyuan Songb, Tingting Yuc, Dahai Liua*, Chuang Dengb,* +a School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang +330063, China +b Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada +c School of Aviation and Mechanical Engineering, Changzhou Institute of Technology, Changzhou, +Jiangsu 213032, China. +* Corresponding author: dhliu@nchu.edu.cn(D. Liu), Chuang.Deng@umanitoba.ca (C. Deng) + + +Graphical abstract + + +acceleration and size effect +mechanism +50 +80 +30 +Lx- Ly Lz +25 +2Lx +Lx- 2 0L, 2 0Lz +40 +20 +2L-3L,-3Lz +60 +8Lx +20 +aDisplacement (n) +30 +(eV) +40 +20 +20 +1010 +Original +-10 +Fitted +-20 +-15 +:2 +0 +4 +6 +10 +12 +0 +200 +400 +0 +50 +100 +Time (ps) +Time (ps) +Displacement (nm)150 +12 +140 +topfullyfixed +第0ps +120 +120- +nm) +100e +90 +bottomfree +80 +er +E +ien +60. +0g +2L +40 +30 +20x +0 +0 +0 +20 +40 +60 +80 +0 +200 +400 +600 +800 +Time (ps) +Time (ps) +attempt to alleviate acceleration and size effect +2 +Abstract +Grain boundary (GB) migration is widely believed to maintain a linear relation between its +displacement and time under a constant driving force. In this study, we investigated the migration +behaviors of a set of GBs in Ni by applying the synthetic driving force and shear stress via atomistic +simulations. It was found that the displacements of some shear-coupling GBs do not exhibit a linear or +approximately linear relation with the time, as widely assumed, but evidently exhibit an acceleration +tendency. Moreover, the boundary velocity significantly decreases when increasing the bicrystal size +perpendicular to the GB plane. These behaviors were verified to be independent of the magnitude and +type of driving force but closely related to the temperature and revealed to be unique to +shear-coupling GBs exhibiting a rise in the kinetic energy component along the shear direction. +Moreover, after many attempts, we found that the acceleration in migration and size effect can be +largely alleviated by adopting one specific kind of boundary condition. Nevertheless, the continuous +rise of kinetic energy still exists and leads to the true driving force for GB migration lower than the +nominally applied value. For that reason, a technique is proposed to extract the true driving force +based on a quantitative analysis of the work-energy relation in the bicrystal system. Accordingly, the +calculated true mobility shows that the recently proposed mobility tensor may not be symmetric at +relatively large driving forces. +Keywords: grain boundary migration; shear-coupling; size effect; mobility; atomistic simulation. + +1. Introduction +Grain boundary (GB) migration is crucial to a variety of behaviors (e.g., grain growth, +recrystallization, and plastic deformation) and mechanical properties (e.g., strength and ductility) in +polycrystalline materials [1]. Up to now abundant and deep insights have been gained into the +characters and underlying mechanisms of GB migration based on theoretical and experimental +investigations [2-9]. Thereinto, dramatic attentions were paid to GB mobility M, which can be defined +as the coefficient relating to migration velocity v and driving force P (i.e., M = v/P) and is considered +an intrinsic GB property (i.e., only depending on material parameters, temperature, and boundary + + +3 +crystallography) [10-12]. +Nevertheless, computational studies [13-15] have revealed that the magnitude of driving force +can leave significant influences on mobility, owing to the force-induced variation of boundary +structure and/or migration mechanism. For example, Deng and Schuh [13] found that for both +symmetrical and inclined Ni Σ5 100 tilt GBs, their mobilities agree well with the intrinsic values +obtained by the thermal fluctuation method [16] only when the applied driving force is sufficiently +low; increasing the driving force will lead to diffusive-to-ballistic transition in the migration +mechanism and enlarge the discrepancy between the extracted and intrinsic mobility values. Moreover, +for shear-coupling migration GBs (i.e., simultaneous translation in GB plane during the migration +along the boundary normal direction), Han and coworkers [7,17,18] demonstrated that both GB +mobility and shear-coupling factor (ratio of GB sliding and migration rates) do not only strongly +depend on the magnitude but also the source of driving force (stress or a jump in chemical potential +across the boundary). They further revealed that the mobility traditionally defined as a scalar should +be a symmetrical second-rank tensor [18]. The tensor components can be extracted by applying +driving forces in the directions perpendicular and tangent to the boundary plane, respectively [18]. +In addition to the driving force, GB motion may also strongly depend on the size of simulation +cell. Zhou et al. [19] reported that the mobility of a Ni 100 tilt GB decreased monotonically with +decreasing the cell thickness (the size along the tilt axis), due to the interference between the free +surface and the collective rearrangement of atoms during boundary motion driven by an external +stress. A similar size-dependency of mobility was also observed in Ref. [20]. Race et al. [21] revealed +that the boundary area of a flat 111 tilt GB should reach the meso-scale or a large-enough value to +yield a converged migration velocity under the synthetic driving force (SDF). Meanwhile, simulations +for stress-driven [22,23] and SDF-driven [24] migration further discovered that the energy barrier for +disconnection nucleation or the driving force for GB migration would converge when the boundary +area was large enough for shear-coupling GBs. The energy barrier was also found to firstly decrease +and then keep steady with the increase of cell size in GB normal direction for 53.1º Σ5 100 tilt GB +[24]. Existing studies concerning the size effect on GB migration overall reached a consensus that the + + +4 +system size should be large enough to yield physically reliable results and conclusions. This agrees +well with the general understanding related to the size effect in modeling and simulation. +Although GB migration has been reported to suffer influences from various factors (e.g., +crystallography [11,25], temperature [13,26], driving force [10,14], pressure [27,28] and impurity +[29,30]), the mobility values extracted from M = v/P in these studies were all based on a basic premise +that the boundary velocity will maintain constant or approximately constant during the whole +migration process under a fixed driving force, i.e., the boundary displacement exhibiting a linear or +approximately linear relation with the migration time. This character has been widely observed in +existing research concerning GB migration (e.g., Refs. [3,31-34]). Nevertheless, in this study, we +found that velocities of some GBs did not keep constant during migration but exhibited an unusual +acceleration feature, i.e., the velocity varying significantly with GB relative position in the simulation +cell. This signifies a strong dependency of migration on the cell size in the direction perpendicular to +GB plane, which has not yet been discovered before. +The first effort of the present work was therefore to investigate the underlying mechanisms for +the acceleration in GB migration and effects of model size in GB normal direction on migration, +based on atomistic simulations of several GBs driven by the external stress and SDF. After the +corresponding conditions and mechanisms for these two phenomena were clarified, attentions were +paid to effectively alleviate the size effect and to extract the true driving force and mobility in the +presence of acceleration. +2. Methodology +In this study, the acceleration in GB migration, size effect and other related contents were +investigated based on atomistic simulations of several GBs in Ni. First, we simulated the migration of +Ni Σ5 100{310}, Σ29 100 {10 4 0} and Σ55 211 {952} symmetrical tilt GBs, which correspond +to P1, P148 and P233 GBs in the 388 GBs dataset constructed by Olmsted et al. [11], to reveal the +phenomena of acceleration and dependency of migration on the cell size in the direction perpendicular +to GB plane. These simulations were performed at 500 K and an SDF of 0.06 eV when increasing the +normal cell size for each GB. Additional simulations were carried out at lower SDF (i.e., 0.025 eV for + + +5 +P1 and 0.003–0.006 eV for P233) and under the external shear stress τext (τext = 250 MPa for P1 and τext += 50 MPa for P233) to test whether the above phenomena are affected by the magnitude and type of +driving force, respectively. Second, we simulated the migration behaviors for Ni Σ3 110 twist GB +(i.e., P5) at 500 – 1000 K and 0.006 – 0.06 eV, Σ21 210 general GB (i.e., P81) at 1000 K and 0.001 – +0.06 eV, P1 at 1000 K and 0.06 – 0.003 eV. These simulations were aimed to explore the physical +causes for the acceleration and size-dependency from the aspects of shear coupling and work-energy +relation in the bicrystal system. Third, we chose the P1 GB as the representative example to attempt +approaches for inhibiting or alleviating the size effect by performing simulations adopting various +boundary conditions (BCs) and/or manipulating the internal stress in the system. The corresponding +results were compared with those reported in existing studies whenever possible. Finally, we still +chose P1 as the example to discuss how to extract the true driving force for GBs exhibiting +acceleration when the true driving force was not equal to the value nominally applied through the +SDF method or the external shear stress. +All simulations stated above were performed using the Large-scale Atomic/Molecular Massively +Parallel Simulator (LAMMPS) software package [35] with the embedded atom method potential +developed for Ni [36]. As shown in Fig. 1(a), a bicrystal simulation cell was used to construct a flat +GB, which was in y-z plane with y and z directions being periodic. Nevertheless, the boundary +conditions in x direction (parallel to the boundary normal direction) might be quite different, +depending on the simulation tests. For the above first and second groups of simulations, in order to +avoid translation of the whole bicrystal in GB normal direction, a slab of atoms (1nm thickness) near +the bottom surface were partially fixed (i.e., only the velocity and force components along the x +direction being set as zero) while the top surface was set free. For the third group of simulations, the +boundary conditions in x direction might be periodic, fully fixed, free, or one surface free while the +other fixed. When exploring the size effect on boundary migration, the cell size in the GB normal +direction (i.e., the grain size) or in the boundary plane (i.e., the boundary area) was varied accordingly. +The minimum cell size for each above GB was the same as those constructed by Olmsted in Ref. [11]. +For example, the minimum cell size for P1 was Lx = 18.3 nm, Ly = 3.2 nm, and Lz = 3.3 nm. + + +6 + When the bicrystal system of each GB was constructed, its energy was minimized at 0 K +following the scheme introduced in Ref. [11]. Subsequently, the system was elevated to and +sufficiently relaxed at test temperatures (500 K or 1000 K) under the isothermal-isobaric ensemble +(NPT) for about 0.15–7 ns (depending on the temperature, GB, and cell size) with a default time step +of 5 fs. After the system was fully equilibrated, the boundary was driven to migrate under a jump in +chemical potential across the boundary or a shear stress. Thereinto, the former was realized by the +CROP-SDF method [14] while the latter by applying a shear force to individual atoms near the +bottom surface (1 nm thickness in x direction) in Fig. 1(a). The GB displacement under the SDF and +stress was computed by tracking the overall change of the potential energy artificially added to the +bicrystal system and by tracking atoms with the centro-symmetry parameters close to the maximum +value in the system, respectively. For the above four groups of simulations, the NPT ensemble was +also used during the boundary migration for each case, if not otherwise specified, to control the +internal normal stress components as close to 0 GPa as possible. For some specific simulations, the +internal shear stress along the shear direction also needed to be controlled at 0 GPa. The structure of +bicrystal model, if needed, was visualized by Ovito package [37]. Note that some results concerning +this study were presented in the Supplementary Material. +3. Results and discussion +3.1 Acceleration in migration and size effect + Fig. 1(b-e) represents the migration data of Σ5 100 {310} tilt GB (i.e., P1 GB) when increasing +the cell size along different directions while under a constant external driving force. Two features can +be readily observed. First, the boundary velocities (slope of displacement curve) under various cell +sizes all gradually increase with the proceeding of boundary migration, suggesting a clear dependency +of migration velocity on the relative GB position along the boundary normal direction in the +simulation cell. In contrast, the boundary displacement was widely observed and commonly assumed +to exhibit a linear or approximately linear relation with the migration time (e.g., Refs. [3,31]). After a +detailed survey of existing studies, the acceleration in migration was only found in the work by +Coleman et al. [38], who simulated the migration of Ni Σ37 100 symmetrical tilt by applying the + + +7 +synthetic driving force and shear strain at 300 and 400 K. Unfortunately, the focus in Ref. [38] was +the atomic mechanisms of migration and no attention was paid to the acceleration. + +Fig. 1 (a) Schematic of the bicrystal simulation cell. GB displacement vs. time for P1 GB simulated at 500 K +and under a synthetic force of 0.06 eV, when increasing the cell size along (b) x, (c) y, (d) z and (e) all three +directions. To avoid translation of the whole bicrystal along the x direction, the bottom surface was partially +fixed (i.e., setting the velocity and force components along x for atoms near the surface as zero) while the top +surface was set free. + +Second, the velocity is independent of the cell size in GB plane (i.e., the boundary area or lateral +cell size) (see Fig. 1c-e) but strongly and negatively related to the size in the boundary normal +direction (x direction in this study) (see Fig. 1b and e). Nevertheless, the velocity of flat boundary has +been previously reported to show a strong and complex dependence on the boundary area [21]. In +addition to the velocity, the boundary area has also been reported to cause significant influence on GB +mobility [19,20] and the energy barrier of disconnection nucleation for GB migration [22-24]. +Moreover, these properties concerning GB migration exhibited a consistency in their dependency on +the boundary area, i.e., the boundary area should be sufficiently large to yield a converged property +value. The negative dependency of velocity on the cell size along the boundary normal (i.e., vertical +cell size) here is partially similar to the trend regarding the threshold driving force of disconnection +nucleation for Cu Σ5 100 {210} tilt GB (i.e., P6 GB in Ref. [11]) at 10 K revealed by Deng and +Deng [24], who found the threshold driving force, which in practice can be qualitatively regarded as +the reverse of GB mobility [9], overall declines when increasing the vertical cell size. Therefore, the +size effects regarding the boundary area are different between the present and existing studies, but a + + +8 +similarity appears in the dependency on the vertical cell size. +As a typical of low-period and high-angle CSL boundary, the migration behaviors of P1 GB have +been widely studied through atomistic simulations [6,10,11,13,14,17,34,39-41], but why the above +two features were not reported before? This can be attributed to multiple factors. First of all, there has +been no research adopting various vertical cell sizes for this GB up to now, and accordingly no insight +into the size effect was obtained. In studies adopting fixed vertical size [10,11,14,17,34,39], the +acceleration might also exist, though the displacement-time data was not directly provided in these +studies. Nevertheless, we deem that the acceleration might have been disregarded on the grounds that +the main attentions and efforts were focusing on exploring the intended objectives of individual +studies, as in our previous work [14,34]. Meanwhile, the disappearance of acceleration can also be +attributed to the relatively high temperatures (e.g., 1000, 1200 and 1400 K) tested in Refs. [10,11,39] +(see discussion in Section 3.2). In addition, the periodic boundary condition imposed along the +boundary normal direction will prevent the presence of acceleration in work [13,40]. This boundary +condition has been confirmed to inhibit the shear-coupling migration [21,41], which is a necessary but +not a sufficient condition for acceleration migration (see following discussion concerning Figs. 2 and +3). When exploring the shear coupling migration of Cu P1 GB, Cahn et al. [6] directly presented a +displacement-time data up to 1 nm, simulated by applying a constant shear strain 1 m/s at 800 K (see +Fig. 6 in Ref. [6]). To our understanding, 1 nm data may not be sufficiently long to evidently illustrate +the acceleration feature, in comparison with the displacement data in Fig. 1. Another +displacement-time data (up to 6 nm) was provided by Schartt and Mohles [41], who simulated the +migration of Ni P1 under 300–1000 K with free end boundary conditions and a synthetic force of 0.06 +eV imposed through the ECO-SDF method. Nevertheless, the acceleration feature was still not +observed in Ref. [41] though it shows up in our re-tests of simulations in Fig. 1 by using the +ECO-SDF method. Therefore, the discrepancy concerning the acceleration should not be attributed to +the different versions of SDF method (i.e., CROP-SDF [14] or ECO-SDF [41]) utilized in the present +study and Ref. [41]. Since the temperature dependency of migration velocity and shear coupling +factor  in the range of 300-700 K in [41] also appears different from those previously reported +[6,9,13,17], we deem that the discrepancy may be resulted from the difference in the metastable + + +9 +structures for P1 GB adopted for various studies. +To evaluate whether the acceleration in migration or the corresponding size effect is unique to P1 +GB or not, Fig. 2 show the results simulated for some other GBs or under simulation settings different +from Fig. 1. As shown in Fig. 2(a) and (b), the two features can also be observed for P148 and P233 +GBs when adopting the same settings as for P1 in Fig. 1. Since the driving force applied in Figs. 1, +2(a) and (b) is a relatively high value (i.e., 0.06 eV ≈ 0.87 GPa) in comparison with experimentally +applied values, we tested lower forces for P1 and P233 GBs. It can be seen from in Fig. 2(c) and (d) +that these features still hold on for P1 GB at 0.025 eV (lower than KT = 0.043 eV, K Boltzmann +constant and T temperature) and for P233 GB at 0.003 eV which approaches typical experimental +values. Note that lower forces have also been tried for P1 but failed to yield continuous boundary +migration, agreeing with the threshold driving force of boundary migration determined for this GB at +500 K by Yu et al. [9]. It is important to note from Fig. 2(e) and (f) that the acceleration and size effect +still show up for P1 and P233 GBs when applying the external shear stress to drive the GB migration. +Therefore, while the shear coupling mode may be strongly influenced by both the magnitude and type +of the driving force [17], the acceleration and size effect does not exhibit such dependency. +The above analysis suggests that the acceleration in migration and negative dependency on the +vertical cell size are relatively common features for force-driven GB migration. The following content +will further reveal that the latter feature is resulted from the former one. These two features extend our +current understandings of size-effect on GB migration, which almost all focused on the size in the +boundary plane [19-23]. They also remind us that attentions should be taken for GBs exhibiting such +features when extracting the boundary velocity or mobility under a constant driving force, during +which the migration displacement and time were almost always assumed to keep a linear relation. + + +10 + +Fig. 2 Other results simulated at 500 K supporting the acceleration in migration and size effect. Displacement +data in (a-d) were all simulated under the applied synthetic force while (e, f) under the external shear stress. +The shear stress was applied to a slab of atoms (1 nm thickness) near the top surface, as illustrated in Fig. 1, +while a slab of atoms near the bottom surface was set as a grid body. +3.2 Underlying mechanism for acceleration and size effects +The acceleration in boundary migration and vertical size effect have been revealed in the above +section, then for what types of GB or under what kinds of condition that such phenomenon will occur? +A preliminary analysis of the three GBs tested in Figs. 1 and 2 indicates that they are all +shear-coupling migration GBs and with  > 0.5. Fig. 3(a) chooses P5 (Ni Σ3 110 twist) GB as an +example to show the displacement-time (S-t) data and size-dependency of GBs without shear-coupling. +It can be seen that the displacement is linearly related to the migration time and the corresponding +velocities are the same under different vertical sizes, irrespective of temperature and magnitude of +driving force. These results seemingly indicate that only shear-coupling GBs will exhibit acceleration + + +11 +migration and negative size-dependency. However, as shown in Fig. 3(b) for P81 GB, the linear S-t +relation and constant v under different sizes can be observed also for shear-coupling GBs at various +temperatures and driving forces. Furthermore, the acceleration migration and size effect observed at +500 K for P1 GB (Fig. 1) unexpectedly transfers into uniform migration when raising the temperature +to 1000 K at which the shear coupling still exists, regardless of the magnitude of driving force (see +Fig. 3(c)). This kind of transition induced by the temperature also occurs for other shear-coupling GBs +(see Fig. s1 in the Supplementary file). To our understanding, the transition can be attributed to the +temperature-induced variation of disconnections mediated for GB migration, which has been widely +observed [7,23]. Based on these analyses, we conclude that the shear coupling is a necessary but not a +sufficient condition for the acceleration in migration and therefore the size effect, which may suffer +strong influence from the temperature. + +Fig. 3 Examples of uniform migration for GBs with or without shear-coupling: (a) P5; (b) P81; (c) P1. (d) and +(e) presents kinetic energy E for P1 with cell size Lx, simulated at 500 K and 1000 K, respectively. + +To further explore the fundamental mechanisms for acceleration, Fig. 3(d) compares the relative + + +12 +variation of kinetic energy (Ei, i = x, y or z) to the initial state for P1 GB at 500 and 1000 K, at which +the boundary exhibits accelerated (Fig. 1(b)) and uniform (Fig. 3(c)) migration, respectively. At 1000 +K, all three components of the kinetic energy remain almost unchanged (i.e., Ex = Ey = Ez ≈ 0) +with the proceeding of migration. In contrast, at 500 K, although Ex and Ey still remain unchanged, +Ez firstly increases and then decreases (the final Ez is still much higher than zero). Note that the +shear movement is parallel to z direction. The comparison suggests that the work (Wext) done by the +external driving force (Pext) only contributes to shear-coupling migration at 1000 K, but to both +shear-coupling migration and a rise in the shear kinetic energy (Ez) at 500 K. This difference reminds +us that the accelerated and uniform migration can be qualitatively justified from the aspect of true +driving force (Ptrue) for boundary migration, which can be influenced by Wext and Ez. +During the process of boundary migration, the work and kinetic energy for the bicrystal system +meet the relation of Wext = Wtrue + E. Wtrue is the work done by Ptrue, i.e., Wtrue = PtrueꞏSꞏAGB, S and AGB +stand for the GB displacement and area, respectively. Considering Ex = Ey ≈ 0 in the case of both +accelerated and uniform migration, the relation can be given as Wext = Wtrue + Ez. At one specific +moment of migration, the work-energy relation can be further described as dPextꞏdS = dPtrueꞏdS + +d(Ez)/AGB, and the instant true driving force is dPtrue = dPext – (d(Ez)/dS)/AGB, where dPext is a fixed +value for the SDF method. Therefore, dPtrue will be constant when Ez keeps unchanged (e.g., 1000 K +at Fig. 3(d)), i.e., dPtrue = dPext. In such case, the boundary will accordingly exhibit uniform migration +(i.e., a linear S-t relation) and thus consistent velocities when adopting distinct vertical sizes but the +same Pext (e.g., Fig. 3(c)). Nevertheless, in the case of accelerated migration (i.e., Ez ≠ 0, see Fig. +3(d)), dPtrue depends on both dPext and d(Ez)/dS. From the d(Ez)/dS vs. S curve (the red curve) at +500 K for P1 GB shown in Fig. 4, we can observe that d(Ez)/dS continuously descends with the +boundary migration, suggesting a continuous rise in dPtrue and thus in migration velocity. Moreover, it +is conceivable that when applying the same dPext to bicrystal systems with different vertical sizes (e.g., +Fig. 1(b)), dPtrue will be lower (i.e., lower velocities) for larger systems due to higher d(Ez), which +can be further attributed to more atoms involving shear movement for larger systems. When applying +this interpretation to justify the size effect for systems with different sizes in the boundary plane (e.g., + + +13 +Fig. 1(c)), the contribution of GB area to dPtrue must also be considered. In summary, the above +analysis suggests that the acceleration in migration and negative dependency of velocity on the +vertical size are unique to shear-coupling GBs exhibiting a rise in the kinetic energy component along +the shear direction and can be justified from the aspect of true driving force based on the work-energy +relation in the bicrystal system. +80 +30 + +Fig. 4 Variation of d(Ez)/dS with the boundary displacement for P1 GB, calculated based on the Ez-S curve +(blue curve) obtained by the least-square fitting of the original data at 500 K and 0.06 eV, given in Fig. 3(d). +3.3 Attempts to alleviate acceleration +Although the acceleration in migration has been demonstrated as a relatively common feature for +force-driven migration of flat GBs in Section 3.1, it is undesired if the purpose is to compute a GB +mobility by assuming v = MP. Then, is it possible to inhibit or alleviate the acceleration? For this +purpose, we have carried out a series of simulations by manipulating the boundary conditions and +internal stress in each bicrystal system (see Figs. 5-7). +Firstly, we tried to adopt periodic boundary condition in the GB normal direction (x direction in +Fig. 1(a)), which is one kind of boundary conditions widely used in previous studies [13,31,41]. It can +be seen from Fig. 5(a) that the boundary displacements under various vertical sizes all nearly exhibit a +linear relation with the time; the velocity nevertheless keeps increasing when enlarging the vertical +size, in contrast to a negative size dependency of velocity in Fig. 1. Meanwhile, in comparison with +0.06 and 0.025 eV applied for the boundary condition adopted in Figs. 1 and 2c, much larger driving +force (i.e., 0.15eV) must be applied to initiate the boundary movement for all cell sizes, suggesting a + + +14 +significant effect of boundary condition. Under the present boundary condition, the internal shear +stress sharply increases with the initiation of GB migration, and then experiences a short descending +and finally fluctuates around a very high value (see Fig. 5(b)). Furthermore, the shear stress is +obviously lower under larger cell size. The higher velocity under larger size in Fig. 5(a) is therefore +resulted from this tendency of shear stress, which can be attributed to more elastic energy released +with the boundary migration due to larger space along the GB normal. As shown in Fig. 5(c), the +periodic surface strongly inhibits the overall relative shear movement between two grains, as revealed +in Refs. [13,41]), but only enables local shear movement which is more evident for larger cells. This +precisely accounts for the linear S-t relation under various sizes in Fig. 5(a), according to the +discussion concerning mechanism for acceleration in Section 3.2. Evidently, the periodic boundary +can effectively eliminate the acceleration but not the size effect by significantly constraining the shear +movement. + +Fig. 5 Migration results vs. vertical size for P1 GB, simulated at 500 K and 0.15 eV by adopting periodic +boundary conditions along the GB normal direction: (a) displacement data; (b) shear stress; (c) snapshot of +boundary migration. Red and white arrows in (c) illustrate the migration and shear directions, respectively. +Atoms are colored by atom type to visualize the shear-coupling migration using the Ovito software [37]. + +Secondly, we tried to set the top and bottom surfaces to be fully fixed. It can be seen from Fig. 6 +that most of the results are similar to those under periodic boundary in Fig. 5, e.g., much larger +driving force, linear S-t and very high shear stress. Additionally, displacements are nearly independent +of the system size (Fig. 6(a)), though the normal stress continues to rise with the boundary movement +(see the example shown for cell size of Lx in Fig. 6(b)). The corresponding velocity is much lower +than that at 0.06 eV in Fig. 1 while close to that at 0.15 eV and Lx in Fig. 5. In consistency with the + +Initialnitia] +2L +8L +20LC +GBGB +15 +periodic boundary, the boundary condition of fixed ends also inhibits the global shear movement and +enables only local shear. Fig. 6(c) shows that the local shear-coupling mode may change even switch +under fixed ends, as already observed in Ref. [42]. +0 +50 +100 +150 +200 +250 +0 +5 +10 +15 +20 +Time (ps) +Displacement (nm) +Lx +2Lx +8Lx +20Lx +500 K, 0 15 eV +. +25 +(a) +Lx + + +Fig. 6 Migration results for P1 GB simulated at 500 K and 0.15 eV while setting the top and bottom surfaces +to be fully fixed: (a) displacement data; (b) shear stress and normal stress under cell size Lx; (c) snapshot of +boundary migration. The red and yellow atoms colored in (c) are aimed to visualize the local shear movement +and shear-switching. + +Thirdly, considering the impeding effect of the high internal shear stress on shear movement, we +performed simulations that controlling the stress as close to 0 GPa as possible under periodic +boundary (see Fig. 7). Although shear stresses are well controlled especially for larger systems (Fig. +7(b)), acceleration and size effect still exist (Fig. 7(a)). Moreover, the boundary stagnates long before +reaching the other end, and the final displacement value under each size is nearly half of the feasibly +maximum value (compare Figs. 5(a) and 7(a)). The inclined line displayed by the yellow atoms in +Figs. 7(c) illustrates that the shear movement is inhomogeneous along the GB normal direction, and +the top and bottom parts of the blue grain make shear along two opposite directions (see the white +arrows). These results should be resulted from the cell inclination when controlling shear stress (Fig. +7(c)), which leads to the variation of crystallographic orientation and thus erroneous exertion of the +orientation-dependent driving force for shear-coupling migration. We have also tried to control shear +stress for fixed and free boundary conditions but as well obtained cell inclination and other results +similar to those by using periodic boundary. Evidently, the above three attempts all failed to achieve +our anticipated objectives of effectively alleviating the acceleration and size effect. + + +CGB +200ps: +150ps +150psInitiaInitial +27 +8L +201 +16 + + +Fig. 7 Migration results vs. vertical cell size for P1 GB, simulated at 500 K and 0.06 eV by adopting periodic +boundary while controlling shear stress close to 0 GPa: (a) displacement data; (b) shear stress; (c) snapshot of +boundary migration. +Finally, considering the inhibition of overall shear movement by periodic and fixed boundaries, +we performed simulations adopting two free boundaries or setting one boundary as free while the +other as fixed. It can be seen from Fig. 8(a) that the displacement data under two free boundaries and +one free while another partially fixed are consistent and exhibiting an acceleration tendency (as +observed in Fig. 1(b-e)), because the two grains across the GB are free to shear under these two +boundary conditions. When the bottom surface is fully fixed (i.e., the grain near this surface is not +allowed to shear), significant acceleration is observed through the whole migration process (the blue +curve in Fig. 8(a)). Nevertheless, in the case of fully fixed top surface, acceleration is only significant +at the early migration stage and gradually turns into uniform migration at the later stage (see the green +curve and black dashed line in Fig. 8(a)), suggesting a gradual weakening of acceleration. This +tendency can be justified from the similarity between the variations of Ez–t and S-t curves in Fig. 8(b) +and from the relation dPtrue = dPext – (d(Ez)/dS)/AGB. Moreover, as illustrated by the inset snapshots +in Fig. 8(b), the overall shear and coupling factor are not influenced by this boundary condition. +Furthermore, Fig. 8(c) presents the comparison of the linear S-t segments extracted from the complete +displacement data under various sizes. Interestingly, velocities are consistent for different cell sizes. +Therefore, the size effect for shear-coupling GB can be considered as being eliminated if only +focusing on the uniform migration stage. With these attempts, we may conclude that setting the top +surface (i.e., in the forward direction of GB migration) of the bicrystal as fully fixed while the bottom +surface (the backward direction of migration) as free is a relatively effective way to largely alleviate + +0 ps +5 ps +15 ps +25 ps +17 +acceleration migration and thus size-dependency. + + +0 +200 +400 +600 +800 +0 +20 +40 +60 +80 +100 +120 +140 +Time (ps) +Displacement (nm) + + +Lx +2Lx +8Lx +20Lx +(c) + +Fig. 8 Attempts to alleviate acceleration by adopting one or two free boundaries for P1 GB with normal size +Lx, simulated at 500 K and 0.06 eV. (a) Comparison of displacement data under different boundary conditions. +(b) Variation of displacement, kinetic energy and snapshot with the time when setting the bottom surface as +free while the top one as fully fixed. (c) Comparison of the linear S-t segments extracted from the complete +displacement data under various cell sizes, simulated by adopting the same boundary condition as in (b). +When the surface is partially (as in Fig. 1) or fully (setting all force and velocity components for atoms near +the surface as zero) fixed, the grain near this surface can or can not make overall shear movement. The +complete displacement data for (c) can be found in Fig. s2 in the Supplementary file. +3.4 Extraction of true driving force and mobility +Although the accelerated migration and size effect for shear-coupling GBs can be effectively +weaken by adopting one special boundary condition, the kinetic energy of the system Ez still continues +to rise during the boundary migration (see Fig. 8(b)), and thus the true driving force Ptrue does not +equal to the externally applied value Pext and depends on the variation of Ez (see discussion in Section +3.2). Therefore, efforts should be paid to extract Ptrue and the corresponding true mobility Mtrue. As +already discussed in Section 3.2, Ptrue can be determined based on a quantitative analysis of the + +(b) 150 +12 +top fullv fixedps +120Isp +58 ps90 +acen +bottom freee +60nm300 +20 +00 +80 +Time (ps) +18 +work-energy relation in the bicrystal system. Meanwhile, considering the continuous rise of Ez with +boundary migration, we may not obtain a constant but a time dependent Ptrue(t), and therefore the +quantitative analysis should be carried out for individual steps of GB migration. The artificial energy +added to the system by the SDF method in time interval dt can be written as: +dE = eꞏdn (1) +where e denotes the maximum potential energy added to a single atom (e.g., 0.06 eV in Fig. 1) and +dn indicates the number of atoms whose corresponding crystallography changed as the boundary +migrates. dn can be calculated by dn = (N/Lx)ꞏv(t)ꞏdt. Here, N and v(t) represent the total number of +atoms in the system and the instant migration velocity perpendicular to GB plane, respectively. +However, due to the continuous rise of Ez, the actual energy to drive boundary movement is +dE' = dE – dEz = dE – czꞏdt (2) +where cz is variation rate of dEz with respect to time. We can thus deduce the true maximum energy +imposed on per atom e'(t) as: + e'(t) = (dE – dEz)/dn = e – cz/(vꞏN/Lx) (3) +For the SDF method, e is normally considered as Pext [11,13,41] and thus e'(t) can also be treated as +Ptrue(t), which then can be further used to calculate Mtrue. +From the internal stress data in Fig. 9(a), one can observe that the normal stress fluctuates around +0 GPa while the internal shear stress τxz roughly keeps a positive value and gradually declines when +adopting the boundary condition of a fully fixed top surface. This means that the normal stress leaves +no influence on GB migration, but one part of dE may be used to overcome the impeding effect of τxz +on migration, which can be quantitatively described in the form of shear strain energy Ess = +0.5Vꞏ +2 +xz + +/G. V and G stand for the volume and shear modulus for bicrystal system, respectively. +However, Ess is essentially a type of elastic energy and will be dynamically stored and released with +the continuous migration of the GB, as supported by Fig. 9(b) which indicates that this energy +increment dEss also fluctuates around 0 eV (i.e., the long-time average of dEss equaling zero). +Therefore, Ess should make no contribution to the overall work-energy relation in the system and the +potential influence of τxz on GB migration does not need to be considered. Accordingly, Eq. (3) still + + +19 +holds for extracting Ptrue. + +Fig. 9 Variation of (a) the internal stress and (b) dEss with the migration time for P1 GB simulated in Fig. 8(b) +Table 1 presents the calculated Ptrue and Mtrue for P1 GB at 500 K according to Eq. (3). In contrast +to the early migration stage, the velocity, driving force and mobility at the later stage (e.g., t > 50 ps +for Lx and t > 400 ps for 8Lx) all only rise slightly and are nearly consistent under Lx and 8Lx systems. +If we calculate the average value for v, Ptrue and Mtrue at the later stage, we can get the average v, Ptrue +and Mtrue for Lx system as 0.163 nm/ps, 0.0486 eV and 3.341nm/(ps eV) while v = 0.162 nm/ps, Ptrue = +0.0485 eV and Mtrue = 3.353 nm/(ps eV) for 8Lx. The relative differences of these three data between +the two systems are all lower than 1%. These results and comparisons again emphasize that the +acceleration and size-effect in the GB migration velocity have been nearly eliminated through +applying one special boundary condition. More importantly, they also signify that we can obtain +consistent true mobility values for systems with distinct vertical sizes if further considering the +correction of true driving force. +It should be noted that the above principle of correcting the driving force should also be +applicable to the shear-coupled migration driven by an external shear stress τext (i.e., applicable to Fig. +2(e) and (f)). As shown in the inset of Fig. 10(a) for τext-driven migration, Ex and Ey remain +unchanged while Ez overall increases linearly with time, in consistency with the tendency shown for +the SDF-driven migration in Fig. 8(b). The corresponding correction equation of Ptrue can be given as: +z +true +ext +z +GB +c +P +v +A + + + + + (4) +where vz stands for the shear velocity of GB along the z direction. + + +20 +Table 1 True driving forces and mobility values extracted based on Eq. (3) for P1 GB, simulated at 500 K and +0.06 eV when setting the bottom surface as free while the top one as fully fixed. Both cz and v were obtained by +the least-square fitting into discrete Ek and S data. +Lx +8Lx +t +(ps) +v +(nm ps‒1) +Ptrue +(eV) +Mtrue +(nm ps‒1 eV‒1) +t +(ps) +v +(nm ps‒1) +Ptrue +(eV) +Mtrue +(nm ps‒1 eV‒1) +5* +0.030 +-0.0016 -18.906 +40* +0.035 +0.0068 +5.187 +10 +0.061 +0.0299 +2.054 +80 +0.065 +0.0313 +2.090 +15 +0.086 +0.0386 +2.237 +120 +0.089 +0.0390 +2.292 +20 +0.106 +0.0426 +2.489 +160 +0.108 +0.0427 +2.538 +25 +0.121 +0.0447 +2.707 +200 +0.123 +0.0447 +2.747 +30 +0.133 +0.0461 +2.879 +240 +0.134 +0.0460 +2.914 +35 +0.141 +0.0469 +3.010 +280 +0.142 +0.0468 +3.041 +40 +0.148 +0.0475 +3.109 +320 +0.149 +0.0474 +3.137 +45 +0.152 +0.0479 +3.182 +360 +0.153 +0.0477 +3.208 +50 +0.156 +0.0481 +3.237 +400 +0.157 +0.0480 +3.261 +55 +0.159 +0.0483 +3.279 +440 +0.159 +0.0482 +3.301 +60 +0.161 +0.0485 +3.314 +480 +0.161 +0.0484 +3.334 +65 +0.163 +0.0486 +3.345 +520 +0.163 +0.0485 +3.361 +70 +0.165 +0.0488 +3.375 +560 +0.164 +0.0486 +3.385 +75 +0.167 +0.0489 +3.405 +600 +0.166 +0.0487 +3.406 +80 +0.168 +0.0490 +3.435 +640 +0.167 +0.0487 +3.423 +* Extracted Ptrue and Mtrue are erroneous due to the low velocity and kinetic energy at the very early stage of +migration + +To compute the true mobility under an external shear stress, Fig. 10 still chooses the P1 GB as an +example and the shear stress has been carefully tuned so that the GB migrated at the same velocity as +that under the SDF, as shown in Fig. 10(a). Firstly, in comparison with Fig. 2(e), the acceleration in +migration is only significant at the early migration stage and turns into uniform migration at the later +stage for the current simulation (see blue curve in Fig. 10(a)). Secondly, the true mobility calculated +based on the corrected Ptrue according to Eq. (4) experiences a continuous rise and then becomes +nearly stable (see the blue square data in Fig. 10(b)). These results do reveal that the accelerated +migration under external shear stress can also be largely alleviated by the boundary condition as +utilized in Fig. 8(b) for the SDF-driven migration, and that the principle of correcting Ptrue is also +applicable to the case of τext-driven migration. +What is more, although the displacements or velocities are nearly the same under the SDF and +shear stress, the corresponding mobility values under the two types of driving force are significantly + + +21 +different (see Fig. 10(b)). Mtrue by SDF is on average 57.6% lower than that by τext. According to the +theory recently proposed by Chen et al. [18], the mobility values extracted by applying the driving +forces perpendicular and parallel to GB plane can be unified in a mobility tensor as following: + +y + + + + + + +xx +xy +xz +x +y +x +yy +yz +y +z +zx +zy +zz +z +M +M +M +v +v +M +M +M +v +M +M +M + + + + + + + + + + + + + + + +  + + + + + + + + + + + + + + + + + + (5) +Here x is perpendicular to GB plane while y and z are parallel to GB plane, φ is the driving force applied +along GB normal (e.g., the synthetic driving force [14,41]), and τy and τz are shear stresses. Moreover, +the GB mobility tensor should be symmetric according to the Onsager relation [18], i.e., Mxz = Mzx. + + +Fig. 10 Migration results for P1 GB at 500 K when driven under τext : (a) displacement along GB normal; (b) +true mobility calculated based on the corrected Ptrue. The inset in (a) presents the variation of kinetic energy +Ei (i = x, y, z). To provide a better comparison of true mobility values between SDF-driven and τext-driven +migration, the magnitude of τext was chosen deliberately to ensure the corresponding displacement data as +close to that by SDF in Fig. 8(b) as possible. The shear stress was applied by adding a shear force 0.0037 +eV/Å to a slab of atoms (1 nm thickness) near the bottom surface along z direction, while the top surface was +set as fully fixed. + +As shown in Fig. 10(a),  equals 1.0 under both SDF and shear stress, indicating that vx = vz holds +under both types of driving force. In such case, we can deduce Mxx = Mzx for SDF and Mzz = Mxz for +shear stress. Nevertheless, the mobility data presented in Fig. 10(b) reveals Mzz ≈ 1.6Mxx, i.e., Mxz ≈ +1.6Mzx. Therefore, the symmetry of mobility tensor does not hold in the present study, seemingly +contradicting to the conclusion in Ref. [18]. Recently, we have systematically investigated the GB +mobility tensor and the effects of temperature and external driving force on its symmetry based on + + +22 +atomistic simulations of force-driven and force-free migration for the twist Ni Σ15 (2 1 1) (P14) GB +[43]. It is found that the symmetry holds at low driving force limit while fails at high driving forces. +Therefore, the non-equal Mxz and Mzx as shown in Fig. 10(b) is due to the relatively large driving +forces that have been applied in the current study, which is also consistent with the previous studies +that large driving forces can significantly change the underlying mechanisms for GB migration [13]. +4. Conclusions +In this work, we investigated the migration behaviors of several GBs in Ni by atomistic +simulation. Based on that, the acceleration in GB migration and vertical size effects were revealed. +Then attentions were paid to reveal the concerning mechanisms for these phenomena and to +effectively alleviate them through manipulating the boundary conditions and internal stress in the +bicrystal system. At last, a method was proposed to extract the true driving force and true mobility, +based on which the symmetry of mobility tensor was discussed. The following conclusions can be +drawn based on this study: +(1) The migration displacements of some GBs driven under a constant external driving force are not +linearly related to the migration time, as widely assumed. The corresponding velocity nevertheless +gradually increases with the proceeding of boundary migration and is overall negatively related to +the cell size perpendicular to GB plane, irrespective of the magnitude and type of driving force. +These tendencies suggest that some previously published migration results for such GBs without +considering the acceleration and size effect may need to be calibrated. +(2) The acceleration and vertical size effect in migration are unique to shear-coupling GBs exhibiting +a rise in the kinetic energy component along the shear direction. It is precisely the rise of kinetic +energy that results in the true driving force for GB migration being lower than the nominally +applied value but continuing to increase, and thus leads to the accelerated migration. With the +increase of tested temperatures, the acceleration will transfer into uniform migration and the size +effect will accordingly disappear. +(3) Among the various attempts to eliminate or alleviate the acceleration and size-dependency in +migration, setting the cell surface in the forward direction of GB migration as being fully fixed + + +23 +while the surface in the backward direction as free is a relatively sample but effective way, which +is applicable to both shear stress and SDF-driven migration and does not change the +shear-coupling behaviors and migration mechanism. +(4) Based on a quantitative analysis of the work-energy relation in the bicrystal system, the true +driving force can be determined for GBs exhibiting accelerated migration. Under the coupling +effects of adopting one specific kind of boundary condition and correcting the true driving force, +we can obtain consistent true mobility values for such GBs with distinct vertical sizes. +Furthermore, the calculated true mobility suggests that the symmetry of the mobility tensor may +fail when the driving forces are too large. + +Acknowledgment +The authors thank Dr. David L Olmsted for sharing the 388 Ni GB structure database. This +research was supported by the National Natural Science Foundation of China (Grant No. +52065045) and NSERC Discovery Grant (RGPIN-2019-05834), Canada, and the use of +computing resources provided by Compute/Calcul Canada. +References: +[1] G. Gottstein, L.S. Shvindlerman, Grain Boundary Migration in metals: thermo-dynamics, kinetics, +Applications, CRC press, 2009. +[2] M. Upmanyu, D.J. Srolovitz, L.S. Shvindlerman, G. 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Deng, Computing the intrinsic grain boundary mobility tensor, +https://doi.org/10.48550/arXiv.2212.11462. + + +1 +Supplemental Material: +Unusual acceleration and size effects in grain boundary migration with shear coupling + +Liang Yanga, Xinyuan Songb, Tingting Yuc, Dahai Liua*, Chuang Dengb,* +a School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang +330063, China +b Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada +c School of Aviation and Mechanical Engineering, Changzhou Institute of Technology, Changzhou, +Jiangsu 213032, China. +* Corresponding author: dhliu@nchu.edu.cn(D. Liu), Chuang.Deng@umanitoba.ca (C. Deng) + +1. Supporting results for the transition of acceleration migration into uniform migration +In Fig. 1 of the main text, we can readily observe the accelerated migration and negative +dependency of velocity on the vertical cell size for P1 GB at 500 K. Nevertheless, when raising the +temperature to 1000 K, both the acceleration and size effect disappear though the shear coupling still +exists. The results in Fig. s1 for P148 GB suggest that the transition from accelerated into uniform +migration, induced by the temperature, does also exist for other shear-coupling GBs, regardless of the +magnitude of driving force. +0 +100 +200 +300 +0 +10 +20 +30 +40 +50 +(a) +Time (ps) +Displacement (nm) + + +Lx +2Lx +8Lx +500 K +0.060 eV +β = 0.86 +0 +200 +400 +0 +10 +20 +30 +40 +50 +60 +(b) +Time (ps) + + +1000 K +0.060 eV +β = 0.22 +0 +2000 +4000 +6000 +8000 +0 +10 +20 +30 +40 +50 +60 +(c) +Time (ps) + + +1000 K +0.006 eV +β = 0.22 + +Fig. s1 Variation of GB displacement with the vertical size for P148 GB, simulated under +distinct temperatures and driving forces. Concerning simulation settings were the same as +Fig. 1 in the main text. + + +2 +2. Supporting results for the effective alleviation of acceleration in migration +When adopting one special kind of boundary condition for GBs exhibiting acceleration, i.e., +setting the bottom surface as free while the top one as fully fixed, the acceleration will only be +significant at the early migration stage and gradually turn into uniform migration at the later stage. In +such case, both the acceleration in migration and size effect can be effectively alleviated. Nevertheless, +we only present the complete displacement and kinetic energy data for Lx system due to the limited +scope of the main text. Here Fig. s2 shows the complete data for all four distinct systems considered +in the main text to support the above conclusion. + + + + + +Fig. s2 GB displacement and shear kinetic energy data for P1 GB with different vertical sizes, +simulated under 500 K and 0.06 eV when setting the bottom surface as free while the top one +as fully fixed. + + +(a) 150 +12(b) 300 +25ZL +250- +x +20 +Lsp) +200-15 +ace10100- +(nm50-0 +80 +120 +160 +Time (ps)(c) 1200 +100 +81x +120-OL +1000- +x +80 +D1spl +800-60 +cem +600N +40007200- +200 +0 +0 +200 +400 +600 +800100 +Time (ps)(d) 3500 +2013000 +xispl2500 +1200 +spl +lac150 +eme +e +len +500N +E +100 +nm +100050 +5000 +00 +800 +1200 +1600 +2000 +Time (ps)90 +ace +e +en60.(nm300 +20 +60 +80 +Time (ps) \ No newline at end of file diff --git a/59AyT4oBgHgl3EQfpfhu/content/tmp_files/load_file.txt b/59AyT4oBgHgl3EQfpfhu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb67849a28fedc6bf5f7f733cf239f465d969a48 --- /dev/null +++ b/59AyT4oBgHgl3EQfpfhu/content/tmp_files/load_file.txt @@ -0,0 +1,1001 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf,len=1000 +page_content='1 Unusual acceleration and size effects in grain boundary migration with shear coupling Liang Yanga, Xinyuan Songb, Tingting Yuc, Dahai Liua*, Chuang Dengb,* a School of Aeronautical Manufacturing Engineering, Nanchang Hangkong University, Nanchang 330063, China b Department of Mechanical Engineering, University of Manitoba, Winnipeg, MB R3T 2N2, Canada c School of Aviation and Mechanical Engineering, Changzhou Institute of Technology, Changzhou, Jiangsu 213032, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Corresponding author: dhliu@nchu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='cn(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Liu), Chuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='Deng@umanitoba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='ca (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Deng) Graphical abstract acceleration and size effect mechanism 50 80 30 Lx- Ly Lz 25 2Lx Lx- 2 0L, 2 0Lz 40 20 2L-3L,-3Lz 60 8Lx 20 aDisplacement (n) 30 (eV) 40 20 20 1010 Original 10 Fitted 20 15 :2 0 4 6 10 12 0 200 400 0 50 100 Time (ps) Time (ps) Displacement (nm)150 12 140 topfullyfixed 第0ps 120 120- nm) 100e 90 bottomfree 80 er E ien 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 0g 2L 40 30 20x 0 0 0 20 40 60 80 0 200 400 600 800 Time (ps) Time (ps) attempt to alleviate acceleration and size effect 2 Abstract Grain boundary (GB) migration is widely believed to maintain a linear relation between its displacement and time under a constant driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In this study, we investigated the migration behaviors of a set of GBs in Ni by applying the synthetic driving force and shear stress via atomistic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It was found that the displacements of some shear-coupling GBs do not exhibit a linear or approximately linear relation with the time, as widely assumed, but evidently exhibit an acceleration tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, the boundary velocity significantly decreases when increasing the bicrystal size perpendicular to the GB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These behaviors were verified to be independent of the magnitude and type of driving force but closely related to the temperature and revealed to be unique to shear-coupling GBs exhibiting a rise in the kinetic energy component along the shear direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, after many attempts, we found that the acceleration in migration and size effect can be largely alleviated by adopting one specific kind of boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, the continuous rise of kinetic energy still exists and leads to the true driving force for GB migration lower than the nominally applied value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For that reason, a technique is proposed to extract the true driving force based on a quantitative analysis of the work-energy relation in the bicrystal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Accordingly, the calculated true mobility shows that the recently proposed mobility tensor may not be symmetric at relatively large driving forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Keywords: grain boundary migration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' shear-coupling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' size effect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' mobility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' atomistic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Introduction Grain boundary (GB) migration is crucial to a variety of behaviors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', grain growth, recrystallization, and plastic deformation) and mechanical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', strength and ductility) in polycrystalline materials [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Up to now abundant and deep insights have been gained into the characters and underlying mechanisms of GB migration based on theoretical and experimental investigations [2-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Thereinto, dramatic attentions were paid to GB mobility M, which can be defined as the coefficient relating to migration velocity v and driving force P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', M = v/P) and is considered an intrinsic GB property (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', only depending on material parameters, temperature, and boundary 3 crystallography) [10-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, computational studies [13-15] have revealed that the magnitude of driving force can leave significant influences on mobility, owing to the force-induced variation of boundary structure and/or migration mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For example, Deng and Schuh [13] found that for both symmetrical and inclined Ni Σ5 \uf0e1100\uf0f1 tilt GBs, their mobilities agree well with the intrinsic values obtained by the thermal fluctuation method [16] only when the applied driving force is sufficiently low;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' increasing the driving force will lead to diffusive-to-ballistic transition in the migration mechanism and enlarge the discrepancy between the extracted and intrinsic mobility values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, for shear-coupling migration GBs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', simultaneous translation in GB plane during the migration along the boundary normal direction), Han and coworkers [7,17,18] demonstrated that both GB mobility and shear-coupling factor (ratio of GB sliding and migration rates) do not only strongly depend on the magnitude but also the source of driving force (stress or a jump in chemical potential across the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' They further revealed that the mobility traditionally defined as a scalar should be a symmetrical second-rank tensor [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The tensor components can be extracted by applying driving forces in the directions perpendicular and tangent to the boundary plane, respectively [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In addition to the driving force, GB motion may also strongly depend on the size of simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [19] reported that the mobility of a Ni \uf0e1100\uf0f1 tilt GB decreased monotonically with decreasing the cell thickness (the size along the tilt axis), due to the interference between the free surface and the collective rearrangement of atoms during boundary motion driven by an external stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' A similar size-dependency of mobility was also observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Race et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [21] revealed that the boundary area of a flat \uf0e1111\uf0f1 tilt GB should reach the meso-scale or a large-enough value to yield a converged migration velocity under the synthetic driving force (SDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Meanwhile, simulations for stress-driven [22,23] and SDF-driven [24] migration further discovered that the energy barrier for disconnection nucleation or the driving force for GB migration would converge when the boundary area was large enough for shear-coupling GBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The energy barrier was also found to firstly decrease and then keep steady with the increase of cell size in GB normal direction for 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='1º Σ5 \uf0e1100\uf0f1 tilt GB [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Existing studies concerning the size effect on GB migration overall reached a consensus that the 4 system size should be large enough to yield physically reliable results and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This agrees well with the general understanding related to the size effect in modeling and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Although GB migration has been reported to suffer influences from various factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', crystallography [11,25], temperature [13,26], driving force [10,14], pressure [27,28] and impurity [29,30]), the mobility values extracted from M = v/P in these studies were all based on a basic premise that the boundary velocity will maintain constant or approximately constant during the whole migration process under a fixed driving force, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the boundary displacement exhibiting a linear or approximately linear relation with the migration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This character has been widely observed in existing research concerning GB migration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [3,31-34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, in this study, we found that velocities of some GBs did not keep constant during migration but exhibited an unusual acceleration feature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the velocity varying significantly with GB relative position in the simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This signifies a strong dependency of migration on the cell size in the direction perpendicular to GB plane, which has not yet been discovered before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The first effort of the present work was therefore to investigate the underlying mechanisms for the acceleration in GB migration and effects of model size in GB normal direction on migration, based on atomistic simulations of several GBs driven by the external stress and SDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' After the corresponding conditions and mechanisms for these two phenomena were clarified, attentions were paid to effectively alleviate the size effect and to extract the true driving force and mobility in the presence of acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Methodology In this study, the acceleration in GB migration, size effect and other related contents were investigated based on atomistic simulations of several GBs in Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' First, we simulated the migration of Ni Σ5 \uf0e1100\uf0f1{310}, Σ29 \uf0e1100\uf0f1 {10 4 0} and Σ55 \uf0e1211\uf0f1 {952} symmetrical tilt GBs, which correspond to P1, P148 and P233 GBs in the 388 GBs dataset constructed by Olmsted et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [11], to reveal the phenomena of acceleration and dependency of migration on the cell size in the direction perpendicular to GB plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These simulations were performed at 500 K and an SDF of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV when increasing the normal cell size for each GB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Additional simulations were carried out at lower SDF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='025 eV for 5 P1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='003–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='006 eV for P233) and under the external shear stress τext (τext = 250 MPa for P1 and τext = 50 MPa for P233) to test whether the above phenomena are affected by the magnitude and type of driving force, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Second, we simulated the migration behaviors for Ni Σ3 \uf0e1110\uf0f1 twist GB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', P5) at 500 – 1000 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='006 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV, Σ21 \uf0e1210\uf0f1 general GB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', P81) at 1000 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='001 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV, P1 at 1000 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='003 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These simulations were aimed to explore the physical causes for the acceleration and size-dependency from the aspects of shear coupling and work-energy relation in the bicrystal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Third, we chose the P1 GB as the representative example to attempt approaches for inhibiting or alleviating the size effect by performing simulations adopting various boundary conditions (BCs) and/or manipulating the internal stress in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The corresponding results were compared with those reported in existing studies whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Finally, we still chose P1 as the example to discuss how to extract the true driving force for GBs exhibiting acceleration when the true driving force was not equal to the value nominally applied through the SDF method or the external shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' All simulations stated above were performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software package [35] with the embedded atom method potential developed for Ni [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(a), a bicrystal simulation cell was used to construct a flat GB, which was in y-z plane with y and z directions being periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, the boundary conditions in x direction (parallel to the boundary normal direction) might be quite different, depending on the simulation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For the above first and second groups of simulations, in order to avoid translation of the whole bicrystal in GB normal direction, a slab of atoms (1nm thickness) near the bottom surface were partially fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', only the velocity and force components along the x direction being set as zero) while the top surface was set free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For the third group of simulations, the boundary conditions in x direction might be periodic, fully fixed, free, or one surface free while the other fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' When exploring the size effect on boundary migration, the cell size in the GB normal direction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the grain size) or in the boundary plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the boundary area) was varied accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The minimum cell size for each above GB was the same as those constructed by Olmsted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For example, the minimum cell size for P1 was Lx = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='3 nm, Ly = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2 nm, and Lz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='3 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6 When the bicrystal system of each GB was constructed, its energy was minimized at 0 K following the scheme introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Subsequently, the system was elevated to and sufficiently relaxed at test temperatures (500 K or 1000 K) under the isothermal-isobaric ensemble (NPT) for about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='15–7 ns (depending on the temperature, GB, and cell size) with a default time step of 5 fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' After the system was fully equilibrated, the boundary was driven to migrate under a jump in chemical potential across the boundary or a shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Thereinto, the former was realized by the CROP-SDF method [14] while the latter by applying a shear force to individual atoms near the bottom surface (1 nm thickness in x direction) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The GB displacement under the SDF and stress was computed by tracking the overall change of the potential energy artificially added to the bicrystal system and by tracking atoms with the centro-symmetry parameters close to the maximum value in the system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For the above four groups of simulations, the NPT ensemble was also used during the boundary migration for each case, if not otherwise specified, to control the internal normal stress components as close to 0 GPa as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For some specific simulations, the internal shear stress along the shear direction also needed to be controlled at 0 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The structure of bicrystal model, if needed, was visualized by Ovito package [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Note that some results concerning this study were presented in the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Results and discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='1 Acceleration in migration and size effect Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(b-e) represents the migration data of Σ5 \uf0e1100\uf0f1 {310} tilt GB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', P1 GB) when increasing the cell size along different directions while under a constant external driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Two features can be readily observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' First, the boundary velocities (slope of displacement curve) under various cell sizes all gradually increase with the proceeding of boundary migration, suggesting a clear dependency of migration velocity on the relative GB position along the boundary normal direction in the simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In contrast, the boundary displacement was widely observed and commonly assumed to exhibit a linear or approximately linear relation with the migration time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [3,31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' After a detailed survey of existing studies, the acceleration in migration was only found in the work by Coleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [38], who simulated the migration of Ni Σ37 \uf0e1100\uf0f1 symmetrical tilt by applying the 7 synthetic driving force and shear strain at 300 and 400 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Unfortunately, the focus in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [38] was the atomic mechanisms of migration and no attention was paid to the acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 (a) Schematic of the bicrystal simulation cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' GB displacement vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' time for P1 GB simulated at 500 K and under a synthetic force of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV, when increasing the cell size along (b) x, (c) y, (d) z and (e) all three directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' To avoid translation of the whole bicrystal along the x direction, the bottom surface was partially fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', setting the velocity and force components along x for atoms near the surface as zero) while the top surface was set free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Second, the velocity is independent of the cell size in GB plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the boundary area or lateral cell size) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1c-e) but strongly and negatively related to the size in the boundary normal direction (x direction in this study) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1b and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, the velocity of flat boundary has been previously reported to show a strong and complex dependence on the boundary area [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In addition to the velocity, the boundary area has also been reported to cause significant influence on GB mobility [19,20] and the energy barrier of disconnection nucleation for GB migration [22-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, these properties concerning GB migration exhibited a consistency in their dependency on the boundary area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the boundary area should be sufficiently large to yield a converged property value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The negative dependency of velocity on the cell size along the boundary normal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', vertical cell size) here is partially similar to the trend regarding the threshold driving force of disconnection nucleation for Cu Σ5 \uf0e1100\uf0f1 {210} tilt GB (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', P6 GB in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [11]) at 10 K revealed by Deng and Deng [24], who found the threshold driving force, which in practice can be qualitatively regarded as the reverse of GB mobility [9], overall declines when increasing the vertical cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, the size effects regarding the boundary area are different between the present and existing studies, but a 8 similarity appears in the dependency on the vertical cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As a typical of low-period and high-angle CSL boundary, the migration behaviors of P1 GB have been widely studied through atomistic simulations [6,10,11,13,14,17,34,39-41], but why the above two features were not reported before?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This can be attributed to multiple factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' First of all, there has been no research adopting various vertical cell sizes for this GB up to now, and accordingly no insight into the size effect was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In studies adopting fixed vertical size [10,11,14,17,34,39], the acceleration might also exist, though the displacement-time data was not directly provided in these studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, we deem that the acceleration might have been disregarded on the grounds that the main attentions and efforts were focusing on exploring the intended objectives of individual studies, as in our previous work [14,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Meanwhile, the disappearance of acceleration can also be attributed to the relatively high temperatures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 1000, 1200 and 1400 K) tested in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [10,11,39] (see discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In addition, the periodic boundary condition imposed along the boundary normal direction will prevent the presence of acceleration in work [13,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This boundary condition has been confirmed to inhibit the shear-coupling migration [21,41], which is a necessary but not a sufficient condition for acceleration migration (see following discussion concerning Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' When exploring the shear coupling migration of Cu P1 GB, Cahn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [6] directly presented a displacement-time data up to 1 nm, simulated by applying a constant shear strain 1 m/s at 800 K (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' To our understanding, 1 nm data may not be sufficiently long to evidently illustrate the acceleration feature, in comparison with the displacement data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Another displacement-time data (up to 6 nm) was provided by Schartt and Mohles [41], who simulated the migration of Ni P1 under 300–1000 K with free end boundary conditions and a synthetic force of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV imposed through the ECO-SDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, the acceleration feature was still not observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [41] though it shows up in our re-tests of simulations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 by using the ECO-SDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, the discrepancy concerning the acceleration should not be attributed to the different versions of SDF method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', CROP-SDF [14] or ECO-SDF [41]) utilized in the present study and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Since the temperature dependency of migration velocity and shear coupling factor \uf062 in the range of 300-700 K in [41] also appears different from those previously reported [6,9,13,17], we deem that the discrepancy may be resulted from the difference in the metastable 9 structures for P1 GB adopted for various studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' To evaluate whether the acceleration in migration or the corresponding size effect is unique to P1 GB or not, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2 show the results simulated for some other GBs or under simulation settings different from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2(a) and (b), the two features can also be observed for P148 and P233 GBs when adopting the same settings as for P1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Since the driving force applied in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1, 2(a) and (b) is a relatively high value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='87 GPa) in comparison with experimentally applied values, we tested lower forces for P1 and P233 GBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It can be seen from in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2(c) and (d) that these features still hold on for P1 GB at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='025 eV (lower than KT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='043 eV, K Boltzmann constant and T temperature) and for P233 GB at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='003 eV which approaches typical experimental values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Note that lower forces have also been tried for P1 but failed to yield continuous boundary migration, agreeing with the threshold driving force of boundary migration determined for this GB at 500 K by Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It is important to note from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2(e) and (f) that the acceleration and size effect still show up for P1 and P233 GBs when applying the external shear stress to drive the GB migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, while the shear coupling mode may be strongly influenced by both the magnitude and type of the driving force [17], the acceleration and size effect does not exhibit such dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The above analysis suggests that the acceleration in migration and negative dependency on the vertical cell size are relatively common features for force-driven GB migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The following content will further reveal that the latter feature is resulted from the former one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These two features extend our current understandings of size-effect on GB migration, which almost all focused on the size in the boundary plane [19-23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' They also remind us that attentions should be taken for GBs exhibiting such features when extracting the boundary velocity or mobility under a constant driving force, during which the migration displacement and time were almost always assumed to keep a linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2 Other results simulated at 500 K supporting the acceleration in migration and size effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Displacement data in (a-d) were all simulated under the applied synthetic force while (e, f) under the external shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The shear stress was applied to a slab of atoms (1 nm thickness) near the top surface, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1, while a slab of atoms near the bottom surface was set as a grid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2 Underlying mechanism for acceleration and size effects The acceleration in boundary migration and vertical size effect have been revealed in the above section, then for what types of GB or under what kinds of condition that such phenomenon will occur?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' A preliminary analysis of the three GBs tested in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 and 2 indicates that they are all shear-coupling migration GBs and with \uf062 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(a) chooses P5 (Ni Σ3 \uf0e1110\uf0f1 twist) GB as an example to show the displacement-time (S-t) data and size-dependency of GBs without shear-coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It can be seen that the displacement is linearly related to the migration time and the corresponding velocities are the same under different vertical sizes, irrespective of temperature and magnitude of driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These results seemingly indicate that only shear-coupling GBs will exhibit acceleration 11 migration and negative size-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(b) for P81 GB, the linear S-t relation and constant v under different sizes can be observed also for shear-coupling GBs at various temperatures and driving forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Furthermore, the acceleration migration and size effect observed at 500 K for P1 GB (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1) unexpectedly transfers into uniform migration when raising the temperature to 1000 K at which the shear coupling still exists, regardless of the magnitude of driving force (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This kind of transition induced by the temperature also occurs for other shear-coupling GBs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' s1 in the Supplementary file).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' To our understanding, the transition can be attributed to the temperature-induced variation of disconnections mediated for GB migration, which has been widely observed [7,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Based on these analyses, we conclude that the shear coupling is a necessary but not a sufficient condition for the acceleration in migration and therefore the size effect, which may suffer strong influence from the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3 Examples of uniform migration for GBs with or without shear-coupling: (a) P5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (b) P81;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (c) P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (d) and (e) presents kinetic energy \uf044E for P1 with cell size Lx, simulated at 500 K and 1000 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' To further explore the fundamental mechanisms for acceleration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(d) compares the relative 12 variation of kinetic energy (\uf044Ei, i = x, y or z) to the initial state for P1 GB at 500 and 1000 K, at which the boundary exhibits accelerated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(b)) and uniform (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(c)) migration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' At 1000 K, all three components of the kinetic energy remain almost unchanged (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', \uf044Ex = \uf044Ey = \uf044Ez ≈ 0) with the proceeding of migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In contrast, at 500 K, although \uf044Ex and \uf044Ey still remain unchanged, \uf044Ez firstly increases and then decreases (the final \uf044Ez is still much higher than zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Note that the shear movement is parallel to z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The comparison suggests that the work (Wext) done by the external driving force (Pext) only contributes to shear-coupling migration at 1000 K, but to both shear-coupling migration and a rise in the shear kinetic energy (\uf044Ez) at 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This difference reminds us that the accelerated and uniform migration can be qualitatively justified from the aspect of true driving force (Ptrue) for boundary migration, which can be influenced by Wext and \uf044Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' During the process of boundary migration, the work and kinetic energy for the bicrystal system meet the relation of Wext = Wtrue + \uf044E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Wtrue is the work done by Ptrue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Wtrue = PtrueꞏSꞏAGB, S and AGB stand for the GB displacement and area, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Considering \uf044Ex = \uf044Ey ≈ 0 in the case of both accelerated and uniform migration, the relation can be given as Wext = Wtrue + \uf044Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' At one specific moment of migration, the work-energy relation can be further described as dPextꞏdS = dPtrueꞏdS + d(\uf044Ez)/AGB, and the instant true driving force is dPtrue = dPext – (d(\uf044Ez)/dS)/AGB, where dPext is a fixed value for the SDF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, dPtrue will be constant when \uf044Ez keeps unchanged (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 1000 K at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(d)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', dPtrue = dPext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In such case, the boundary will accordingly exhibit uniform migration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', a linear S-t relation) and thus consistent velocities when adopting distinct vertical sizes but the same Pext (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, in the case of accelerated migration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', \uf044Ez ≠ 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(d)), dPtrue depends on both dPext and d(\uf044Ez)/dS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' From the d(\uf044Ez)/dS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' S curve (the red curve) at 500 K for P1 GB shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 4, we can observe that d(\uf044Ez)/dS continuously descends with the boundary migration, suggesting a continuous rise in dPtrue and thus in migration velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, it is conceivable that when applying the same dPext to bicrystal systems with different vertical sizes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(b)), dPtrue will be lower (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', lower velocities) for larger systems due to higher d(\uf044Ez), which can be further attributed to more atoms involving shear movement for larger systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' When applying this interpretation to justify the size effect for systems with different sizes in the boundary plane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(c)), the contribution of GB area to dPtrue must also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In summary, the above analysis suggests that the acceleration in migration and negative dependency of velocity on the vertical size are unique to shear-coupling GBs exhibiting a rise in the kinetic energy component along the shear direction and can be justified from the aspect of true driving force based on the work-energy relation in the bicrystal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 80 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 4 Variation of d(\uf044Ez)/dS with the boundary displacement for P1 GB, calculated based on the \uf044Ez-S curve (blue curve) obtained by the least-square fitting of the original data at 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV, given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='3 Attempts to alleviate acceleration Although the acceleration in migration has been demonstrated as a relatively common feature for force-driven migration of flat GBs in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='1, it is undesired if the purpose is to compute a GB mobility by assuming v = MP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Then, is it possible to inhibit or alleviate the acceleration?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' For this purpose, we have carried out a series of simulations by manipulating the boundary conditions and internal stress in each bicrystal system (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Firstly, we tried to adopt periodic boundary condition in the GB normal direction (x direction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(a)), which is one kind of boundary conditions widely used in previous studies [13,31,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5(a) that the boundary displacements under various vertical sizes all nearly exhibit a linear relation with the time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' the velocity nevertheless keeps increasing when enlarging the vertical size, in contrast to a negative size dependency of velocity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Meanwhile, in comparison with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='025 eV applied for the boundary condition adopted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 and 2c, much larger driving force (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='15eV) must be applied to initiate the boundary movement for all cell sizes, suggesting a 14 significant effect of boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Under the present boundary condition, the internal shear stress sharply increases with the initiation of GB migration, and then experiences a short descending and finally fluctuates around a very high value (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Furthermore, the shear stress is obviously lower under larger cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The higher velocity under larger size in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5(a) is therefore resulted from this tendency of shear stress, which can be attributed to more elastic energy released with the boundary migration due to larger space along the GB normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5(c), the periodic surface strongly inhibits the overall relative shear movement between two grains, as revealed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [13,41]), but only enables local shear movement which is more evident for larger cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This precisely accounts for the linear S-t relation under various sizes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5(a), according to the discussion concerning mechanism for acceleration in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Evidently, the periodic boundary can effectively eliminate the acceleration but not the size effect by significantly constraining the shear movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5 Migration results vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' vertical size for P1 GB, simulated at 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='15 eV by adopting periodic boundary conditions along the GB normal direction: (a) displacement data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (b) shear stress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (c) snapshot of boundary migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Red and white arrows in (c) illustrate the migration and shear directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Atoms are colored by atom type to visualize the shear-coupling migration using the Ovito software [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Secondly, we tried to set the top and bottom surfaces to be fully fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6 that most of the results are similar to those under periodic boundary in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', much larger driving force, linear S-t and very high shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Additionally, displacements are nearly independent of the system size (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6(a)), though the normal stress continues to rise with the boundary movement (see the example shown for cell size of Lx in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The corresponding velocity is much lower than that at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 while close to that at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='15 eV and Lx in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In consistency with the Initialnitia] 2L 8L 20LC GBGB 15 periodic boundary, the boundary condition of fixed ends also inhibits the global shear movement and enables only local shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6(c) shows that the local shear-coupling mode may change even switch under fixed ends, as already observed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 0 50 100 150 200 250 0 5 10 15 20 Time (ps) Displacement (nm) Lx 2Lx 8Lx 20Lx 500 K, 0 15 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 25 (a) Lx Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 6 Migration results for P1 GB simulated at 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='15 eV while setting the top and bottom surfaces to be fully fixed: (a) displacement data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (b) shear stress and normal stress under cell size Lx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (c) snapshot of boundary migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The red and yellow atoms colored in (c) are aimed to visualize the local shear movement and shear-switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Thirdly, considering the impeding effect of the high internal shear stress on shear movement, we performed simulations that controlling the stress as close to 0 GPa as possible under periodic boundary (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Although shear stresses are well controlled especially for larger systems (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 7(b)), acceleration and size effect still exist (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 7(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, the boundary stagnates long before reaching the other end, and the final displacement value under each size is nearly half of the feasibly maximum value (compare Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 5(a) and 7(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The inclined line displayed by the yellow atoms in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 7(c) illustrates that the shear movement is inhomogeneous along the GB normal direction, and the top and bottom parts of the blue grain make shear along two opposite directions (see the white arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These results should be resulted from the cell inclination when controlling shear stress (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 7(c)), which leads to the variation of crystallographic orientation and thus erroneous exertion of the orientation-dependent driving force for shear-coupling migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' We have also tried to control shear stress for fixed and free boundary conditions but as well obtained cell inclination and other results similar to those by using periodic boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Evidently, the above three attempts all failed to achieve our anticipated objectives of effectively alleviating the acceleration and size effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' CGB 200ps: 150ps 150psInitiaInitial 27 8L 201 16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 7 Migration results vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' vertical cell size for P1 GB, simulated at 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV by adopting periodic boundary while controlling shear stress close to 0 GPa: (a) displacement data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (b) shear stress;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (c) snapshot of boundary migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Finally, considering the inhibition of overall shear movement by periodic and fixed boundaries, we performed simulations adopting two free boundaries or setting one boundary as free while the other as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(a) that the displacement data under two free boundaries and one free while another partially fixed are consistent and exhibiting an acceleration tendency (as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1(b-e)), because the two grains across the GB are free to shear under these two boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' When the bottom surface is fully fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the grain near this surface is not allowed to shear), significant acceleration is observed through the whole migration process (the blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, in the case of fully fixed top surface, acceleration is only significant at the early migration stage and gradually turns into uniform migration at the later stage (see the green curve and black dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(a)), suggesting a gradual weakening of acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This tendency can be justified from the similarity between the variations of \uf044Ez–t and S-t curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b) and from the relation dPtrue = dPext – (d(\uf044Ez)/dS)/AGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, as illustrated by the inset snapshots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b), the overall shear and coupling factor are not influenced by this boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(c) presents the comparison of the linear S-t segments extracted from the complete displacement data under various sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Interestingly, velocities are consistent for different cell sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, the size effect for shear-coupling GB can be considered as being eliminated if only focusing on the uniform migration stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' With these attempts, we may conclude that setting the top surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', in the forward direction of GB migration) of the bicrystal as fully fixed while the bottom surface (the backward direction of migration) as free is a relatively effective way to largely alleviate 0 ps 5 ps 15 ps 25 ps 17 acceleration migration and thus size-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 0 200 400 600 800 0 20 40 60 80 100 120 140 Time (ps) Displacement (nm) Lx 2Lx 8Lx 20Lx (c) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8 Attempts to alleviate acceleration by adopting one or two free boundaries for P1 GB with normal size Lx, simulated at 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (a) Comparison of displacement data under different boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (b) Variation of displacement, kinetic energy and snapshot with the time when setting the bottom surface as free while the top one as fully fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (c) Comparison of the linear S-t segments extracted from the complete displacement data under various cell sizes, simulated by adopting the same boundary condition as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' When the surface is partially (as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1) or fully (setting all force and velocity components for atoms near the surface as zero) fixed, the grain near this surface can or can not make overall shear movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The complete displacement data for (c) can be found in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' s2 in the Supplementary file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='4 Extraction of true driving force and mobility Although the accelerated migration and size effect for shear-coupling GBs can be effectively weaken by adopting one special boundary condition, the kinetic energy of the system Ez still continues to rise during the boundary migration (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b)), and thus the true driving force Ptrue does not equal to the externally applied value Pext and depends on the variation of Ez (see discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, efforts should be paid to extract Ptrue and the corresponding true mobility Mtrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As already discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2, Ptrue can be determined based on a quantitative analysis of the (b) 150 12 top fullv fixedps 120Isp 58 ps90 acen bottom freee 60nm300 20 00 80 Time (ps) 18 work-energy relation in the bicrystal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Meanwhile, considering the continuous rise of \uf044Ez with boundary migration, we may not obtain a constant but a time dependent Ptrue(t), and therefore the quantitative analysis should be carried out for individual steps of GB migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The artificial energy added to the system by the SDF method in time interval dt can be written as: dE = \uf044eꞏdn (1) where \uf044e denotes the maximum potential energy added to a single atom (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1) and dn indicates the number of atoms whose corresponding crystallography changed as the boundary migrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' dn can be calculated by dn = (N/Lx)ꞏv(t)ꞏdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Here, N and v(t) represent the total number of atoms in the system and the instant migration velocity perpendicular to GB plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=" However, due to the continuous rise of Ez, the actual energy to drive boundary movement is dE' = dE – dEz = dE – czꞏdt (2) where cz is variation rate of dEz with respect to time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=" We can thus deduce the true maximum energy imposed on per atom \uf044e'(t) as: \uf044e'(t) = (dE – dEz)/dn = \uf044e – cz/(vꞏN/Lx) (3) For the SDF method, \uf044e is normally considered as Pext [11,13,41] and thus \uf044e'(t) can also be treated as Ptrue(t), which then can be further used to calculate Mtrue." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' From the internal stress data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 9(a), one can observe that the normal stress fluctuates around 0 GPa while the internal shear stress τxz roughly keeps a positive value and gradually declines when adopting the boundary condition of a fully fixed top surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This means that the normal stress leaves no influence on GB migration, but one part of dE may be used to overcome the impeding effect of τxz on migration, which can be quantitatively described in the form of shear strain energy Ess = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='5Vꞏ 2 xz \uf074 /G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' V and G stand for the volume and shear modulus for bicrystal system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' However, Ess is essentially a type of elastic energy and will be dynamically stored and released with the continuous migration of the GB, as supported by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 9(b) which indicates that this energy increment dEss also fluctuates around 0 eV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the long-time average of dEss equaling zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, Ess should make no contribution to the overall work-energy relation in the system and the potential influence of τxz on GB migration does not need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Accordingly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (3) still 19 holds for extracting Ptrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 9 Variation of (a) the internal stress and (b) dEss with the migration time for P1 GB simulated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b) Table 1 presents the calculated Ptrue and Mtrue for P1 GB at 500 K according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In contrast to the early migration stage, the velocity, driving force and mobility at the later stage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', t > 50 ps for Lx and t > 400 ps for 8Lx) all only rise slightly and are nearly consistent under Lx and 8Lx systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' If we calculate the average value for v, Ptrue and Mtrue at the later stage, we can get the average v, Ptrue and Mtrue for Lx system as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='163 nm/ps, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='0486 eV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='341nm/(ps eV) while v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='162 nm/ps, Ptrue = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='0485 eV and Mtrue = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='353 nm/(ps eV) for 8Lx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The relative differences of these three data between the two systems are all lower than 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These results and comparisons again emphasize that the acceleration and size-effect in the GB migration velocity have been nearly eliminated through applying one special boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' More importantly, they also signify that we can obtain consistent true mobility values for systems with distinct vertical sizes if further considering the correction of true driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It should be noted that the above principle of correcting the driving force should also be applicable to the shear-coupled migration driven by an external shear stress τext (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', applicable to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2(e) and (f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(a) for τext-driven migration, \uf044Ex and \uf044Ey remain unchanged while \uf044Ez overall increases linearly with time, in consistency with the tendency shown for the SDF-driven migration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The corresponding correction equation of Ptrue can be given as: z true ext z GB c P v A \uf074 \uf03d \uf02d \uf0d7 (4) where vz stands for the shear velocity of GB along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 20 Table 1 True driving forces and mobility values extracted based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (3) for P1 GB, simulated at 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV when setting the bottom surface as free while the top one as fully fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Both cz and v were obtained by the least-square fitting into discrete \uf044Ek and S data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Lx 8Lx t (ps) v (nm ps‒1) Ptrue (eV) Mtrue (nm ps‒1 eV‒1) t (ps) v (nm ps‒1) Ptrue (eV) Mtrue (nm ps‒1 eV‒1) 5* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='0016 -18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='906 40* 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='0487 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='423 Extracted Ptrue and Mtrue are erroneous due to the low velocity and kinetic energy at the very early stage of migration To compute the true mobility under an external shear stress, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10 still chooses the P1 GB as an example and the shear stress has been carefully tuned so that the GB migrated at the same velocity as that under the SDF, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Firstly, in comparison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2(e), the acceleration in migration is only significant at the early migration stage and turns into uniform migration at the later stage for the current simulation (see blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Secondly, the true mobility calculated based on the corrected Ptrue according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (4) experiences a continuous rise and then becomes nearly stable (see the blue square data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These results do reveal that the accelerated migration under external shear stress can also be largely alleviated by the boundary condition as utilized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b) for the SDF-driven migration, and that the principle of correcting Ptrue is also applicable to the case of τext-driven migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' What is more, although the displacements or velocities are nearly the same under the SDF and shear stress, the corresponding mobility values under the two types of driving force are significantly 21 different (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Mtrue by SDF is on average 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='6% lower than that by τext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' According to the theory recently proposed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [18], the mobility values extracted by applying the driving forces perpendicular and parallel to GB plane can be unified in a mobility tensor as following: y xx xy xz x y x yy yz y z zx zy zz z M M M v v M M M v M M M \uf06a \uf074 \uf074 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf03d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 \uf0e8 \uf0f8 (5) Here x is perpendicular to GB plane while y and z are parallel to GB plane, φ is the driving force applied along GB normal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', the synthetic driving force [14,41]), and τy and τz are shear stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Moreover, the GB mobility tensor should be symmetric according to the Onsager relation [18], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Mxz = Mzx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10 Migration results for P1 GB at 500 K when driven under τext : (a) displacement along GB normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (b) true mobility calculated based on the corrected Ptrue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The inset in (a) presents the variation of kinetic energy \uf044Ei (i = x, y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' To provide a better comparison of true mobility values between SDF-driven and τext-driven migration, the magnitude of τext was chosen deliberately to ensure the corresponding displacement data as close to that by SDF in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 8(b) as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The shear stress was applied by adding a shear force 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='0037 eV/Å to a slab of atoms (1 nm thickness) near the bottom surface along z direction, while the top surface was set as fully fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(a), \uf062 equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='0 under both SDF and shear stress, indicating that vx = vz holds under both types of driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In such case, we can deduce Mxx = Mzx for SDF and Mzz = Mxz for shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, the mobility data presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(b) reveals Mzz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='6Mxx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', Mxz ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='6Mzx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, the symmetry of mobility tensor does not hold in the present study, seemingly contradicting to the conclusion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Recently, we have systematically investigated the GB mobility tensor and the effects of temperature and external driving force on its symmetry based on 22 atomistic simulations of force-driven and force-free migration for the twist Ni Σ15 (2 1 1) (P14) GB [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It is found that the symmetry holds at low driving force limit while fails at high driving forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Therefore, the non-equal Mxz and Mzx as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 10(b) is due to the relatively large driving forces that have been applied in the current study, which is also consistent with the previous studies that large driving forces can significantly change the underlying mechanisms for GB migration [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Conclusions In this work, we investigated the migration behaviors of several GBs in Ni by atomistic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Based on that, the acceleration in GB migration and vertical size effects were revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Then attentions were paid to reveal the concerning mechanisms for these phenomena and to effectively alleviate them through manipulating the boundary conditions and internal stress in the bicrystal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' At last, a method was proposed to extract the true driving force and true mobility, based on which the symmetry of mobility tensor was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The following conclusions can be drawn based on this study: (1) The migration displacements of some GBs driven under a constant external driving force are not linearly related to the migration time, as widely assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The corresponding velocity nevertheless gradually increases with the proceeding of boundary migration and is overall negatively related to the cell size perpendicular to GB plane, irrespective of the magnitude and type of driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' These tendencies suggest that some previously published migration results for such GBs without considering the acceleration and size effect may need to be calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (2) The acceleration and vertical size effect in migration are unique to shear-coupling GBs exhibiting a rise in the kinetic energy component along the shear direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' It is precisely the rise of kinetic energy that results in the true driving force for GB migration being lower than the nominally applied value but continuing to increase, and thus leads to the accelerated migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' With the increase of tested temperatures, the acceleration will transfer into uniform migration and the size effect will accordingly disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (3) Among the various attempts to eliminate or alleviate the acceleration and size-dependency in migration, setting the cell surface in the forward direction of GB migration as being fully fixed 23 while the surface in the backward direction as free is a relatively sample but effective way, which is applicable to both shear stress and SDF-driven migration and does not change the shear-coupling behaviors and migration mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (4) Based on a quantitative analysis of the work-energy relation in the bicrystal system, the true driving force can be determined for GBs exhibiting accelerated migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Under the coupling effects of adopting one specific kind of boundary condition and correcting the true driving force, we can obtain consistent true mobility values for such GBs with distinct vertical sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Furthermore, the calculated true mobility suggests that the symmetry of the mobility tensor may fail when the driving forces are too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Acknowledgment The authors thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' David L Olmsted for sharing the 388 Ni GB structure database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' This research was supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 52065045) and NSERC Discovery Grant (RGPIN-2019-05834), Canada, and the use of computing resources provided by Compute/Calcul Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' References: [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Gottstein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Shvindlerman, Grain Boundary Migration in metals: thermo-dynamics, kinetics, Applications, CRC press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' [2] M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='11462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 Supplemental Material: Unusual acceleration and size effects in grain boundary migration with shear coupling Liang Yanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Xinyuan Songb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Tingting Yuc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Dahai Liua*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Chuang Dengb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='* a School of Aeronautical Manufacturing Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nanchang Hangkong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nanchang 330063,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' China b Department of Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' University of Manitoba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Winnipeg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' MB R3T 2N2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Canada c School of Aviation and Mechanical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Changzhou Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Changzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Jiangsu 213032,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Corresponding author: dhliu@nchu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='cn(D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Liu), Chuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='Deng@umanitoba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='ca (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Deng) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Supporting results for the transition of acceleration migration into uniform migration In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 of the main text, we can readily observe the accelerated migration and negative dependency of velocity on the vertical cell size for P1 GB at 500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, when raising the temperature to 1000 K, both the acceleration and size effect disappear though the shear coupling still exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' s1 for P148 GB suggest that the transition from accelerated into uniform migration, induced by the temperature, does also exist for other shear-coupling GBs, regardless of the magnitude of driving force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 0 100 200 300 0 10 20 30 40 50 (a) Time (ps) Displacement (nm) Lx 2Lx 8Lx 500 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='060 eV β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='86 0 200 400 0 10 20 30 40 50 60 (b) Time (ps) 1000 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='060 eV β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='22 0 2000 4000 6000 8000 0 10 20 30 40 50 60 (c) Time (ps) 1000 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='006 eV β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' s1 Variation of GB displacement with the vertical size for P148 GB, simulated under distinct temperatures and driving forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Concerning simulation settings were the same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 1 in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Supporting results for the effective alleviation of acceleration in migration When adopting one special kind of boundary condition for GBs exhibiting acceleration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=', setting the bottom surface as free while the top one as fully fixed, the acceleration will only be significant at the early migration stage and gradually turn into uniform migration at the later stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' In such case, both the acceleration in migration and size effect can be effectively alleviated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Nevertheless, we only present the complete displacement and kinetic energy data for Lx system due to the limited scope of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Here Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' s2 shows the complete data for all four distinct systems considered in the main text to support the above conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' s2 GB displacement and shear kinetic energy data for P1 GB with different vertical sizes, simulated under 500 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content='06 eV when setting the bottom surface as free while the top one as fully fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (a) 150 12(b) 300 25ZL 250- x 20 Lsp) 200-15 ace10100- (nm50-0 80 120 160 Time (ps)(c) 1200 100 81x 120-OL 1000- x 80 D1spl 800-60 cem 600N 40007200- 200 0 0 200 400 600 800100 Time (ps)(d) 3500 2013000 xispl2500 1200 spl lac150 eme e len 500N E 100 nm 100050 5000 00 800 1200 1600 2000 Time (ps)90 ace e en60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} +page_content=' (nm300 20 60 80 Time (ps)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59AyT4oBgHgl3EQfpfhu/content/2301.00526v1.pdf'} diff --git a/69FAT4oBgHgl3EQfnx3V/content/tmp_files/2301.08631v1.pdf.txt b/69FAT4oBgHgl3EQfnx3V/content/tmp_files/2301.08631v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..826dac03583607e497aef7f8e84968e86799bc93 --- /dev/null +++ b/69FAT4oBgHgl3EQfnx3V/content/tmp_files/2301.08631v1.pdf.txt @@ -0,0 +1,830 @@ +Superconductivity in type II layered Weyl semi-metals +B. Rosenstein1 and B. Ya. +Shapiro2 +1Department of Electrohysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan, R.O.C. +2Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel. +Novel ”quasi two dimensional” typically layered (semi) metals offer a unique opportunity to +control the density and even the topology of the electronic matter. In intercalated MoTe2 type +II Weyl semi - metal the tilt of the dispersion relation cones is so large that topologically of the +Fermi surface is distinct from a more conventional type I. Superconductivity observed recently in +this compound [Zhang et al, 2D Materials 9, 045027 (2022)] demonstrated two puzzling phenomena: +the gate voltage has no impact on critical temperature, Tc, in wide range of density, while it is very +sensitive to the inter - layer distance. The phonon theory of pairing in a layered Weyl material +including the effects of Coulomb repulsion is constructed and explains the above two features in +MoTe2. +The first feature turns out to be a general one for any type II topological material, while +the second reflects properties of the intercalated materials affecting the Coulomb screening. +arXiv:2301.08631v1 [cond-mat.supr-con] 20 Jan 2023 + +2 +INTRODUCTION. +The 3D and 2D topological quantum materials, such as topological insulators and Weyl semi - metals (WSM), +attracted much interests due to their rich physics and promising prospects for application in electronic and spinotronic +devices. The band structure in the so called type I WSM like graphene[1], is characterized by appearance linear +dispersion relation (cones around several Dirac points) due to the band inversion. This is qualitatively distinct from +conventional metals, semi - metals or semiconductors, in which bands are typically parabolic. In type-II WSM [2], the +cones have such a strong tilt, κ, so that they exhibit a nearly flat band and the Fermi surface ”encircles” the Brillouin +zone, Fig.1b, Fig.1c. It is topologically distinct from conventional ”pockets”, see Fig.1a. This in turn leads to exotic +electronic properties different from both the those in both the conventional and in the type I WSM. Examples include +the collapse of the Landau level spectrum in magnetoresistance [3], and novel quantum oscillations [4]. +The type II topology of the Fermi surface was achieved in particular in transition metal dichalcogenides [5]. Very +recentlyMoTe2 layers intercalated by ionic liquid cations were studied[6]. The tilt value was estimated to as high +as κ = 1.3 that places it firmly within the type II WSM class. The measurements included the Hall effect and the +resistivity at low temperatures demonstrating appearance of superconductivity. They discovered two intriguing facts +that are currently under discussion. First changing the gate voltage (chemical potential) surprisingly has no impact +on critical temperature, Tc, in wide range of density of the electron gas. Second Tc turned out to be very sensitive to +the inter - layer distance d: it increases from 10.5A to 11.7A, while the critical temperature jumps from 4.2K to 7K. +In the present paper we propose a theoretical explanation of these observations based on appropriate generalization +of the conventional superconductivity theory applied to these materials. +Although early on unconventional mechanisms of superconductivity in WSM have been considered, accumulated +experimental evidence points towards the conventional phonon mediated one [7–9]. In the previous paper[11] and a +related work[10] a continuum theory of conventional superconductivity in WSM was developed. Magnetic response +in the superconducting state was calculated[10][12]. The model was too ”mesoscopic” to describe the type II phase +since the global topology of the Brillouin zone was beyond the scope of the continuum approach. Therefore we go +beyond the continuum model in the present paper by modeling a type II layered WSM using a tight binding approach. +The in-plane electron liquid model is similarl to that of graphene oxide[13] and other 2D WSM. It possesses a chiral +symmetry between two Brave sublattices for all values of the tilt parameter κ, but lacks hexagonal symmetry. The +second necessary additional feature is inclusion of Coulomb repulsion. +It turns out that the screened Coulomb repulsion significantly opposes the phonon mediated pairing. Consequently +a detailed RPA theory of screening in a layered material[14] is applied. We calculate the superconducting critical +temperature taking into consideration the modification of the Coulomb interaction due to the dielectric constant of +intercalator material and the inter-layered spacing d. The Gorkov equations for the two sublattices system are solved +without resorting to the mesoscopic approach. Moreover since screening of Coulomb repulsion plays a much more +profound role in quasi 2D materials the pseudo-potential simplification developed by McMillan[15] is not valid. +Rest of the paper is organized as follows. In Section II the microscopic model of the layered WSM is described. The +RPA calculation of both the intra- and inter - layer screening is presented. In Section III the Gorkov equations for the +optical phonon mediated intra- layer pairing for a multiband system including the Coulomb repulsion is derived and +solved numerically. In Section IV the phonon theory of pairing including the Coulomb repulsion for a layered material +is applied to recent extensive experiments on MoTe2. The effect of intercalation and density on superconductivity is +studied. This explains the both remarkable features of Tc observed[6] in MoTe2. The last Section contains conclusions +and discussion. +A ”GENERIC” LATTICE MODEL OF LAYERED WEYL SEMI-METALS +Intra- layer hopping +A great variety of tight binding models were used to describe Weyl (Dirac) semimetals in 2D. Historically the +first was graphene (type I, κ = 0) , in which electrons hope between the neighboring cites of the honeycomb lattice. +We restrict the discussion to systems with the minimal two cones of opposite chirality and negligible spin orbit +coupling. The two Dirac cones appear in graphene at K and K′ crystallographic points in BZ. Upon modification +(more complicated molecules like graphene oxide, stress, intercalation) the hexagonal symmetry is lost, however a +discrete chiral symmetry between two sublattices, denoted by I = A, B, ensures the WSM. The tilted type I and even +type II (for which typically κ > 1) crystals can be described by the same Hamiltonian with the tilt term added. This + +3 +FIG. 1. Two distict topologies of the Fermi surface in 2D. Topology of the 2D Brillouin zone is that of the surface of 3D +torroid. On the left the “conventional” type I pocket is shown. In the ceter and on the right the type II topology is shown +schematically. The filled states are in blue and envelop the torus. Despite the large difference in density of the two the Fermi +surface properties like density of states are the same. +2D model is extended to a layered system with inter - layer distance d. Physically the 2D WSM layers are separated +by a dielectric material with inter - layer hopping neglected, so that they are coupled electromagnetically only[14]. +The lateral atomic coordinates are still considered on the honeycomb lattice are rn = n1a1 + n2a2, where lattice +vectors are: +a1 = a +2 +� +1, +√ +3 +� +; a2 = a +2 +� +1, − +√ +3 +� +, +(1) +despite the fact that hopping energies are different for jumps between nearest neighbors. Each site has three neighbors +separated by δ1 = 1 +3 (a1 − a2) , δ2 = − 1 +3 (2a1 + a2) and δ3 = 1 +3 (a1 + 2a2), in different directions. The length of the +lattice vectors a will be taken as the length unit and we set ℏ = 1. The hopping Hamiltonian including the tilt term +is[13, 16]: +K = +√ +3 +4 +� +nl +� +γ +� +ψsA† +nl ψsB +rn+δ1,l + ψsA† +nl ψsB +rn+δ2,l + tψsA† +nl ψsB +rn+δ3,l +� ++ h.c. − κψsI† +nl ψsI +rn+a1,l − µnn,l +� +. +(2) +Here an integer l labels the layers. Operator ψsA† +nl +is the creation operators with spin s =↑, ↓, while the density +operator is defined as nnl = ψsI† +nl ψsI +nl. The chemical potential is µ, while γ is the hopping energy for two neighbors +at δ1, δ2 . Since the the system does not possesses hexagonal symmetry (only the chiral one), the third jump has the +different hopping[13] tγ. Dimensionless parameter κ determines the tilt of the Dirac cones along the a1direction[16]. +In the 2D Fourier space, ψsA +nl = N −2 +s +� +k ψsA +kl e−ik·rn, one obtains for Hamiltonian (for finite discrete reciprocal lattice +Ns × Ns): +K = N −2 +s +� +kl ψs† +klMkψs +kl. +(3) +Here k = k1 +Ns b1 + k2 +Ns b2 (reciprocal lattice vectors are given in Appendix A) and matrix Mk = dxσx + dyσy + d0I in +terms of Pauli matrices has components: +dx = 2t +√ +3 cos +� 2π +3Ns +(k1 − k2) +� ++ 4 +√ +3 cos +� π +Ns +(k1 + k2) +� +cos +� +− π +3Ns +(k1 − k2) +� +; +(4) +dy = − 2t +√ +3 sin +� 2π +3Ns +(k1 − k2) +� ++ 4 +√ +3 cos +� π +Ns +(k1 + k2) +� +sin +� π +3Ns +(k1 − k2) +� +; +d0 = +2 +√ +3 +� +−κ cos +� 2π +Ns +k1 +� +− µ +� +. + +4 +Using γ as our energy unit from now on, the free electrons part of the Matsubara action for Grassmanian fields +ψ∗sI +kln is: +Se = 1 +T +� +kln ψ∗sI +kln +�� +−iωn + d0 +k +� +δIJ + σIJ +i di +k +� +ψsJ +kln. +(5) +where ωn = πT (2n + 1) is the Matsubara frequency. The Green Function of free electrons has the matrix form +gkn = +�� +−iωn + d0 +k +� +I + σidi +k +�−1 = +� +−iωn + d0 +k +� +I − σidi +k +(iωn − d0 +k)2 − dx2 +k − dy2 +k +. +(6) +Now we turn to the spectrum of this model. +FIG. 2. The topological phase diagram of the Weyl semimetal at large tilt parameter (κ = 1.3). Chemical potential (in units +of γ = 500 meV) is marked on each contour. The electron type I topology at low values of µ undergoes transition to the type +II at µ = µ1 +c = 0.8 meV. At yet larger µ > µ2 +c = 1.35. the Fermi surface becomes again type I. This time the excitations are +hole rather than electrons. +The range of the topological type II phase at large κ +The spectrum of Hamiltonian of Eqs.(4) consists of two branches. The upper branch for µ = 0.9eV is given in Fig. +2. The lower branch for a reasonable choice of parameters appropriate to MoTe2 is significantly below the Fermi +surface and is not plotted. Blue regions represent the filled electron states. One observes a ”river” from one boundary +to the other of the Brillouin zone (in coordinates k1 and k2, in terms of the original kx, ky it is a rhomb) characteristic +to type II Fermi surface. Topologically this is akin to Fig.1b. +In Fig. 3 the Fermi surfaces in a wide range of densities n = 7.×1013−4.5×1014cm−2 are given. Topologically they +separate into three phases. At chemical potentials below µ1 +c = 0.796 eV , corresponding to densities n < n1 +c = 8.×1013 +cm−2 , the Fermi surface consists of one compact electron pocket similar to Fig.1a, so that the electronic matter +is of the (”customary”) topological type I. The density is determined from the (nearly linear) relation between the + +5 +0.5 +0.6975 +0.6975 +0.9 +0.9 +1.2 +1.2 +1.2 +1.2 +1.35 +1.35 +1.35 +1.35 +1.5 +1.5 +1.5 +1.5 +I (electron) +II +II +II +I (hole) +k1 (π/a) +k2 (π/a) +FIG. 3. Dispersion relation of WSM with κ = 1.3. The blue plane corresponds to chemical potentia lµ = 0.8 eV so that the +Fermi surface has the type II topology. +chemical potential and density given in Fig. 4 (blue line, scale on the right). In the range µ1 +c < µ < µ2 +c = 1.35eV +the Fermi surface consists of two banks of a ”river” (blue color represents filled electron states) in Fig.2 and can be +viewed topologically as in Fig.1b and Fig1c. The second critical density is n2 +c = 3.6 × 1014 cm−2. In this range the +shape of both pieces of the Fermi surface largely does not depend on the density that is proportional to the area of +the blue part of the surface. +To make this purely topological observation quantitative, we present in Fig. 4 (green line, scale on the left) the +density of states (DOS) as a function of chemical potential. One observes that it nearly constant away from the two +topological I to II transitions where it peaks. +Coulomb repulsion +The electron-electron repulsion in the layered WSM can be presented in the form, +V = e2 +2 +� +nln′l′ nnlvC +n−n′,l−l′nn′l′ = +e2 +2N 2s +� +qll′ nqln−ql′vC +q,l−l′, +(7) +where vC +n−n′,l−l′ is the ”bare” Coulomb interaction between electrons with Fourier transform vC +q,l−l′ = v2D +q e−dq|l−l′|, +v2D +q += 2πe2/qϵ. Here ϵ is the dielectric constant of the intercalator material +The long range Coulomb interaction is effectively taken into account using the RPA approximation. +SCREENING IN LAYERED WSM. +The screening in the layered system can be conveniently partitioned into the screening within each layer described +by the polarization function Πqn and electrostatic coupling to carriers in other layers. We start with the former. + +6 +I +II +I +0.6 +0.8 +1.0 +1.2 +1.4 +0 +1 +2 +3 +4 +chemical potential (ev) +DOS(2 1014cm-2 ev-1) +density (1014cm-2) +FIG. 4. Electron density and DOS as function of the chemical potential µ.of WSM with κ = 1.3. DOS has cusps at both I to +II transitions. Between the transitions it is nearly constant in the range of densities from 1.1 × 1014/cm2 to 4. × 1014/cm2. +Polarization function of the electron gas in Layered WSM +In a simple Fermi theory of the electron gas in normal state with Coulomb interaction between the electrons in +RPA approximation the Matsubara polarization is calculated as a simple minus ”fish” diagram [14] in the form: +Πqn = − +� +−2TTr +� +pm gpmgp+q,m+n +� +. +(8) +Using the GF of Eq.(6), one obtain: +Πqn = 4T +� +pm +(iωm + A) (iωm + B) + C +� +(iωm + A)2 − α2 +� � +(iωm + B)2 − β2 +�, +(9) +where +A = −d0 +p; B = iωn − d0 +p+q; +C = dx +pdx +p+q + dy +pdy +p+q; +(10) +α2 = dx2 +p + dy2 +p ; +β2 = dx2 +p+q + dy2 +p+q. +Performing summation over m, one obtains: +Πqn = − +� +p +� +� +� +α2−α(A−B)+C +α[(A−B−α)2−β2] tanh α−A +2T ++ +a2+α(A−B)+C +α[(A−B+α)2−β2] tanh α+A +2T ++ β2+β(A−B)+C +β[(A−B+β)2−α2] tanh β−B +2T ++ +β2−β(A−B)+C +β[(A−B−β)2−α2] tanh β+B +2T +� +� +� . +(11) +Now we turn to screening due to other layers. +Screening in a layered system +Coulomb repulsion between electrons in different layers l and l′ within the RPA approximation is determined by +the following integral equation: + +7 +V RP A +q,l−l′,n = vC +q,l−l′ + Πqn +� +l′′ vC +q,l−l′′V RP A +q,l′′−l′,n. +(12) +The polarization function Πqn in 2D was calculated in the previous subsection. This set of equations is decoupled by +the Fourier transform in the z direction, +V RP A +q,qz,n = +vC +q,qz +1 − ΠqnvC +q,qz +, +(13) +where +vC +q,qz = +� +l v2D +q eiqzl−qd|l| = v2D +q +sinh [qd] +cosh [qd] − cos [dqz]. +(14) +The screened interaction in a single layer therefore is is given by the inverse Fourier transform [14]: +V RP A +q,l−l′,n = d +2π +� π/d +qz=−π/d +eiqzd(l−l′) +vC +q,qz +1 − ΠqnvC +q,qz +. +(15) +Considering screened Coulomb potential at the same layer l = l′, the integration gives, +V RP A +qn += v2D +q +sinh [qd] +� +b2qn − 1 +, +(16) +where bqn = cosh [dq] − v2D +q Πqn sinh [dq]. This formula is reliable only away from plasmon region bqn > 1. It turns +out that to properly describe superconductivity, one can simplify the calculation at low temperature by considering +the static limit Πqn ≃ Πq0. Consequently the potential becomes static: V RP A +q +≡ V RP A +q,n=0. +SUPERCONDUCTIVITY +Superconductivity in WSM is caused by a conventional phonon pairing. The leading mode is an optical phonon +mode assumed to be dispersionless. with energy Ω. The effective electron-electron attraction due to the electron - +phonon attraction opposed by Coulomb repulsion (pseudo - potential) mechanism creates pairing below Tc. Further +we assume the singlet s-channel electron-phonon interaction and neglect the inter-layers electrons pairing. +In order +to describe superconductivity, one should ”integrate out” the phonon and the spin fluctuations degrees of freedom to +calculate the effective electron - electron interaction. We start with the phonons. The Matsubara action for effective +electron-electron interaction via in-plane phonons and direct Coulomb repulsion calculated in the previous Section. +It important to note that unlike in metal superconductors where a simplified pseudo - potential approach due to +McMillan and other [15], in 2D and layered WSM, one have to resort to a more microscopic approach. +Effective attraction due to phonon exchange opposed by the effective Coulomb repulsion +The free and the interaction parts of the effective electron action (”integrating phonons”+RPA Coulomb interaction) +in the quasi - momentum - Matzubara frequency representation, S = Se + Sint, +Sint = 1 +2T +� +qll′mm′ nqln +� +δll′V ph +q,m−m′ + V RP A +q,l−l′ +� +n−q,−l′,−n′. +(17) +Here nqln = � +p ψ∗sI +plnψsI +q−p,l,n the Fourier transform of the electron density and Se was defined in Eq.(5). The effective +electron - electron coupling due to phonons is: +V ph +qm = − +�√ +3 +2 +�2 +g2Ω +ωb2 +m + Ω2 , +(18) +where the bosonic frequencies are ωb +m = 2πmT. + +8 +Gorkov Green’s functions and the s-wave gap equations +Normal and anomalous (Matsubara) intra - layer Gorkov Green’s functions are defined by expectation value of the +fields, +� +ψIs +knlψ∗s′J +knl +� += δss′GIJ +kn and +� +ψIs +knlψJs′ +−k,−n,l +� += εss′F IJ +kn, while the gap function is +∆IJ +qn = +� +pm Vq−p,n−mF IJ +pm, +(19) +where Vqn = V ph +qn + V RP A +qn +is a sublattice scalar. The gap equations in the sublattice matrix form are derived from +Gorkov equations in Appendix B: +∆qn = − +� +pm Vq−p,n−mgpm +� +I + ∆pmgt +−p,−m∆∗ +−p,−mgpm +�−1 ∆pmgt +−p,−m. +(20) +In numerical simulation the gap equation was solved iteratively. Relatively large space cutoff Ns = 256 is required. +The frequency cutoff Nt = 128 was required due to low temperatures approached. Typically 15 − 25 iterations were +required. The parameters used were Ω = 16meV . The electron - phonon coupling g = 20meV . Now we turn to +results concentrating on two puzzling experimental results of ref.[6]. +Independence of Tc on density in topological type II phase +In Fig.5 the critical temperature for various values of density are plotted. The blue points are for dielectric +constant[6], ε = 16, describing the intercalated imidazole cations [C2MIm] [17]. +The inter - layer distance was +kept at d = 10.5A. +ϵ=50 +ϵ=∞ +ϵ=16 +0.8 +0.9 +1.0 +1.1 +1.2 +0 +5 +10 +15 +chemical potential (ev) +Tc (K) +FIG. 5. Critical temperature of transition to superconducting state in type II layered WSM is shown as function of chemical +potential (can be translated into carrier density via Fig.4). Three values of dielectric constant of the intercalant for fixed +interlayer distance are shown. Parameters of the electron gas are the same as in previous figures. +The significance and generatiozation of the observation are discussed below. +Increase of Tc with dielectric constant of intercalator materials +The main idea of the paper is that the difference in Tc between different intercalators is attributed not to small +variations in the inter - layer spacing d, but rather to large differences in the dielectric constant of the intercalating +materials due to its effect on the screening. In experiment of ref.[6] the imidazole cations [C2MIm]+ (1- ethyl - 3 - +methyl - imidazolium) are short molecules[17] have ϵ = 16, while [C6MIm]+ (1- hexy l - 3 - methy l - imidazolium) +are long molecules[18] with a larger value ϵ ≃ 50. The inter - layer distance d is slightly dependent intercalators + +9 +changing from 10.5˚ +A to 11.7˚ +A . The blue points in Fig 5 describe a material with dielectric constant ε = 16 should. +This is contrasted[18] with the ε = 50 material, see the red point. Neglecting the Coulomb repulsion, see the green +points, critical temperature (a much simpler calculation of Tc in this case similar to that in ref.[11] is needed in +this case) becomes yet higher. This demonstrate the importance of the Coulomb repulsion in a quasi 2D system. +Superconductivity is weaker for monolayer on substrate since both air and substrate have smaller dielectric constants +and hence weaker screen the Coulomb repulsion. +DISCUSSION AND CONCLUSION +To summarize we have developed a theory of superconductivity in layered type II Weyl semi-metals that properly +takes into account the Coulomb repulsion. The generalization goes beyond the simplistic pseudo - potential approach +due to McMillan[15] and others and depends essentially on the intercalating material. The theory allows to explain +the two puzzling phenomena observed recently in layered intercalated MoTe2 WSW compound [6] +The first experimental observation is that the gate voltage (changes in the chemical potential or equivalently in +density) has no impact on critical temperature Tc. For the 3D density range 8. × 1020cm−3 − 3.6 × 1021cm−3 the +temperature changes within 5%. For the intercalating material [C2MIm]+ with inter - layer distance d = 10.5A the +2D density range translates into 8.4×1013cm−2 −3.8×1014cm−2 wth slightly larger spacings d = 11.7A shown in Figs +2-5. This feature is explained purely topologically, see schematic Fig.1. In the type II density range the shape of both +pieces of the Fermi surface (the blue - yellow boundaries in Fig1b and Fig1c) largely does not depend on the density +(that is proportional to the area of the blue part of the surface) leading, see Fig. 4 to approximate independence of +the density of states (DOS) N (0) of chemical potential µ. This feature is akin to the DOS independence on µ for a +parabolic (topologically type I like in Fig.1a) band in purely 2D materials, but has completely difficult origin. +Using the somewhat naive BCS formula +Tc ≃ Ω e−N(0)g2 +eff . +(21) +Here Ω is the phonon frequency and geff the effective electron - phonon coupling. Assuming that both Ω and g do +not depend on the density one arrives at a conclusion that in the type II topological phase the critical temperature is +density independent . +The second experimental observation[6] was that Tc is in fact very sensitive to the intercalating material. For +imidazole cations [C2MIm]+ the critical temperature is Tc = 4.2K, while for [C6MIm]+ the temperature jumps +to Tc = 6.6K or 6.9K depending on the intercalation method. The inter - layer distance d is slightly dependent +intercaltors increasing from 10.5˚ +A to 11.7˚ +A . Our calculation demonstrates that the difference in Tc between different +intercalators cannot be attributed to small variations in the inter - layer spacing d. On the contrary there are large +differences in the dielectric constant of the intercalating materials. +While [C2MIm]+ have[17] a relatively small +dielectric constant ϵ = 16, [C6MIm]+ is estimated[18] in the range ϵ = 40 − 60. Our theory accounts the difference +in Tc due to changes in the screening of the Coulomb potential due to the inter - layer insulator. +ACKNOWLEDGEMENTS. +This work was supported by NSC of R.O.C. Grants No. 101-2112-M-009-014-MY3. +APPENDIX A. DETAILS OF THE MODEL +The system considered in the paper is fitted for the following values of the hipping and the tilt parameter. The +dimensionless tilt parameter was taken from ref.[6] κ = 1.3. The hopping γ = 500 meV and t = 2. The calculations +were performed on the discrete reciprocal lattice k1, k2 = 1, ...Ns with Ns = 256. Reciprocal lattice basis vectors are, +b1 = 2π +� +1, 1 +√ +3 +� +; +b2 = 2π +� +1, − 1 +√ +3 +� +, +(22) + +10 +so that a convenient representation is k = k1 +Ns b1 + k2 +Ns b2 with +kx = 2π +Ns +(k1 + k2) , +ky = +2π +√ +3Ns +(k1 − k2) . +(23) +APPENDIX B. DERIVATION OF THE TWO SUBLATTICE GAP EQUATION +Green’s functions and the s-wave Gorkov equations +We derive the Gorkov’s equations (GE) within the functional integral approach[19] starting from the effective +electron action for grassmanian fields ψ∗X, ψY . +S = 1 +T +� +ψ∗X � +G−1 +0 +�XY ψY + 1 +2ψ∗Y ψY V Y Xψ∗XψX +� +, +(24) +where X, Y denote space coordinate, sublattices (pseudospin) and spin of the electron. Finite temperature properties +of the condensate are described at temperature T by the normal and the anomalous Matsubara Greens functions for +spin singlet state. +The GE in functional form are: +� +ψAψ∗B� +δ +δψ∗C +� δS +δψ∗B +� ++ +� +ψAψB� +δ +δψ∗C +� δS +δψB +� += 0; +(25) +� +ψAψ∗B� +δ +δψC +� δS +δψ∗B +� ++ +� +ψAψB� +δ +δψC +� δS +δψB +� += δAC. +(26) +Performing the calculations and using the normal and anomalous Green functions in the form F AB = +� +ψAψB� +; GAB = +� +ψAψ∗B� +, one obtains: +F AX �� +G−1 +0 +�CX − vXCGCX + vCXGXX� ++ GAXvXCF XC = 0. +(27) +Skipping second and third terms in bracket in this expression and defining, superconducting gap ∆AB = vABF AB, +one rewrites as a matrix products: +� +G−1 +0 +�CX F XA = GAX∆XC. +(28) +The first GE (multiplied from left by G0) is, +F AB = −GAXGBY +0 +∆XY , +(29) +while the second GE similarly is: +GAB − GAX∆XY GZY +0 +∆∗ZUGUB +0 += GAB +0 +. +(30) +Frequency-quasi-momentum and the spin-sublattice decomposition +The generalized index A contains the space variables (space + Matsubara time, a), spin s and the sublattice I. +After performing the Fourier series with combined quasi - momentum - frequency α: +F s1s2IJ +ab += ϵs1s2 � +α eiα(a−b)F IJ +α ; +∆s1s2IJ +ab += ϵs1s2 � +α eiα(a−b)∆IJ +α ; +(31) +Gs1s2IJ +0ab += δs1s2 � +α eiα(a−b)gIJ +α ; +V s1s2IJ +ab += +� +α eiα(a−b)vα. + +11 +Substituting spins into Eq.(29,30), one obtains in the sublattice matrix form +Fα = −Gα∆αgt +−α; +(32) +Gα = G0α +� +I + ∆αgt +−α∆∗ +−αG0α +�−1 , +Convoluting the first GE by vν one obtains: +∆ω = − +� +ν vω−νGν∆νgt +−ν. +(33) +The solution of the second GE for G is: +Gα = gα +� +I + ∆αgt +−α∆∗ +−αgα +�−1 . +(34) +Substituting into the first GE one obtain Eq.(20) in the text. +[1] Katsnelson M.I. 2012 The Physics of Graphene, (Cambridge, Cambridge University Press). +[2] Soluyanov A. A. , Gresch D. ,Wang Z. ,Wu Q. ,Troyer M. ,Dai X. & Bernevig B. A. 2015, Nature 527, 495. +[3] Yu Z.-M. , Yao Y. , and Yang S. A. 2016 Phys. Rev. Lett. 117, 077202. +[4] O’Brien T. E. ,Diez M. , and Beenakker C. W. J. , 2016 Phys.Rev. Lett. 116, 236401. +[5] Wang C. et al. 2018 Nature, 555, 231; Lin, Z. et al. 2018 Nature, 562, 254; Huang H.,Zhou S. and DuanW., 2016 Phys.Rev.B +94 121117; Yan M. et al, Nature Comm. 2017 8, 257; Furue Y. 2021 et al Phys. Rev. B 104,144510. +[6] Zhang H. , Rousuli A. ,Zhang K. ,Zhong H. ,Wu Y. ,Yu P. ,Zhou S. 2022 2D Materials 9 045027. +[7] Das Sarma S. andLi Q. 2013 Phys. Rev. B 88, 081404(R);Brydon P.M.R. ,Das Sarma S. ,Hui H.-Y. and Sau J. D. 2014 +Phys. Rev. B 90, 184512;Li D. ,Rosenstein B. ,Shapiro B. Ya. , andShapiro I. 2014 Phys. Rev. B 90 054517. +[8] Fu L. and Berg E. 2010 Phys. Rev. Lett. 105, 097001. +[9] Zhang J.-L. et al. 2012 Front. Phys., 7, 193. +[10] Alidoust M. , Halterman K. , and Zyuzin A. A. 2017 Phys. Rev. B 95, 155124. +[11] Li D. , Rosenstein B. , Shapiro B. Ya. , and Shapiro I. 2017 Phys. Rev. B 95, 094513. +[12] Shapiro B Ya , Shapiro I , Li D. and Rosenstein B. 2018 J. Phys.:Condens. Matter 30 335403. +[13] Wang S. T. et al. 2012 Appl. Phys. Let. 101, 183110. +[14] Hawrylak P. , Eliasson G. , and Quinn J. J., 1988 Phys. Rev. B 37 10187. +[15] Bilbro G. and McMillan L. 1976 Phys. Rev. B 14 1887. +[16] Katayama S. ,Kobayashi A. ,Suzumura Y. 2006 J. Phys. Soc. Japan 75, 054705;Goerbig M. O. , Fuchs J. -N.,Montambaux +G. , Pi´echon F. 2008 Phys. Rev. B 78, 045415; Hirata M. et al. 2016 Nature Commun. 7 12666. +[17] Beal A R and Hughes H P 1979 J. Phys. C: Solid State Phys. 12 881 . +[18] Yang L., Fishbine B. H. , Migliori A. and Pratt L.R., 2010 J. Chem. Phys. 132 044701. +[19] Negele J.W. and Orlando H. , Quantum Many Particle Systems, 1998 Aspen, Advanced Book Classics, Westview Press. + diff --git a/69FAT4oBgHgl3EQfnx3V/content/tmp_files/load_file.txt b/69FAT4oBgHgl3EQfnx3V/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6264b664fe5a3ed5b994ca68c4ca46b3ebed53ee --- /dev/null +++ b/69FAT4oBgHgl3EQfnx3V/content/tmp_files/load_file.txt @@ -0,0 +1,491 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf,len=490 +page_content='Superconductivity in type II layered Weyl semi-metals B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Rosenstein1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Shapiro2 1Department of Electrohysics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 2Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Novel ”quasi two dimensional” typically layered (semi) metals offer a unique opportunity to control the density and even the topology of the electronic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In intercalated MoTe2 type II Weyl semi - metal the tilt of the dispersion relation cones is so large that topologically of the Fermi surface is distinct from a more conventional type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Superconductivity observed recently in this compound [Zhang et al, 2D Materials 9, 045027 (2022)] demonstrated two puzzling phenomena: the gate voltage has no impact on critical temperature, Tc, in wide range of density, while it is very sensitive to the inter - layer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The phonon theory of pairing in a layered Weyl material including the effects of Coulomb repulsion is constructed and explains the above two features in MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The first feature turns out to be a general one for any type II topological material, while the second reflects properties of the intercalated materials affecting the Coulomb screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='08631v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='supr-con] 20 Jan 2023 2 INTRODUCTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The 3D and 2D topological quantum materials, such as topological insulators and Weyl semi - metals (WSM), attracted much interests due to their rich physics and promising prospects for application in electronic and spinotronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The band structure in the so called type I WSM like graphene[1], is characterized by appearance linear dispersion relation (cones around several Dirac points) due to the band inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This is qualitatively distinct from conventional metals, semi - metals or semiconductors, in which bands are typically parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In type-II WSM [2], the cones have such a strong tilt, κ, so that they exhibit a nearly flat band and the Fermi surface ”encircles” the Brillouin zone, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1b, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' It is topologically distinct from conventional ”pockets”, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This in turn leads to exotic electronic properties different from both the those in both the conventional and in the type I WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Examples include the collapse of the Landau level spectrum in magnetoresistance [3], and novel quantum oscillations [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The type II topology of the Fermi surface was achieved in particular in transition metal dichalcogenides [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Very recentlyMoTe2 layers intercalated by ionic liquid cations were studied[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The tilt value was estimated to as high as κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='3 that places it firmly within the type II WSM class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The measurements included the Hall effect and the resistivity at low temperatures demonstrating appearance of superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' They discovered two intriguing facts that are currently under discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' First changing the gate voltage (chemical potential) surprisingly has no impact on critical temperature, Tc, in wide range of density of the electron gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Second Tc turned out to be very sensitive to the inter - layer distance d: it increases from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5A to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='7A, while the critical temperature jumps from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2K to 7K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In the present paper we propose a theoretical explanation of these observations based on appropriate generalization of the conventional superconductivity theory applied to these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Although early on unconventional mechanisms of superconductivity in WSM have been considered, accumulated experimental evidence points towards the conventional phonon mediated one [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In the previous paper[11] and a related work[10] a continuum theory of conventional superconductivity in WSM was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Magnetic response in the superconducting state was calculated[10][12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The model was too ”mesoscopic” to describe the type II phase since the global topology of the Brillouin zone was beyond the scope of the continuum approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Therefore we go beyond the continuum model in the present paper by modeling a type II layered WSM using a tight binding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The in-plane electron liquid model is similarl to that of graphene oxide[13] and other 2D WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' It possesses a chiral symmetry between two Brave sublattices for all values of the tilt parameter κ, but lacks hexagonal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The second necessary additional feature is inclusion of Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' It turns out that the screened Coulomb repulsion significantly opposes the phonon mediated pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Consequently a detailed RPA theory of screening in a layered material[14] is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' We calculate the superconducting critical temperature taking into consideration the modification of the Coulomb interaction due to the dielectric constant of intercalator material and the inter-layered spacing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The Gorkov equations for the two sublattices system are solved without resorting to the mesoscopic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Moreover since screening of Coulomb repulsion plays a much more profound role in quasi 2D materials the pseudo-potential simplification developed by McMillan[15] is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In Section II the microscopic model of the layered WSM is described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The RPA calculation of both the intra- and inter - layer screening is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In Section III the Gorkov equations for the optical phonon mediated intra- layer pairing for a multiband system including the Coulomb repulsion is derived and solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In Section IV the phonon theory of pairing including the Coulomb repulsion for a layered material is applied to recent extensive experiments on MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The effect of intercalation and density on superconductivity is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This explains the both remarkable features of Tc observed[6] in MoTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The last Section contains conclusions and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' A ”GENERIC” LATTICE MODEL OF LAYERED WEYL SEMI-METALS Intra- layer hopping A great variety of tight binding models were used to describe Weyl (Dirac) semimetals in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Historically the first was graphene (type I, κ = 0) , in which electrons hope between the neighboring cites of the honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' We restrict the discussion to systems with the minimal two cones of opposite chirality and negligible spin orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The two Dirac cones appear in graphene at K and K′ crystallographic points in BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Upon modification (more complicated molecules like graphene oxide, stress, intercalation) the hexagonal symmetry is lost, however a discrete chiral symmetry between two sublattices, denoted by I = A, B, ensures the WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The tilted type I and even type II (for which typically κ > 1) crystals can be described by the same Hamiltonian with the tilt term added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Two distict topologies of the Fermi surface in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Topology of the 2D Brillouin zone is that of the surface of 3D torroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' On the left the “conventional” type I pocket is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In the ceter and on the right the type II topology is shown schematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The filled states are in blue and envelop the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Despite the large difference in density of the two the Fermi surface properties like density of states are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 2D model is extended to a layered system with inter - layer distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Physically the 2D WSM layers are separated by a dielectric material with inter - layer hopping neglected, so that they are coupled electromagnetically only[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The lateral atomic coordinates are still considered on the honeycomb lattice are rn = n1a1 + n2a2, where lattice vectors are: a1 = a 2 � 1, √ 3 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' a2 = a 2 � 1, − √ 3 � , (1) despite the fact that hopping energies are different for jumps between nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Each site has three neighbors separated by δ1 = 1 3 (a1 − a2) , δ2 = − 1 3 (2a1 + a2) and δ3 = 1 3 (a1 + 2a2), in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The length of the lattice vectors a will be taken as the length unit and we set ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The hopping Hamiltonian including the tilt term is[13, 16]: K = √ 3 4 � nl � γ � ψsA† nl ψsB rn+δ1,l + ψsA† nl ψsB rn+δ2,l + tψsA† nl ψsB rn+δ3,l � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' − κψsI† nl ψsI rn+a1,l − µnn,l � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (2) Here an integer l labels the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Operator ψsA† nl is the creation operators with spin s =↑, ↓, while the density operator is defined as nnl = ψsI† nl ψsI nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The chemical potential is µ, while γ is the hopping energy for two neighbors at δ1, δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Since the the system does not possesses hexagonal symmetry (only the chiral one), the third jump has the different hopping[13] tγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Dimensionless parameter κ determines the tilt of the Dirac cones along the a1direction[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In the 2D Fourier space, ψsA nl = N −2 s � k ψsA kl e−ik·rn, one obtains for Hamiltonian (for finite discrete reciprocal lattice Ns × Ns): K = N −2 s � kl ψs† klMkψs kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (3) Here k = k1 Ns b1 + k2 Ns b2 (reciprocal lattice vectors are given in Appendix A) and matrix Mk = dxσx + dyσy + d0I in terms of Pauli matrices has components: dx = 2t √ 3 cos � 2π 3Ns (k1 − k2) � + 4 √ 3 cos � π Ns (k1 + k2) � cos � − π 3Ns (k1 − k2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (4) dy = − 2t √ 3 sin � 2π 3Ns (k1 − k2) � + 4 √ 3 cos � π Ns (k1 + k2) � sin � π 3Ns (k1 − k2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' d0 = 2 √ 3 � −κ cos � 2π Ns k1 � − µ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 4 Using γ as our energy unit from now on, the free electrons part of the Matsubara action for Grassmanian fields ψ∗sI kln is: Se = 1 T � kln ψ∗sI kln �� −iωn + d0 k � δIJ + σIJ i di k � ψsJ kln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (5) where ωn = πT (2n + 1) is the Matsubara frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The Green Function of free electrons has the matrix form gkn = �� −iωn + d0 k � I + σidi k �−1 = � −iωn + d0 k � I − σidi k (iωn − d0 k)2 − dx2 k − dy2 k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (6) Now we turn to the spectrum of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The topological phase diagram of the Weyl semimetal at large tilt parameter (κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Chemical potential (in units of γ = 500 meV) is marked on each contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The electron type I topology at low values of µ undergoes transition to the type II at µ = µ1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='8 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' At yet larger µ > µ2 c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' the Fermi surface becomes again type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This time the excitations are hole rather than electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The range of the topological type II phase at large κ The spectrum of Hamiltonian of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (4) consists of two branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The upper branch for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='9eV is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The lower branch for a reasonable choice of parameters appropriate to MoTe2 is significantly below the Fermi surface and is not plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Blue regions represent the filled electron states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' One observes a ”river” from one boundary to the other of the Brillouin zone (in coordinates k1 and k2, in terms of the original kx, ky it is a rhomb) characteristic to type II Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Topologically this is akin to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 3 the Fermi surfaces in a wide range of densities n = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='×1013−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5×1014cm−2 are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Topologically they separate into three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' At chemical potentials below µ1 c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='796 eV , corresponding to densities n < n1 c = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='×1013 cm−2 , the Fermi surface consists of one compact electron pocket similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1a, so that the electronic matter is of the (”customary”) topological type I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The density is determined from the (nearly linear) relation between the 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='6975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='6975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5 I (electron) II II II I (hole) k1 (π/a) k2 (π/a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Dispersion relation of WSM with κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The blue plane corresponds to chemical potentia lµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='8 eV so that the Fermi surface has the type II topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' chemical potential and density given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 4 (blue line, scale on the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In the range µ1 c < µ < µ2 c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='35eV the Fermi surface consists of two banks of a ”river” (blue color represents filled electron states) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 and can be viewed topologically as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1b and Fig1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The second critical density is n2 c = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='6 × 1014 cm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In this range the shape of both pieces of the Fermi surface largely does not depend on the density that is proportional to the area of the blue part of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' To make this purely topological observation quantitative, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 4 (green line, scale on the left) the density of states (DOS) as a function of chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' One observes that it nearly constant away from the two topological I to II transitions where it peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Coulomb repulsion The electron-electron repulsion in the layered WSM can be presented in the form, V = e2 2 � nln′l′ nnlvC n−n′,l−l′nn′l′ = e2 2N 2s � qll′ nqln−ql′vC q,l−l′, (7) where vC n−n′,l−l′ is the ”bare” Coulomb interaction between electrons with Fourier transform vC q,l−l′ = v2D q e−dq|l−l′|, v2D q = 2πe2/qϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Here ϵ is the dielectric constant of the intercalator material The long range Coulomb interaction is effectively taken into account using the RPA approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' SCREENING IN LAYERED WSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The screening in the layered system can be conveniently partitioned into the screening within each layer described by the polarization function Πqn and electrostatic coupling to carriers in other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' We start with the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 6 I II I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='4 0 1 2 3 4 chemical potential (ev) DOS(2 1014cm-2 ev-1) density (1014cm-2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Electron density and DOS as function of the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='of WSM with κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' DOS has cusps at both I to II transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Between the transitions it is nearly constant in the range of densities from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1 × 1014/cm2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' × 1014/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Polarization function of the electron gas in Layered WSM In a simple Fermi theory of the electron gas in normal state with Coulomb interaction between the electrons in RPA approximation the Matsubara polarization is calculated as a simple minus ”fish” diagram [14] in the form: Πqn = − � −2TTr � pm gpmgp+q,m+n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (8) Using the GF of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (6), one obtain: Πqn = 4T � pm (iωm + A) (iωm + B) + C � (iωm + A)2 − α2 � � (iωm + B)2 − β2 �, (9) where A = −d0 p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' B = iωn − d0 p+q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' C = dx pdx p+q + dy pdy p+q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (10) α2 = dx2 p + dy2 p ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' β2 = dx2 p+q + dy2 p+q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Performing summation over m, one obtains: Πqn = − � p � � � α2−α(A−B)+C α[(A−B−α)2−β2] tanh α−A 2T + a2+α(A−B)+C α[(A−B+α)2−β2] tanh α+A 2T + β2+β(A−B)+C β[(A−B+β)2−α2] tanh β−B 2T + β2−β(A−B)+C β[(A−B−β)2−α2] tanh β+B 2T � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (11) Now we turn to screening due to other layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Screening in a layered system Coulomb repulsion between electrons in different layers l and l′ within the RPA approximation is determined by the following integral equation: 7 V RP A q,l−l′,n = vC q,l−l′ + Πqn � l′′ vC q,l−l′′V RP A q,l′′−l′,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (12) The polarization function Πqn in 2D was calculated in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This set of equations is decoupled by the Fourier transform in the z direction, V RP A q,qz,n = vC q,qz 1 − ΠqnvC q,qz , (13) where vC q,qz = � l v2D q eiqzl−qd|l| = v2D q sinh [qd] cosh [qd] − cos [dqz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (14) The screened interaction in a single layer therefore is is given by the inverse Fourier transform [14]: V RP A q,l−l′,n = d 2π � π/d qz=−π/d eiqzd(l−l′) vC q,qz 1 − ΠqnvC q,qz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (15) Considering screened Coulomb potential at the same layer l = l′, the integration gives, V RP A qn = v2D q sinh [qd] � b2qn − 1 , (16) where bqn = cosh [dq] − v2D q Πqn sinh [dq].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This formula is reliable only away from plasmon region bqn > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' It turns out that to properly describe superconductivity, one can simplify the calculation at low temperature by considering the static limit Πqn ≃ Πq0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Consequently the potential becomes static: V RP A q ≡ V RP A q,n=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' SUPERCONDUCTIVITY Superconductivity in WSM is caused by a conventional phonon pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The leading mode is an optical phonon mode assumed to be dispersionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' with energy Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The effective electron-electron attraction due to the electron - phonon attraction opposed by Coulomb repulsion (pseudo - potential) mechanism creates pairing below Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Further we assume the singlet s-channel electron-phonon interaction and neglect the inter-layers electrons pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In order to describe superconductivity, one should ”integrate out” the phonon and the spin fluctuations degrees of freedom to calculate the effective electron - electron interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' We start with the phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The Matsubara action for effective electron-electron interaction via in-plane phonons and direct Coulomb repulsion calculated in the previous Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' It important to note that unlike in metal superconductors where a simplified pseudo - potential approach due to McMillan and other [15], in 2D and layered WSM, one have to resort to a more microscopic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Effective attraction due to phonon exchange opposed by the effective Coulomb repulsion The free and the interaction parts of the effective electron action (”integrating phonons”+RPA Coulomb interaction) in the quasi - momentum - Matzubara frequency representation, S = Se + Sint, Sint = 1 2T � qll′mm′ nqln � δll′V ph q,m−m′ + V RP A q,l−l′ � n−q,−l′,−n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (17) Here nqln = � p ψ∗sI plnψsI q−p,l,n the Fourier transform of the electron density and Se was defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The effective electron - electron coupling due to phonons is: V ph qm = − �√ 3 2 �2 g2Ω ωb2 m + Ω2 , (18) where the bosonic frequencies are ωb m = 2πmT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 8 Gorkov Green’s functions and the s-wave gap equations Normal and anomalous (Matsubara) intra - layer Gorkov Green’s functions are defined by expectation value of the fields, � ψIs knlψ∗s′J knl � = δss′GIJ kn and � ψIs knlψJs′ −k,−n,l � = εss′F IJ kn, while the gap function is ∆IJ qn = � pm Vq−p,n−mF IJ pm, (19) where Vqn = V ph qn + V RP A qn is a sublattice scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The gap equations in the sublattice matrix form are derived from Gorkov equations in Appendix B: ∆qn = − � pm Vq−p,n−mgpm � I + ∆pmgt −p,−m∆∗ −p,−mgpm �−1 ∆pmgt −p,−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (20) In numerical simulation the gap equation was solved iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Relatively large space cutoff Ns = 256 is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The frequency cutoff Nt = 128 was required due to low temperatures approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Typically 15 − 25 iterations were required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The parameters used were Ω = 16meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The electron - phonon coupling g = 20meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Now we turn to results concentrating on two puzzling experimental results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Independence of Tc on density in topological type II phase In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5 the critical temperature for various values of density are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The blue points are for dielectric constant[6], ε = 16, describing the intercalated imidazole cations [C2MIm] [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The inter - layer distance was kept at d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' ϵ=50 ϵ=∞ ϵ=16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2 0 5 10 15 chemical potential (ev) Tc (K) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Critical temperature of transition to superconducting state in type II layered WSM is shown as function of chemical potential (can be translated into carrier density via Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Three values of dielectric constant of the intercalant for fixed interlayer distance are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Parameters of the electron gas are the same as in previous figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The significance and generatiozation of the observation are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Increase of Tc with dielectric constant of intercalator materials The main idea of the paper is that the difference in Tc between different intercalators is attributed not to small variations in the inter - layer spacing d, but rather to large differences in the dielectric constant of the intercalating materials due to its effect on the screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In experiment of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' [6] the imidazole cations [C2MIm]+ (1- ethyl - 3 - methyl - imidazolium) are short molecules[17] have ϵ = 16, while [C6MIm]+ (1- hexy l - 3 - methy l - imidazolium) are long molecules[18] with a larger value ϵ ≃ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The inter - layer distance d is slightly dependent intercalators 9 changing from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5˚ A to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='7˚ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The blue points in Fig 5 describe a material with dielectric constant ε = 16 should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This is contrasted[18] with the ε = 50 material, see the red point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Neglecting the Coulomb repulsion, see the green points, critical temperature (a much simpler calculation of Tc in this case similar to that in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' [11] is needed in this case) becomes yet higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This demonstrate the importance of the Coulomb repulsion in a quasi 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Superconductivity is weaker for monolayer on substrate since both air and substrate have smaller dielectric constants and hence weaker screen the Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' DISCUSSION AND CONCLUSION To summarize we have developed a theory of superconductivity in layered type II Weyl semi-metals that properly takes into account the Coulomb repulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The generalization goes beyond the simplistic pseudo - potential approach due to McMillan[15] and others and depends essentially on the intercalating material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The theory allows to explain the two puzzling phenomena observed recently in layered intercalated MoTe2 WSW compound [6] The first experimental observation is that the gate voltage (changes in the chemical potential or equivalently in density) has no impact on critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' For the 3D density range 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' × 1020cm−3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='6 × 1021cm−3 the temperature changes within 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' For the intercalating material [C2MIm]+ with inter - layer distance d = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5A the 2D density range translates into 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='4×1013cm−2 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='8×1014cm−2 wth slightly larger spacings d = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='7A shown in Figs 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This feature is explained purely topologically, see schematic Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' In the type II density range the shape of both pieces of the Fermi surface (the blue - yellow boundaries in Fig1b and Fig1c) largely does not depend on the density (that is proportional to the area of the blue part of the surface) leading, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 4 to approximate independence of the density of states (DOS) N (0) of chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This feature is akin to the DOS independence on µ for a parabolic (topologically type I like in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='1a) band in purely 2D materials, but has completely difficult origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Using the somewhat naive BCS formula Tc ≃ Ω e−N(0)g2 eff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (21) Here Ω is the phonon frequency and geff the effective electron - phonon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Assuming that both Ω and g do not depend on the density one arrives at a conclusion that in the type II topological phase the critical temperature is density independent .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The second experimental observation[6] was that Tc is in fact very sensitive to the intercalating material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' For imidazole cations [C2MIm]+ the critical temperature is Tc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='2K, while for [C6MIm]+ the temperature jumps to Tc = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='6K or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='9K depending on the intercalation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The inter - layer distance d is slightly dependent intercaltors increasing from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='5˚ A to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='7˚ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Our calculation demonstrates that the difference in Tc between different intercalators cannot be attributed to small variations in the inter - layer spacing d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' On the contrary there are large differences in the dielectric constant of the intercalating materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' While [C2MIm]+ have[17] a relatively small dielectric constant ϵ = 16, [C6MIm]+ is estimated[18] in the range ϵ = 40 − 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Our theory accounts the difference in Tc due to changes in the screening of the Coulomb potential due to the inter - layer insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' ACKNOWLEDGEMENTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' This work was supported by NSC of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 101-2112-M-009-014-MY3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' DETAILS OF THE MODEL The system considered in the paper is fitted for the following values of the hipping and the tilt parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The dimensionless tilt parameter was taken from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' [6] κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The hopping γ = 500 meV and t = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The calculations were performed on the discrete reciprocal lattice k1, k2 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content='Ns with Ns = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Reciprocal lattice basis vectors are, b1 = 2π � 1, 1 √ 3 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' b2 = 2π � 1, − 1 √ 3 � , (22) 10 so that a convenient representation is k = k1 Ns b1 + k2 Ns b2 with kx = 2π Ns (k1 + k2) , ky = 2π √ 3Ns (k1 − k2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (23) APPENDIX B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' DERIVATION OF THE TWO SUBLATTICE GAP EQUATION Green’s functions and the s-wave Gorkov equations We derive the Gorkov’s equations (GE) within the functional integral approach[19] starting from the effective electron action for grassmanian fields ψ∗X, ψY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' S = 1 T � ψ∗X � G−1 0 �XY ψY + 1 2ψ∗Y ψY V Y Xψ∗XψX � , (24) where X, Y denote space coordinate, sublattices (pseudospin) and spin of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' Finite temperature properties of the condensate are described at temperature T by the normal and the anomalous Matsubara Greens functions for spin singlet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' The GE in functional form are: � ψAψ∗B� δ δψ∗C � δS δψ∗B � + � ψAψB� δ δψ∗C � δS δψB � = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (25) � ψAψ∗B� δ δψC � δS δψ∗B � + � ψAψB� δ δψC � δS δψB � = δAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (26) Performing the calculations and using the normal and anomalous Green functions in the form F AB = � ψAψB� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' GAB = � ψAψ∗B� , one obtains: F AX �� G−1 0 �CX − vXCGCX + vCXGXX� + GAXvXCF XC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (27) Skipping second and third terms in bracket in this expression and defining, superconducting gap ∆AB = vABF AB, one rewrites as a matrix products: � G−1 0 �CX F XA = GAX∆XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (28) The first GE (multiplied from left by G0) is, F AB = −GAXGBY 0 ∆XY , (29) while the second GE similarly is: GAB − GAX∆XY GZY 0 ∆∗ZUGUB 0 = GAB 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (30) Frequency-quasi-momentum and the spin-sublattice decomposition The generalized index A contains the space variables (space + Matsubara time, a), spin s and the sublattice I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' After performing the Fourier series with combined quasi - momentum - frequency α: F s1s2IJ ab = ϵs1s2 � α eiα(a−b)F IJ α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' ∆s1s2IJ ab = ϵs1s2 � α eiα(a−b)∆IJ α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (31) Gs1s2IJ 0ab = δs1s2 � α eiα(a−b)gIJ α ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' V s1s2IJ ab = � α eiα(a−b)vα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' 11 Substituting spins into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (29,30), one obtains in the sublattice matrix form Fα = −Gα∆αgt −α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (32) Gα = G0α � I + ∆αgt −α∆∗ −αG0α �−1 , Convoluting the first GE by vν one obtains: ∆ω = − � ν vω−νGν∆νgt −ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (33) The solution of the second GE for G is: Gα = gα � I + ∆αgt −α∆∗ −αgα �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (34) Substituting into the first GE one obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69FAT4oBgHgl3EQfnx3V/content/2301.08631v1.pdf'} +page_content=' (20) in the text.' metadata={'source': 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b/6NAzT4oBgHgl3EQff_yv/content/tmp_files/2301.01462v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1486b6a3a9a4ee8945e6fa36a91a34c4545b7f82 --- /dev/null +++ b/6NAzT4oBgHgl3EQff_yv/content/tmp_files/2301.01462v1.pdf.txt @@ -0,0 +1,1754 @@ +1 +A Stochastic Multi-Objective Optimization +Framework for Planning and Scheduling of +Community Energy Storage Systems +K.B.J. Anuradha, Student Member, IEEE, and Chathurika P. Mediwaththe, Member, IEEE +Abstract—This paper explores a methodology to optimize the +planning and the scheduling of a community energy storage +(CES) considering the uncertainty of real power consumption +and solar photovoltaic (SPV) generation of the customers in low +voltage (LV) distribution networks. To this end, we develop a +stochastic multi-objective optimization framework which mini- +mizes the investment and the operation costs of the CES provider, +and the social costs of the customers (i.e. cost of customers for +trading energy with the grid and the CES). The uncertainty +of SPV generation and real power consumption are modelled +to follow the beta and normal distributions, respectively. Then, +the roulette wheel mechanism (RWM) is exploited to formulate a +scenario-based stochastic program. The initial scenarios obtained +from the RWM, are then reduced by using the K-Means clus- +tering algorithm, to keep the problem tractability. A case study +highlights our model provides 10-21% more cumulative economic +benefits for the customers and the CES provider, compared with +the models that optimize only the CES scheduling. Also, the +simulation results for different energy price schemes of the CES +provider reflect, the customers change their power exchange with +the CES and the grid significantly, to minimize their social costs. +Index +Terms—Community +energy +storage +(CES), +multi- +objective optimization, planning and scheduling, power flow, +roulette wheel mechanism (RWM), scenarios, uncertainty +I. INTRODUCTION +T +HE integration of solar photovoltaic (SPV) systems in +low voltage (LV) distribution networks, has undergone +a rapid upsurge over the last few decades. However, the +intermittent and non-dispatchable nature of SPV generation, +may restrict their beneficiaries such as the customers from +exploiting the merits of SPV fully. These issues can be +efficiently alleviated by exploiting energy storage systems. +Community energy storage (CES) devices are an emerging +type of battery system, which is gaining increasing interest in +the industry, as they can enable increased community access +and network hosting capacity for renewable energy [1]. +An energy management framework which aims at optimiz- +ing only the scheduling of a CES such as its charging and +discharging pattern, may not deliver the expected rewards +from a CES completely. Hence, it is imperative that the +planning aspects including the location, the capacity and the +rated power of a CES are optimized concurrently with its +K. B. J. Anuradha is with The Australian National University, Canberra, +ACT 0200, Australia (email: u7146121@anu.edu.au). +Chathurika P. Mediwaththe is with The Australian National University, +Canberra, ACT 0200, Australia, and also with the Commonwealth Scientific +and Industrial Research Organisation, Canberra, ACT 2601, Australia (email: +chathurika.mediwaththe@csiro.au). +scheduling. Several studies have presented deterministic opti- +mization frameworks to find the optimal CES planning and/or +the scheduling, and thus achieve the objectives of different +stakeholders [2]–[4]. Here, the authors have assumed both real +power consumption and SPV generation of the customers are +perfectly known ahead from their forecasts. However, due to +the uncertainty of SPV generation and real power consumption +of the customers, their forecast errors can be quite high at +times. Eventually, this may result the optimization models de- +scribed in [2]–[4] unable to achieve the objectives effectively. +Also, different stakeholders have distinct objectives for them. +Thus, a multi-objective optimization framework can reflect the +trade-offs between those objectives comprehensively. +In this paper, we examine the extent to which the optimal +planning and scheduling of a CES benefit different stakehold- +ers. To this end, we develop an energy management framework +between the customers, the CES and the grid, by incorporating +the uncertainty of real power consumption and SPV generation +of the customers. Additionally, we leverage a linearized power +flow model with our energy management framework to formu- +late a mixed integer linear program (MILP). In summary, the +main contributions of this paper can be highlighted as follows. +• We develop a stochastic multi-objective optimization +framework which optimizes both planning and scheduling +of a CES for benefiting (i) the CES provider by minimiz- +ing the investment and the operation costs of the CES, +and (ii) the customers by minimizing their social costs. +• The proposed optimization framework is capable of pro- +viding significantly higher economic benefits for the CES +provider and the customers than in the models which +arbitrarily choose the CES connected node. +• A case study compares our proposed stochastic model +with its corresponding deterministic model for different +energy price schemes of the CES provider. This study +enables to understand how the economic benefits for +the CES provider, and for the customers change due +to the uncertainty of real power consumption and SPV +generation of the customers, and the energy price scheme +of the CES provider. +Prior research work have presented optimization frame- +works which focus on the scheduling of the CES [2], [3], and +both planning and scheduling of CES [4]–[7]. For instance, the +authors of [2] have presented a method for CES scheduling, to +minimize the social costs of the customers while maximizing +the revenue of the CES provider. In [3], an optimization +arXiv:2301.01462v1 [eess.SY] 4 Jan 2023 + +2 +framework is presented to schedule the CES, to minimize the +real energy losses, and energy trading costs with the grid by +the CES provider and the customers. The authors of [4] and +[5] have presented models to optimize the CES planning and +scheduling simultaneously, to enhance the hosting capacity of +LV networks, and to mitigate the voltage excursions in three +phase unbalanced LV networks, respectively. Also, analytical +methods for optimizing the CES planning and scheduling have +been discussed in [6], [7]. A common feature of [2]–[7] is +that the authors have used deterministic models, assuming +the SPV generation and the real power consumption of the +customers are known ahead with no uncertainty. Hence, those +models may not be efficient in providing realistic planning and +operation decisions. +The uncertainty of real power consumption and generation +from SPV have been taken into account in [8], [9] for +CES management problems. For instance, the authors of [8] +have presented a method for optimizing both planning and +scheduling of CES to accomplish multiple objectives. Here, +the authors have used the normal distribution and the RWM +to model the uncertainty of real power consumption and SPV +generation. In [9], the authors of have investigated how the +CES location impacts the voltage profile and real power losses +in LV networks. For this, they have arbitrarily allocated the +CES at different nodes. In contrast to [2]–[7], our paper opti- +mizes both planning and scheduling aspects of the CES taking +into account the uncertainty of real power consumption and +SPV generation. Thus, our approach models the CES planning +and its scheduling problem more realistically. Also, compared +to [8], [9], we present a method to minimize the personal costs +of the CES provider and the customers concurrently. +This paper is structured as follows. Notations used in this +paper are detailed in Section II. The system models of the CES, +the customers and the network power flow are presented in +Section III. The stochastic models of SPV generation and real +power consumption of the customers are given in Section IV. +The formulation of the multi-objective optimization framework +is described in Section V. Section VI presents the validation of +the results, and Section VII gives the conclusion of the paper. +II. NOTATIONS +A. +Stochastic Model-related Notations +To formulate a scenario-based stochastic program, the un- +certainty of SPV generation and real power consumption of +the customers are modelled using known probability density +functions. Here, a scenario represents a possible combination +of SPV generation and real power consumption of all the +customers together with their corresponding probabilities at +a given time. The initial set of scenarios is denoted by +R, and r ∈ R. Due to the computational complexity of +stochastic programs which use scenarios, it is imperative to +use a scenario reduction approach to reduce the number of +scenarios, and keep the problem tractability. Thus, a scenario +reduction technique is used in this paper, and the set composed +by the reduced scenarios is given by S, where s ∈ S. +The implementation of the scenario-based stochastic program +including the scenario reduction are discussed in Section IV. +Fig. 1: Mutual power exchanges between the CES, the grid +and a customer in scenario s at time t +B. Network and Power Flow-related Notations +In this paper, a distribution network with a radial topology +is considered. It is described by the graph G = (V, E), where +V = {0, 1, ..., N} is the set of all nodes, and E = {(i, j)} ⊂ +V×V is the set of all lines in the network. Node 0 (slack node) +represents the secondary side of the distribution transformer. +The resistance and the reactance of line (i, j) are rij (in Ω) +and xij (in Ω), respectively. The set of customers at node j +is represented by Cj, and c ∈ Cj ∀j ∈ V\ {0}. Also, t ∈ T , +where T is the set of time intervals, and ∆t is the difference +between two adjacent time instances (in hours). The real and +reactive power flow from i to j node in scenario s at time t +are represented by Pij,s(t) (in kW) and Qij,s(t) (in kVAR), +respectively. Also, ∀j ∈ V\ {0}, t∈ T , s∈ S, the real power +absorption, reactive power absorption, voltage and squared +voltage are given by pj,s(t) (in kW), qj,s(t) (in kVAR), Vj,s(t) +(in V ) and Uj,s(t) (in V 2), respectively. +III. SYSTEM MODELS +This section first presents the network power flow model, +followed by the power exchange model of the customers, and +the CES model. We consider each customer, the CES and the +grid can exchange power with each entity as shown in Fig. 1. +A. Power Flow Model of the Network +In this work, we consider the power absorption for the +nodes as positive, and the power injections from the nodes as +negative. Additionally, it is considered that there are multiple +customers at each node. To model the network power flows, +we use the linearized power flow equations (1) and (2) from +the Distflow model in [10]. +Pij,s(t) = pj,s(t) + +� +k:j→k +Pjk,s(t) +∀(i, j) ∈ E, , t∈ T , s∈ S +(1) +Qij,s(t) = qj,s(t)+ +� +k:j→k +Qjk,s(t) +∀(i, j) ∈ E, , t∈ T , s∈ S +(2) + +External Grid +Grid +pCES,s(t) < 0 +pCES,s(t) ≥ 0 +0 > (0)sgd +pejs(t) > 0 +pcS(t) > 0 +田田 +pCS (t) < 0 +cth Customer +CES +at node j3 +The real and reactive power absorbed by the node j at time t +in scenario s, can be expressed as (3) and (4). The real power +absorption for the CES installed node is governed by (3a), +while for all the other nodes (except slack node), it is (3b). +Th equation (4) handles the reactive power absorption for all +nodes, and we assume the CES and the SPV devices operate +at unity power factor. The real power consumption, reactive +power consumption and SPV generation of the customer c at +node j at time t in scenario s are given by pL +cj,s(t) (in kW), +qL +cj,s(t) (in kVAR) and pP V +cj,s(t) (in kW), respectively. For j ∈ +V\ {0} , t∈ T , s∈ S, the CES charging and discharging power +are represented by pCES,ch +j,s +(t) and pCES,dis +j,s +(t), respectively. +pj,s(t) = +� +c∈Cj +pL +cj,s(t)− +� +c∈Cj +pP V +cj,s(t)+pCES,ch +j,s +(t)−pCES,dis +j,s +(t) +(3a) +pj,s(t) = +� +c∈Cj +pL +cj,s(t)− +� +c∈Cj +pP V +cj,s(t) +∀j ∈ V\ {0} , t∈ T , s∈ S +(3b) +qj,s(t) = +� +c∈Cj +qL +cj,s(t) +∀j ∈ V\ {0} , t∈ T , s∈ S +(4) +The linearized Distflow equations described in (1)-(2) can +be explicitly written as (5), where U0 = |V0|2 and U=|V(t)|2 +are the vectors of the squared voltage magnitude of the slack +node, and the squared voltage magnitudes of all other nodes, +respectively. 1 symbolizes a vector of all ones. Moreover, p +and q are the vectors of real and reactive power absorption at +each node. The matrices ˜R and ˜X ∈ RN×N have the elements +Rij = 2 � +(m,n)∈Li∩Lj rmn and Xij = 2 � +(m,n)∈Li∩Lj xmn, +respectively, where Li is the set of lines on the path which +connects node 0 and i [2], [10]. Also, (6) ensures the squared +voltage magnitude at each node is within its allowable voltage +magnitude limits. Here, Umin = +��V 2 +min +�� 1 and Umax = +��V 2 +max +�� 1, where Vmin and Vmax are the allowable lower and +upper bound of voltage, respectively. +U = U01 − ˜Rp − ˜Xq +∀t∈ T , s∈ S +(5) +Umin ≤ U ≤ Umax +∀t∈ T , s∈ S +(6) +B. Power Exchange Model of the Customers +According to Fig. 1, pG +cj,s(t) and pCES +cj,s (t) denote the real +power exchange with the grid and the CES by the customer c at +node j at time t in scenario s, respectively. When pG +cj,s(t) > 0, +it suggests a power import from the grid by a customer. On +the contrary, when that customer exports power back to the +grid, it will be pG +cj,s(t) < 0. The same sign convention is used +for customer power exchanges with the CES, and CES power +exchange with the grid pG +CES,s(t) (in kW). +When a customer’s SPV generation is insufficient to fulfill +its real power consumption, that deficit is attained by importing +power from the grid and the CES. This is mathematically +represented by (7a). Nevertheless, the imported power from +each entity should be within the deficit quantity. This is +ensured by (7b) and (7c). Besides, when that customer has +excess SPV generation, it exports its surplus to the grid and +the CES, which is mathematically interpreted by (8a). Similar +to (7b) and (7c), the exported power to the CES and the grid +should not exceed the mismatch of the SPV generation and +the real power consumption. This is demonstrated by (8b) and +(8c). Considering the ability of the CES to exchange power +with the grid and the customers, pG +CES,s(t) can be expressed +in terms of pCES +cj,s , pCES,ch +j,s +(t) and pCES,dis +j,s +(t) as in (9). +If pL +cj,s(t) ≥ pP V +cj,s(t): +0 ≤ pG +cj,s(t) + pCES +cj,s (t) = pL +cj,s(t) − pP V +cj,s(t) +(7a) +0 ≤ pG +cj,s(t) ≤ pL +cj,s(t) − pP V +cj,s(t) +(7b) +0 ≤ pCES +cj,s (t) ≤ pL +cj,s(t) − pP V +cj,s(t) +(7c) +Otherwise: +pG +cj,s(t) + pCES +cj,s (t) = pL +cj,s(t) − pP V +cj,s(t) ≤ 0 +(8a) +pL +cj,s(t) − pP V +cj,s(t) ≤ pG +cj,s(t) ≤ 0 +(8b) +pL +cj,s(t) − pP V +cj,s(t) ≤ pCES +cj,s (t) ≤ 0 +∀j ∈ V\ {0} , c ∈ Cj, t∈ T , s∈ S +(8c) +pG +CES,s(t) = �N +j=1 +�� +c∈Cj pCES +cj,s (t) + pCES,ch +j,s +(t) − pCES,dis +j,s +(t) +� +∀j ∈ V\ {0} , c ∈ Cj, t∈ T , s∈ S +(9) +C. Community Energy Storage Model +In this paper, we assume the CES is owned by a third +party, and we designate the owner as the CES provider. The +planning constraints of the CES are given by (10)-(12), and the +operation of the CES is mathematically modeled by (13)-(16). +The equation (10) finds the optimal CES node, and we use +a binary variable Lj, in which Lj = 1 suggests node j as +the optimal CES node, and Lj = 0 means there is no CES at +node j. Also, (10) guarantees only a single CES is installed in +the network. The inequalities in (11) and (12) find the optimal +CES capacity Ecap +j +(in kWh) and its rated power pRate +j +(in +kW) at node j, respectively. Here, Ecap +min (in kWh) and Ecap +max +(in kWh) are the minimum and maximum CES capacity limits, +and pRate +max is the maximum CES rated power limit. +N +� +j=1 +Lj = 1 +∀j ∈ V\ {0} , Lj ∈ {0, 1} +(10) + +4 +LjECap +min ≤ ECap +j +≤ LjECap +max +∀j ∈ V\ {0} , Lj ∈ {0, 1} +(11) +0 ≤ pRate +j +≤ LjpRate +max +∀j ∈ V\ {0} , Lj ∈ {0, 1} +(12) +The inequality described in (13) avoids simultaneous charg- +ing and discharging of the CES, while guaranteeing the CES +charging and discharging power does not exceed its optimal +rated power pRate +j +. For this, a binary variable Bj and two +additional variables namely, y and z are used. Here, Bj ∈ [0, 1] +and y, z ∈ R+ such that these variables also satisfy (13). When +the CES is charging, Bj = 1. Hence, z = pCES,dis +j,s +(t) = 0 ac- +cording to (13d) and (13e). Also, as stated in (13c), y = pRate +j +and thus, 0 ≤ pCES,ch +j,s +(t) ≤ y = pRate +j +≤ pRate +max . When the +CES discharges, Bj = 0, and hence, y = pCES,ch +j,s +(t) = 0 +according to (13a) and (13b). In this instance, z = pRate +j +, +which generates the inequality 0 ≤ pCES,dis +j,s +(t) ≤ z = +pRate +j +≤ pRate +max . The equation (14) illustrates how the CES +energy level changes with time t, where ECES +j,s +(t) (in kWh) is +the energy level of the CES at node j at time t in scenario s. +Additionally, ηch and ηdis are the charging and discharging +efficiency of the CES, respectively. Furthermore, the CES +energy level should be maintained with in the minimum and +maximum state of charge (SoC) levels. This is regulated by +(15), where ηmin and ηmax represent minimum and maximum +percentage coefficients of the CES capacity, respectively. Also, +it is required to guarantee the continuity of the CES operation +over the next day, and thus, the CES energy level at the end +of the day should be kept approximately same as the initial +energy level at the start of the day . This is managed by (16) +[2], [11]. Here, ε is a small positive number (in kWh), and +tn ∈ TN where TN = {1, 2, ...., |T | /24}. +0 ≤ pCES,ch +j,s +(t) ≤ y +(13a) +pCES,ch +j,s +(t) ≤ y ≤ pRate +max Bj +(13b) +−pRate +max (1 − Bj) ≤ y − pRate +j +≤ 0 +(13c) +0 ≤ pCES,dis +j,s +(t) ≤ z +(13d) +pCES,dis +j,s +(t) ≤ z ≤ pRate +max (1 − Bj) +(13e) +−pRate +max Bj ≤ z − pRate +j +≤ 0 +∀j ∈ V\ {0} , t∈ T , s∈ S +(13f) +ECES +j,s +(t) = ECES +j,s +(t − 1) + (ηchpCES,ch +j,s +(t) +− 1 +ηdis pCES,dis +j,s +(t))∆t +∀j ∈ V\ {0} , t∈ T , s∈ S +(14) +ηminECap +j +≤ ECES +j,s +(t) ≤ ηmaxECap +j +∀j ∈ V\ {0} , t∈ T , s∈ S +(15) +��ECES +j,s +(24tn) − ECES +j,s +(0) +�� ≤ ε +∀j ∈ V\ {0} , tn∈ T N, s∈ S +(16) +IV. STOCHASTIC MODELS +In this section, the uncertainty modeling of real power +consumption and SPV generation are presented. Similar to +[12], [13], the uncertainty of real power consumption and SPV +generation are modelled by the probability density functions +(PDFs) of normal and beta distributions, respectively. +A. Uncertainty of the Real Power Consumption +The uncertainty of real power consumption of the customers +follows the probability density function of normal distribution +PDFL(.) given in (17) [12], [14]. The forecasted real power +consumption of the customer c at node j at time t is consid- +ered as the mean real power consumption µL,t +cj +of PDFL(.) +[12], [14]. The standard deviation and a sample real power +consumption are denoted by σL,t +cj +and XL,t +cj , respectively. +PDFL(X) = +1 +σL,t +cj +√ +2π +e +−0.5 +� +XL,t +cj +−µL,t +cj +σL,t +cj +�2 +∀j ∈ V\ {0} , c ∈ Cj, t∈ T +(17) +B. Uncertainty of the SPV Generation +As mentioned in [12], [13], the uncertainty of SPV gener- +ation of the customers mimic the probability density function +of beta distribution PDFP V (.) as given in (18a). Also, the +forecasted SPV generation of the customer c at node j at time t +is taken as the mean SPV generation µP V,t +cj +of PDFP V (.). The +equations (18a)-(18d) describe the relationship between the +shape parameters αt +cj, βt +cj, a sample SPV generation XP V,t +cj +, +SPV capacity of a customer PVcap,cj, the mean µP V,t +cj +and the +standard deviation σP V,t +cj +of PDFP V (.) [12], [13]. Also, Γ(.) +represents the gamma function. +PDFP V (X) = +� +� +� +� +� +� +� +� +� +� +� +� +� +Γ(α+β) +Γ(α)Γ(β) +� +XP V,t +cj +�αt +cj−1 � +1 − XP V,t +cj +�βt +cj−1 +∀ +0 < XP V,t +cj +< 1 +αt +cj, βt +cj ≥ 0 +0 +Otherwise +(18a) +µP V,t +cj += +αt +cj +αt +cj + βt +cj +(18b) +(σP V,t +cj +)2 = +αt +cjβt +cj +� +αt +cj + βt +cj +�2 � +αt +cj + βt +cj + 1 +� +(18c) +σP V,t +cj += +0.2µP V,t +cj +PVcap,cj ++ 0.21 +∀j ∈ V\ {0} , c ∈ Cj, t∈ T +(18d) + +5 +C. Scenario-based Stochastic Program +Since the normal and beta distributions are continuous +PDFs, they represent infinite number of realizations of the +random variables. Here, a realization refers to a sample real +power consumption or SPV generation of a customer at a given +time. A large number of realizations can model the uncertainty +better at the expense of a large computational burden. But, +continuous PDFs approximated as discrete functions by a +finite number of realizations, as given in [15], can be used +to eliminate the similar and less probable real power con- +sumption or SPV generation values. Hence, the discretization +of continuous PDFs, reduces the complexity of uncertainty +modelling. Thus, both normal and beta PDFs are approximated +as discrete functions by 7 realizations. Here, the approximated +discrete functions are constructed to have 7 intervals, with +every interval having a width of a standard deviation σ. The +midpoint of an interval is a possible realization. For instance, +when the forecasted real power consumption is µ, the intervals +1-7 are centered around the 7 realizations µ, µ + σ, µ − σ, +µ + 2σ, µ − 2σ, µ + 3σ and µ − 3σ as done in [16], [17]. The +steps of scenario generation and reduction are given below. +• Do Step 1 to Step 5 ∀t ∈ T , j ∈ V\ {0}, c ∈ Cj +Step 1: Find the model parameters in (18) namely, σP V,t +cj +αt +cj and βt +cj, by using the forecasted SPV generation µP V,t +cj +. +It is assumed that PVcap,cj ∀j ∈ V\ {0} , c ∈ Cj are known +prior. The standard deviation σL,t +cj +in (17) is taken as 4% of +the mean real power consumption µL,t +cj +of PDFL(.) [12]. +Step 2: Discretize the normal and beta distributions de- +scribed in (17) and (18) into 7 intervals. For this, 7 possible +realizations for each PDF are calculated as µ, µ + σ, µ − σ, +µ + 2σ, µ − 2σ, µ + 3σ and µ − 3σ. Then, their respective +probability densities are calculated from (17) and (18). Once +the probability densities are available, the probability for the +occurrence of each SPV generation and real power consump- +tion is found by taking the product of probability density and +the width of each discrete interval (i.e. σ). +Step 3: Normalize the calculated probabilities of real power +consumption and SPV generation. This is done by taking the +sum of the probabilities, and dividing each probability by the +sum [8], [16], [17]. This should be done for the 7 realizations +obtained from each PDF separately. Since the continuous PDFs +are approximated by discretization, sum of the 7 probabilities +will only be close to unity but not exactly equal to 1. Hence, +the normalization guarantees that the sum of the probabilities +will be precisely equal to unity [8], [16], [17]. +Step 4: Use the roulette wheel mechanism (RWM) explained +in [8], [16], [17] to construct two roulette wheels in the range +[0,1], each having 7 intervals. For this, assign the normalized +7 probabilities obtained from each PDF to [0,1] range. Hence, +each interval has a width of the normalized probability of the +respective real power consumption or SPV generation. +Step 5: Generate Nr = |R| number of random numbers +between 0 and 1, which follow the uniform distribution. Here, +the random numbers are obtained from a uniform distribution, +to guarantee they are generated without any bias. +• Do Step 6 ∀t ∈ T , j ∈ V\ {0}, c ∈ Cj, r ∈ R +Step 6: Assign each random number to the two roulette +wheels according to their magnitudes. Select φL,t +cj,r, φP V,t +cj,r , +pL +cj,r(t) and pP V +cj,r(t) from the roulette wheels corresponding to +the value of the random number, where φL,t +cj,r and φP V,t +cj,r are the +normalized probabilities of pL +cj,r(t) and pP V +cj,r(t), respectively. +In this way, the initial set of scenarios are obtained. +• Do Step 7 ∀t ∈ T , j ∈ V\ {0}, c ∈ Cj +Step 7: A scenario reduction approach is essential in sce- +nario based stochastic programs to keep the problem tractabil- +ity, while sustaining a fair approximation for the uncertainty. +Thus, the initially generated scenarios in R, are then reduced +to Ns = |S| number of scenarios to form a new scenario set +S, by using the K-Means clustering algorithm [8], [18]. This +will generate a new set of values for the probabilities and their +realizations as φL,t +cj,s, φP V,t +cj,s , pL +cj,s(t), pP V +cj,s(t) ∀s ∈ S. The K- +Means clustering method is illustrated in Algorithm 1. +Algorithm 1 K-Means Clustering Algorithm +1: Input φL,t +cj,r ∀j ∈ V\ {0} , c ∈ Cj, t∈ T , r∈ R +2: for each t in T do +3: +for each j in V\ {0} do +4: +for each c in Cj do +5: +Randomly initialize the centroids of K-Means clus- +ters as Z = +� +φL,t +cj,1, ..., φL,t +cj,s, ..., φL,t +cj,Ns +� +, where +|Z| = Ns +6: +for each s in S do +7: +As ← ∅ +8: +end for +9: +while centroids of clusters do not change do +10: +for each r in R do +11: +s∗ ← argmin +s +���φL,t +cj,r − φL,t +cj,s +��� +12: +As∗ ← As∗ ∪ +� +φL,t +cj,r +� +13: +end for +14: +for each s in S do +15: +φL,t +cj,s ← +1 +|As| +� +φL,t +cj,r∈As φL,t +cj,r +16: +end for +17: +end while +18: +end for +19: +end for +20: end for +21: Repeat Step 1 to Step 20 for the inputs φP V,t +cj,r , pL +cj,r(t) and +pP V +cj,r(t), separately ∀j ∈ V\ {0} , c ∈ Cj, t∈ T , r∈ R +22: Return φL,t +cj,s, φP V,t +cj,s , pL +cj,s(t), pP V +cj,s(t) ∀j ∈ V\ {0} , c ∈ +Cj, t∈ T , s∈ S +• Do Step 8 ∀t ∈ T , s ∈ S +Step 8: Calculate the overall probability ωs,t in (19), which +gives the probability for the occurrence of scenario s at time +t. The numerical values found for ωs,t, pL +cj,s(t), and pP V +cj,s(t) +∀j ∈ V\ {0} , c ∈ Cj, t∈ T , s∈ S are then fed into the system +models and optimization framework in Section III and V. +ωs,t = +��N +j=1 +�� +c∈Cj φL,t +cj,sφP V,t +cj,s +�� +�Ns +s=1 +��N +j=1 +�� +c∈Cj φL,t +cj,sφP V,t +cj,s +�� +(19) + +6 +V. MULTI-OBJECTIVE OPTIMIZATION FRAMEWORK +In this paper, it is aimed to minimize the investment cost of +the CES as a planning objective, and minimize the CES opera- +tion cost and the social costs of the customers as the operation +objectives. Thus, a multi-objective function is formulated by +combining both planning and operation objectives. +A. Objective Functions +1) Minimizing the Investment Cost of the CES: The in- +vestment cost for a CES can be expressed as (20) [8], [19]. +The first term of (20) relates the investment cost for the rated +power of the CES, and latter for the capacity of the CES. Since +minimizing the investment cost is a planning objective, it is not +impacted by the uncertainty of the real power consumption and +the SPV generation. Thus, (20) is independent of the scenarios. +fInv,cost = ρCES(CCES,Inv +Rate +pRate +j ++ CCES,Inv +Cap +ECap +j +) (20) +Here, ρCES = +d(1+d)τ +(1+d)τ −1, where ρCES, d, and τ are the +annual cost of the CES, discount rate and the CES life time +(in years), respectively. Also, CCES,Inv +Rate +and CCES,Inv +Cap +are +the CES investment cost per kW (in AUD/kW) and the CES +investment cost per kWh (in AUD/kWh), respectively. +2) Minimizing the Operation Cost of the CES: The cost for +operating the CES is given by (21) [8]. Since (21) illustrates +an operation objective, it is also a function of the scenarios. +fop,cost = +� +t∈T +� Ns +� +s=1 +ωs,t +� +CCES,oppCES +j,s +(t) +� +� +(21) +where pCES +j,s +(t) = ηchpCES,ch +j,s +(t) − +1 +ηdis pCES,dis +j,s +(t) and +CCES,op is the CES operation cost per kW (in AUD/kW). +3) Minimizing the Social Costs of the Customers: Cus- +tomers incur a cost or earn a revenue for trading energy with +the CES and the grid, which is jointly named as the social +costs as given in (22). Its first term denotes the energy trading +cost with the grid, and latter for trading energy with the CES. +Similar to (21), as the social costs of the customers is also an +operation objective, fC,cost is a function of the scenarios. +fC,cost = +� +t∈T +Ns +� +s=1 +ωs,t +� +λG(t) +N +� +j=1 +� +c∈Cj +pG +cj,s(t) ++ λCES(t) +N +� +j=1 +� +c∈Cj +pCES +cj,s (t) +� +∆t +(22) +Here, λG(t) and λCES(t) are the grid energy price and CES +provider’s energy price at time t, respectively. We adopt a one- +for-one non-dispatchable energy buyback method for λG(t), to +value energy imports and exports from/to the grid equally [20]. +B. Optimization Problem +The three objective functions which determine the CES +planning and its operation, are combined together to form +a multi-objective function as given in (23). The objective +functions are normalized by their corresponding nadir and +utopia points to attain a Pareto optimal solution for each +objective compatible with the weights assigned for them [21]. +x∗ = argmin +x∈X +w1 +� +fInv,cost−f utopia +Inv,cost +f Nadir +Inv,cost−f utopia +Inv,cost +� ++ w2 +� +fop,cost−f utopia +op,cost +f Nadir +op,cost−f utopia +op,cost +� ++w3 +� +fC,cost−f utopia +C,cost +f Nadir +C,cost−f utopia +C,cost +� +(23) +Here, +x = (Lj, pRate +j +, ECap +j +, pCES,ch +j +, pCES,dis +j +, pCES +cj +, pG +cj) +(24) +where x is the decision variable vector. Here, Lj, Ecap +j +and pRate +j +are the vectors of the optimal CES location, CES +capacity and its rated power, respectively. Vectors of the CES +charging power, CES discharging power, power exchange with +the CES and the grid by the customers are given by pCES,ch +j +, +pCES,dis +j +, pCES +cj +and pG +cj, respectively. The feasible set is +given by X, which is constrained by (1)-(16). Furthermore, the +calculation of the utopia and nadir values for each objective +function are done in line with the techniques mentioned in +[21]. Also, w1, w2 and w3 are the weight coefficients of +each objective function. The implementation of the overall +optimization framework is succinctly given in Algorithm 2. +Algorithm 2 Algorithm to Run the Stochastic Multi-Objective +Optimization +1: Input µL,t +cj , µP V,t +cj +∀j ∈ V\ {0} , c ∈ Cj, t∈ T +2: Initialize the model parameters used. +3: Execute Step 1 to Step 8 detailed in Section IV-C, in- +cluding the Algorithm 1, to model the uncertainty of the +real power consumption and the SPV generation of the +customers. +4: Return ωs,t, pL +cj,s(t), and pP V +cj,s(t) ∀j ∈ V\ {0} , c ∈ +Cj, t∈ T , s∈ S. +5: Solve the multi-objective function in (23), subject to the +set of constraints (1) - (16), as a MILP. +VI. NUMERICAL AND SIMULATION RESULTS +In the simulations, a radial distribution network with 7- +nodes given in Fig. 2 was considered, and its line data +can be found in [22]. The forecasted SPV generation and +real power consumption data of 30 residential customers in +an Australian community, for a period of 1 year, measured +in 1-hour time intervals were used for simulations [23]. +Here, all the residential customers generate SPV power and +consume real power. Nevertheless, due to the lack of real +data on customers’ reactive power consumption, it was not +considered for the simulations. Also, we randomly assigned +the 30 customers for all the nodes except for the slack + +7 +Fig. 2: The 7-Node LV radial feeder with the number of +customers marked at each node +Fig. 3: Variation of the grid energy price λG(t) with time +node (see Fig. 2). Therefore, �N +j=1 |Cj| = 30. Additionally, +|V0| += +1p.u., |Vmin| += +0.95p.u., |Vmax| += +1.05p.u., +pRate +max = 200kW, ECap +min = 50kWh, ECap +max = 1000kWh, +ηch += 0.98, ηdis += 1.02, ηmin += 0.05, ηmax += 1, +ε = 0.0001kWh, ∆t = 1h, d = 0.1, τ = 12.5 years, +CCES,Inv +Rate += 463AUD/kW, CCES,Inv +Cap += 795AUD/kWh +and CCES,op = 0.69AUD/kW. Moreover, the PV capacities +of the customers (i.e. PVcap,cj) were obtained from [23]. Also, +the values for CCES,Inv +Rate +, CCES,Inv +Cap +and CCES,op were taken +assuming Li-ion as the CES technology [19]. +The three objectives of the multi-objective function in (23), +were weighted according to their importance. For this, we +used the analytic hierarchy process (AHP) illustrated in [24]. +We assigned an equal importance for fInv,cost and fop,cost, +and a strong importance for fC,cost compared to fInv,cost and +fop,cost. Therefore, the values of w1, w2, w3 were calculated +as 1/7, 1/7, 5,7, respectively [24]. After computing the weight +coefficients, the simulations were done over a period of 1 year +(i.e. |T | = 8760) using the CPLEX solver in Python-Pyomo. +A. Case Study I: Comparison of the Proposed Optimization +Framework With its Corresponding Deterministic Model +To understand the impact of uncertainty on the optimal +planning and scheduling decisions of a CES, we compared +the results of the proposed stochastic optimization framework +and its corresponding deterministic model. The simulations +were done for our stochastic model by considering 50 initial +scenarios, which is then reduced to 10 (i.e. Ns = 10) by using +the K-Means clustering algorithm. Simulations for the deter- +ministic model were obtained by neglecting the uncertainty of +real power consumption and SPV generation of the customers. +The same set of constraints and the multi-objective function +used for the proposed stochastic model (i.e. (1)-(16) and (23) ), +were used for the deterministic model as well, while excluding +the scenario dependency of (1)-(9), (13)-(16) and (23). +In the simulations, a time-of-use (TOU) grid energy price +λg(t) shown in Fig. 3 was used [25]. Due to the lack of +accurate real data about the CES provider’s energy price +λCES(t), we assumed three different energy price schemes +for it as (i) λCES(t) = λG(t), (ii) λCES(t) = 0 and (iii) +λCES(t) = λG,avg ,where λG,avg = +�24 +t=1 λG(t) +24 +. A summary +of the numerical results obtained for the two types of the +optimization models are given in Table I. For the three energy +price schemes of λCES(t), in both deterministic and proposed +stochastic models, the optimal CES location is node 7. +The pRate +j +and ECap +j +in the stochastic model are higher than +their values in the deterministic model. In the stochastic model, +due to the impact of the higher values of the realizations with +respect to µL,t +cj , µP V,t +cj +∀j ∈ V\ {0} , c ∈ Cj, t∈ T , both pRate +j +and ECap +j +will be greater than their values in the deterministic +model. Moreover, in both models, the values of the planning +decisions namely, Lj, pRate +j +and ECap +j +for their respective +models have not changed irrespective of the CES provider’s +energy price scheme. This has happened, as the planning +decisions are independent from operation variables, and thus +from λCES(t). Hence, the investment costs in the deterministic +and stochastic models are AUD 32228 and 33695, respectively, +irrespective of the CES provider’s energy price scheme. +The operation objectives consider minimizing the CES +operation cost fop,cost, and the social costs of the customers +fC,cost. According to Table I, fop,cost is same for the deter- +ministic model regardless of the CES provider’s energy price +scheme. This is resulted as fop,cost is a function independent +of λCES(t) (see (21). Besides, when λCES(t) = λG(t) and +λCES(t) = λG,avg(t), fop,cost in the stochastic model is +higher than the corresponding deterministic model values. +Since the stochastic model takes into account the uncer- +tainty of SPV generation and real power consumption of +the customers, the costs in the stochastic model are higher +than the ones in the deterministic model. This behaviour +is seen for fC,cost as well when λCES(t) = λG(t) and +λCES(t) = λG,avg(t). Additionally, as fC,cost < 0 when +λCES(t) = 0, it implies that the customers earn a revenue. The +customers minimize their social costs by importing power only +from the CES as λCES(t) = 0. Also, the customers export +power solely to the grid to maximize their social revenue. This +is the intuition for fC,cost being negative when λCES(t) = 0 +for both deterministic and stochastic models. This is discussed +in detail in Section VI-B-2. In the proposed stochastic model, +fop,cost for the three energy price schemes of λCES(t) are +different from each other. As the set of random numbers +generated during stochastic modelling are unique and different +for every execution of Algorithm 2, a unique set of values +for ωs,t, pL +cj,s(t), and pP V +cj,s(t) ∀j ∈ V\ {0}, c ∈ Cj, t∈ T , +s∈ S are obtained. This results in getting different values for +fop,cost, irrespective of fop,cost being independent of λCES(t). + +2 +6 +External +Transformer +IC2l = 4 +IC6l = 4 +Grid +22/0.4 kvV +0 +1 +3 +4 +L80 +ICil = 3 +IC3/ = 5 +IC4l = 6 +ICzl = 5 +5 +ICsl = 30.55 +0.50 +0.45 +0.40 +0.35 +0.30 +0.25 +T2 +T3 +T4 T5 +0.20 +0 +5 +10 +15 +20 +25 +Time Duration (24 Hours)8 +TABLE I: RESULTS OF THE DETERMINISTIC AND PROPOSED STOCHASTIC MODELS +Optimal +CES Node +Lj +Optimal CES +Rated Power1 +pRate +j +(kW) +Optimal CES +Capacity1 +ECap +j +(kWh) +CES Investment +Cost1 (AUD) +fInv,cost +CES Operation +Cost1(AUD) +fop,cost +Social +Costs1 (AUD) +fC,cost +Deterministic +λCES(t) = λG(t) +7 +72 +240 +32228 +24674 +124830 +Model +λCES(t) = 0 +7 +72 +240 +32228 +24674 +-94681 +λCES(t) = λG,avg +7 +72 +240 +32228 +24674 +70810 +Proposed Stochastic +Model (Ns = 10) +λCES(t) = λG(t) +7 +78 (7.69%) +249 (3.61%) +33695 (4.35%) +27266 (9.51%) +129429 (3.55%) +λCES(t) = 0 +7 +78 (7.69%) +249 (3.61%) +33695 (4.35%) +29712 (16.96%) +-98201 (3.58%) +λCES(t) = λG,avg +7 +78 (7.69%) +249 (3.61%) +33695 (4.35%) +29347 (15.92%) +75263 (5.92%) +1 Increment percentage values are computed with respect to their corresponding values found in the deterministic model +Fig. 4: (a) Total power exchange with the grid and the CES by +customers, (b) CES charging/discharging power and temporal +variation of CES energy - When Ns = 10, λCES(t) = λG(t) +B. Analysis of the Results of CES Scheduling and Mutual +Power Exchanges Between the CES, the Grid and Customers +The results obtained for the proposed stochastic optimiza- +tion framework, for different energy price schemes of the CES +provider are detailed next. Here, we do the analysis for a +randomly selected a day, for a duration of 24 hours. +1) When λCES(t)=λG(t): Fig. 4(a) depicts the total power +exchange with the grid (brown plot) and the CES (orange plot) +by the customers. As the customers do not have sufficient +SPV generation during T1, T3 T4 and T5, they import power +from the grid and the CES. Besides, the customers export their +excess generation to the CES and the grid during T2. Since +λCES(t)=λG(t), the customers do not have any preference +whether to exchange power with the grid or the CES. The +charging and discharging pattern of the CES (red plot), and the +CES energy level variation (green plot) with time are shown +in Fig. 4(b). During T1, the CES charges, and by the end of +T1, it discharges completely. The CES is fully discharged by +the end of T1 to exploit its full capacity to charge from the +excess SPV generation during T2. This is evident as the CES +energy level has reached its full capacity of 249 kWh during +T2. During T3, the CES exports its power to the customers, +and at the end of the day, CES reaches its initial energy level. +Fig. 5: (a) Total power exchange with the grid and the CES by +customers, (b) CES charging/discharging power and temporal +variation of CES energy - When Ns = 10, λCES(t) = 0 +2) When λCES(t)=0: In this case, as λCES(t)=0, the +customers neither incur a cost nor earn a revenue when trading +energy with the CES. According to Fig. 5(a), during T1, the +customers import power only from the CES, as the customers +do not incur a cost for importing power from the CES. The +same trend is followed by the customers during T3, T4 and T5. +During T2, the customers export the excess SPV generation +to the grid. Also, as λCES(t)=0, the customers do not export +power to the CES as they cannot earn a revenue from the +CES provider. Hence, when λCES(t) = 0, the customers do +not incur a cost for all time. Instead, they earn a revenue from +the grid which is shown in Table I as a negative value for +fC,cost. Fig. 5(b) shows the charging and discharging pattern +of the CES, including its energy level variation with time. +3) When λCES(t)=λG,avg: +In this paper, λG,avg += +0.34180 AUD/kWh. Hence, during T3 λCES(t) < λG(t), +and during T1, T2 T4, T5 λCES(t) > λG(t). As seen in Fig. +6(a), during T1, the customers import power only from the +grid. Since λCES(t) > λG(t) during this time period, it is not +economically beneficial for the customers to import expensive +power from the CES. During T2, in which the time period +with high SPV generation, the customers export the excess +SPV generation to the CES as λCES(t) > λG(t). Hence, the +customers can earn a higher revenue from the CES provider. + +(a) +100 +Power (kW) +-100 +0 +10 +15 +oZ +25 +Discharging Power (kW) +(b) + CES Charging and, +50 +CES Energy (kWh) +200 +0 +100 +50 +0 +5 +10 +15 +25 +Time Duration (24 hours) + pa. (t) +/= 1c E G +5=1 + pa5(t) +/= 1cE G +-(a) +Power (kw) +0 +-200 +0 +n +10 +15 +20 +25 +Discharging Power (kW) +(b) +CES Charging and. + 50 + Energy (kWh) +200 +0 +100 +50 +CES +0 +-5 +10 +15 +20 +25 +Time Duration (24 hours) + pa. (t) +5 = 1 +/= 1c E G +5=1 +M +Zpas(t) +5 = 1 +/= 1cE G +5 = 19 +TABLE II: RESULTS OF PROPOSED STOCHASTIC MODEL AND CASE I-IV - (WITH Ns = 10, λCES(t) = λG,avg) +Case / CES Node +Optimal CES +Rated Power1 +pRate +j +(kW) +Optimal CES +Capacity1 ECap +j +(kWh) +CES Investment +Cost1 (AUD) +fInv,cost +CES Operation +Cost1 (AUD) +fop,cost +Social +Costs1 (AUD) +fC,cost +Cumulative Cost1 +(AUD) +Proposed Model- 7 (optimal) +78 +249 +33695 +29347 +75263 +138305 +Case I - 3 (chosen) +124 (37.10%) +380 (34.47%) +51688 (34.81%) +44900 (34.64%) +77703 (3.14%) +174291 (20.65%) +Case II - 4 (chosen) +98 (20.41%) +303 (17.82%) +41217 (18.25%) +36723 (20.09%) +76973 (2.22%) +154913 (10.72%) +Case III - 5 (chosen) +127 (38.58%) +366 (31.97%) +50258 (32.96%) +39093 (24.93%) +77024 (2.29%) +166375 (16.87%) +Case IV - 6 (chosen) +96 (18.75%) +310 (19.68%) +41837 (19.46%) +35578 (17.51%) +76491 (1.61%) +153906 (10.14%) +1 Increment percentage values are computed with respect to their corresponding values found in the proposed model +Fig. 6: (a) Total power exchange with the grid and the CES by +customers, (b) CES charging/discharging power and temporal +variation of CES energy - When Ns = 10, λCES(t) = λG,avg +Since λCES(t) < λG(t) during T3, the customers import +power from the CES, so that they have to pay less for the +imported power. During T4 and T5, the customers import +power only from the grid as λCES(t) > λG(t). Fig. 6(b) +shows the charging and discharging pattern, and the temporal +variation of the CES energy level with time. +C. Case Study II: Proposed Optimization Framework Vs Mod- +els With Arbitrary CES Locations +The merits of a CES may be fully exploited if its both +planning and scheduling are optimized simultaneously. To +test this, we compared our proposed model with four dif- +ferent cases that randomly choose the CES location, with +λCES(t) = λG,avg(t), Ns = 10, and taking into account the +uncertainty of real power consumption and SPV generation. +As given in Table II, the CES is allocated for nodes 3, 4, +5 and 6 which represent Case I, II, III and IV, respectively. +For Case I-IV, we considered the constraints (1)-(16), and +the objective function in (23). Additionally, Lj = 1 where +j ∈ {3, 4, 5, 6} in (10)-(12) for Case I-IV, respectively. In +Case I-IV, pRate +j +and ECap +j +are significantly higher than in our +proposed stochastic model. This has resulted in a substantial +increase of the CES investment and operation costs. But, +the social costs show only a minor increase for Case I-IV +compared with the increase of the investment and operation +costs of the CES. This can be explained in terms of the weight +coefficients used for weighting the objective functions in (20)- +(22). Since w1, w2 and w3 are 1/7, 1/7 and 5/7, respectively, +the highest importance is given for minimizing the social costs. +Hence, the optimization solver tries to maintain the social costs +as much as close to fC,cost obtained in our proposed model. +However, this comes at an expense as fInv,cost and fop,cost +which have a less significance, increase significantly. But, the +cumulative cost (i.e. sum of fInv,cost, fop,cost and fC,cost ) +is the least for the proposed model, while for Case I-IV, it is +about 10-21% higher than the cost in our proposed model. +D. Case Study III: Impact of Scenario Reduction Approaches +In this section, we present a comparison of the results +obtained for our optimization model utilizing two scenario re- +duction methods namely, backward scenario reduction (BSR) +method and K-Means clustering algorithm. In BSR method, +the initial number of scenarios are reduced by minimizing the +Monge-Kantorovich distance between the scenarios in both +initial and reduced scenarios sets. Thereby, the initial scenarios +are eliminated iteratively one by one, until the desired number +of elements in the reduced scenario set is reached. Further +explanation about the BSR method can be found in [12]. +In this case study, we considered λCES(t) = λG,avg, and +the initial number of scenarios as 50. The proposed opti- +mization framework was implemented under the two scenario +reduction approaches by taking Ns = 10 and Ns = 30 for each +method. The numerical results obtained for this case study are +summarized in Table III. Note that, for all the cases, node 7 +was recorded as the optimal CES location. +The costs for all the three objectives are the highest for the +model which did not use a scenario reduction method. This is +because, the model with Ns = 50 captures more uncertainty +of the real power consumption and SPV generation, so that the +optimal CES rated power and the capacity are higher than the +models with a lesser number of scenarios. For both K-Means +and BSR methods, even with a different number of reduced +scenarios (i.e. Ns = 10 and Ns = 30), the CES planning +aspects have not changed. This occurs as the planning aspects +are independent of the number of scenarios. Nevertheless, as +the operation decisions are scenario dependent, the operation +cost of the CES, and the customers’ social costs have changed +according to Ns. This trend is observed in the results obtained +for the model which used the BSR method as well. + +(a) +Power (kW) +100 +0 +100 +-200 +0 +n +10 +15 +20 +25 +Discharging Power (kW) +(b) + CES Charging and, +CES Energy (kWh) +200 +0 +100 +50 +0 +5 +10 +15 +20 +Time Duration (24 hours) + pa. (t) +5= 1 +/= 1c E G +5=1 +MN + pa5(t) +ws, EE5(t) +/=1cE G +-10 +TABLE III: RESULTS FOR DIFFERENT SCENARIO REDUCTION METHODS - (WITH λCES(t) = λG,avg) +No. of +scenarios +Computational +time1 +(min) +Optimal CES +Rated Power1 +pRate +j +(kW) +Optimal CES +Capacity1 +ECap +j +(kWh) +CES Investment +Cost1 (AUD) +fInv,cost +CES Operation +Cost1 (AUD) +fop,cost +Social +Costs1 (AUD) +fC,cost +Without Scenario +Reduction +Ns = 50 +91 +79 +257 +34573 +38434 +91321 +K-Means +Clustering Algorithm +Ns = 10 +38 (58.24%) +78 (1.27%) +249 (3.11%) +33695 (2.54%) +29347 (23.64%) +75263 (17.58%) +Ns = 30 +53 (41.76%) +78 (1.27%) +249 (3.11%) +33695 (2.54%) +30019 (21.89%) +89425 (2.08%) +Backward Scenario +Reduction Method +Ns = 10 +41 (54.95%) +78 (1.27%) +250 (2.72%) +33735 (2.42%) +29785 (22.50%) +75709 (17.10%) +Ns = 30 +57 (37.36%) +78 (1.27%) +250 2.72%) +33735 (2.42%) +30664 (20.22%) +89981 (1.47%) +1 Decrement percentage values are computed with respect to their corresponding values found without scenario reduction +The models with reduced number of scenarios have con- +verged for a solution in a lesser time than the case with +Ns = 50. Scenario reduction methods like K-Means clustering +algorithm and BSR method play a key role in reducing the +computational time while maintaining the problem tractability. +However, according to the results obtained for the models +which used the K-Means and the BSR method, it is not +conclusive to claim which scenario reduction method is better, +as there is no any significant difference between the results. +VII. CONCLUSION & FUTURE WORK +In this paper, we have explored how the optimization of the +planning and scheduling of a community energy storage (CES) +benefit the CES provider by minimizing the CES investment +and operation costs, and the customers by minimizing their +social costs. The uncertainty of real power consumption and +solar photovoltaic (SPV) generation of the customers have +been accounted to formulate a scenario-based stochastic op- +timization program. To reduce the computational burden of +the stochastic model, we have used the K-Means clustering +algorithm. It has been shown that, both the customers and the +CES provider can significantly minimize their personal costs +by optimizing both the CES planning and its scheduling. +Future work includes extending the proposed model for +unbalanced distribution networks, developing optimization +models for networks with multiple CES, and considering the +reactive power regulation capabilities of CES and SPV. +REFERENCES +[1] M. Shaw, B. Sturmberg, C.P. Mediwaththe, H. Ransan-Cooper, D. Tay- +lor and L. Blackhall, “Community batteries: a cost/benefit analysis,” +Technical Report, Australian National University, 2020. +[2] C.P. Mediwaththe, and L. Blackhall, “Network-Aware Demand-Side +Management Framework With A Community Energy Storage System +Considering Voltage Constraints,” IEEE Trans. Power Syst., vol. 36, +no. 2, pp. 1229–1238, 2021. +[3] C.P. Mediwaththe, and L. Blackhall, “Community Energy Storage-based +Energy Trading Management for Cost Benefits and Network Support,” +in Proc. Int. Conf. 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Eng., vol. 13, no. 1, pp. 1–35, 2004. +[25] “Origin, “VIC residential energy price fact sheet,” 2018.” [Online]. +Available: shorturl.at/gkmV5” + diff --git a/6NAzT4oBgHgl3EQff_yv/content/tmp_files/load_file.txt b/6NAzT4oBgHgl3EQff_yv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d9a8198ee9060ba6d220c73911970a1106008fa --- /dev/null +++ b/6NAzT4oBgHgl3EQff_yv/content/tmp_files/load_file.txt @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf,len=817 +page_content='1 A Stochastic Multi-Objective Optimization Framework for Planning and Scheduling of Community Energy Storage Systems K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Anuradha, Student Member, IEEE, and Chathurika P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Mediwaththe, Member, IEEE Abstract—This paper explores a methodology to optimize the planning and the scheduling of a community energy storage (CES) considering the uncertainty of real power consumption and solar photovoltaic (SPV) generation of the customers in low voltage (LV) distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' To this end, we develop a stochastic multi-objective optimization framework which mini- mizes the investment and the operation costs of the CES provider, and the social costs of the customers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' cost of customers for trading energy with the grid and the CES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The uncertainty of SPV generation and real power consumption are modelled to follow the beta and normal distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Then, the roulette wheel mechanism (RWM) is exploited to formulate a scenario-based stochastic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The initial scenarios obtained from the RWM, are then reduced by using the K-Means clus- tering algorithm, to keep the problem tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A case study highlights our model provides 10-21% more cumulative economic benefits for the customers and the CES provider, compared with the models that optimize only the CES scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, the simulation results for different energy price schemes of the CES provider reflect, the customers change their power exchange with the CES and the grid significantly, to minimize their social costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Index Terms—Community energy storage (CES), multi- objective optimization, planning and scheduling, power flow, roulette wheel mechanism (RWM), scenarios, uncertainty I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' INTRODUCTION T HE integration of solar photovoltaic (SPV) systems in low voltage (LV) distribution networks, has undergone a rapid upsurge over the last few decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' However, the intermittent and non-dispatchable nature of SPV generation, may restrict their beneficiaries such as the customers from exploiting the merits of SPV fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' These issues can be efficiently alleviated by exploiting energy storage systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Community energy storage (CES) devices are an emerging type of battery system, which is gaining increasing interest in the industry, as they can enable increased community access and network hosting capacity for renewable energy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' An energy management framework which aims at optimiz- ing only the scheduling of a CES such as its charging and discharging pattern, may not deliver the expected rewards from a CES completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, it is imperative that the planning aspects including the location, the capacity and the rated power of a CES are optimized concurrently with its K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Anuradha is with The Australian National University, Canberra, ACT 0200, Australia (email: u7146121@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Chathurika P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Mediwaththe is with The Australian National University, Canberra, ACT 0200, Australia, and also with the Commonwealth Scientific and Industrial Research Organisation, Canberra, ACT 2601, Australia (email: chathurika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='mediwaththe@csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Several studies have presented deterministic opti- mization frameworks to find the optimal CES planning and/or the scheduling, and thus achieve the objectives of different stakeholders [2]–[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, the authors have assumed both real power consumption and SPV generation of the customers are perfectly known ahead from their forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' However, due to the uncertainty of SPV generation and real power consumption of the customers, their forecast errors can be quite high at times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Eventually, this may result the optimization models de- scribed in [2]–[4] unable to achieve the objectives effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, different stakeholders have distinct objectives for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, a multi-objective optimization framework can reflect the trade-offs between those objectives comprehensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In this paper, we examine the extent to which the optimal planning and scheduling of a CES benefit different stakehold- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' To this end, we develop an energy management framework between the customers, the CES and the grid, by incorporating the uncertainty of real power consumption and SPV generation of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Additionally, we leverage a linearized power flow model with our energy management framework to formu- late a mixed integer linear program (MILP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In summary, the main contributions of this paper can be highlighted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' We develop a stochastic multi-objective optimization framework which optimizes both planning and scheduling of a CES for benefiting (i) the CES provider by minimiz- ing the investment and the operation costs of the CES, and (ii) the customers by minimizing their social costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The proposed optimization framework is capable of pro- viding significantly higher economic benefits for the CES provider and the customers than in the models which arbitrarily choose the CES connected node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A case study compares our proposed stochastic model with its corresponding deterministic model for different energy price schemes of the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This study enables to understand how the economic benefits for the CES provider, and for the customers change due to the uncertainty of real power consumption and SPV generation of the customers, and the energy price scheme of the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Prior research work have presented optimization frame- works which focus on the scheduling of the CES [2], [3], and both planning and scheduling of CES [4]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For instance, the authors of [2] have presented a method for CES scheduling, to minimize the social costs of the customers while maximizing the revenue of the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In [3], an optimization arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='01462v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='SY] 4 Jan 2023 2 framework is presented to schedule the CES, to minimize the real energy losses, and energy trading costs with the grid by the CES provider and the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The authors of [4] and [5] have presented models to optimize the CES planning and scheduling simultaneously, to enhance the hosting capacity of LV networks, and to mitigate the voltage excursions in three phase unbalanced LV networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, analytical methods for optimizing the CES planning and scheduling have been discussed in [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A common feature of [2]–[7] is that the authors have used deterministic models, assuming the SPV generation and the real power consumption of the customers are known ahead with no uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, those models may not be efficient in providing realistic planning and operation decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The uncertainty of real power consumption and generation from SPV have been taken into account in [8], [9] for CES management problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For instance, the authors of [8] have presented a method for optimizing both planning and scheduling of CES to accomplish multiple objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, the authors have used the normal distribution and the RWM to model the uncertainty of real power consumption and SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In [9], the authors of have investigated how the CES location impacts the voltage profile and real power losses in LV networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For this, they have arbitrarily allocated the CES at different nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In contrast to [2]–[7], our paper opti- mizes both planning and scheduling aspects of the CES taking into account the uncertainty of real power consumption and SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, our approach models the CES planning and its scheduling problem more realistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, compared to [8], [9], we present a method to minimize the personal costs of the CES provider and the customers concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Notations used in this paper are detailed in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The system models of the CES, the customers and the network power flow are presented in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The stochastic models of SPV generation and real power consumption of the customers are given in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The formulation of the multi-objective optimization framework is described in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Section VI presents the validation of the results, and Section VII gives the conclusion of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' NOTATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Stochastic Model-related Notations To formulate a scenario-based stochastic program, the un- certainty of SPV generation and real power consumption of the customers are modelled using known probability density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, a scenario represents a possible combination of SPV generation and real power consumption of all the customers together with their corresponding probabilities at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The initial set of scenarios is denoted by R, and r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Due to the computational complexity of stochastic programs which use scenarios, it is imperative to use a scenario reduction approach to reduce the number of scenarios, and keep the problem tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, a scenario reduction technique is used in this paper, and the set composed by the reduced scenarios is given by S, where s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The implementation of the scenario-based stochastic program including the scenario reduction are discussed in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 1: Mutual power exchanges between the CES, the grid and a customer in scenario s at time t B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Network and Power Flow-related Notations In this paper, a distribution network with a radial topology is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' It is described by the graph G = (V, E), where V = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=', N} is the set of all nodes, and E = {(i, j)} ⊂ V×V is the set of all lines in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Node 0 (slack node) represents the secondary side of the distribution transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The resistance and the reactance of line (i, j) are rij (in Ω) and xij (in Ω), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The set of customers at node j is represented by Cj, and c ∈ Cj ∀j ∈ V\\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, t ∈ T , where T is the set of time intervals, and ∆t is the difference between two adjacent time instances (in hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The real and reactive power flow from i to j node in scenario s at time t are represented by Pij,s(t) (in kW) and Qij,s(t) (in kVAR), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, ∀j ∈ V\\ {0}, t∈ T , s∈ S, the real power absorption, reactive power absorption, voltage and squared voltage are given by pj,s(t) (in kW), qj,s(t) (in kVAR), Vj,s(t) (in V ) and Uj,s(t) (in V 2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' SYSTEM MODELS This section first presents the network power flow model, followed by the power exchange model of the customers, and the CES model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' We consider each customer, the CES and the grid can exchange power with each entity as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Power Flow Model of the Network In this work, we consider the power absorption for the nodes as positive, and the power injections from the nodes as negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Additionally, it is considered that there are multiple customers at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' To model the network power flows, we use the linearized power flow equations (1) and (2) from the Distflow model in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Pij,s(t) = pj,s(t) + � k:j→k Pjk,s(t) ∀(i, j) ∈ E, , t∈ T , s∈ S (1) Qij,s(t) = qj,s(t)+ � k:j→k Qjk,s(t) ∀(i, j) ∈ E, , t∈ T , s∈ S (2) External Grid Grid pCES,s(t) < 0 pCES,s(t) ≥ 0 0 > (0)sgd pejs(t) > 0 pcS(t) > 0 田田 pCS (t) < 0 cth Customer CES at node j3 The real and reactive power absorbed by the node j at time t in scenario s, can be expressed as (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The real power absorption for the CES installed node is governed by (3a), while for all the other nodes (except slack node), it is (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Th equation (4) handles the reactive power absorption for all nodes, and we assume the CES and the SPV devices operate at unity power factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The real power consumption, reactive power consumption and SPV generation of the customer c at node j at time t in scenario s are given by pL cj,s(t) (in kW), qL cj,s(t) (in kVAR) and pP V cj,s(t) (in kW), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For j ∈ V\\ {0} , t∈ T , s∈ S, the CES charging and discharging power are represented by pCES,ch j,s (t) and pCES,dis j,s (t), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) = � c∈Cj pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t)− � c∈Cj pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t)+pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='ch j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t)−pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='dis j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) (3a) pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) = � c∈Cj pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t)− � c∈Cj pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (3b) qj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) = � c∈Cj qL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (4) The linearized Distflow equations described in (1)-(2) can be explicitly written as (5),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' where U0 = |V0|2 and U=|V(t)|2 are the vectors of the squared voltage magnitude of the slack node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' and the squared voltage magnitudes of all other nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 1 symbolizes a vector of all ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Moreover, p and q are the vectors of real and reactive power absorption at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The matrices ˜R and ˜X ∈ RN×N have the elements Rij = 2 � (m,n)∈Li∩Lj rmn and Xij = 2 � (m,n)∈Li∩Lj xmn, respectively, where Li is the set of lines on the path which connects node 0 and i [2], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, (6) ensures the squared voltage magnitude at each node is within its allowable voltage magnitude limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, Umin = ��V 2 min �� 1 and Umax = ��V 2 max �� 1, where Vmin and Vmax are the allowable lower and upper bound of voltage, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' U = U01 − ˜Rp − ˜Xq ∀t∈ T , s∈ S (5) Umin ≤ U ≤ Umax ∀t∈ T , s∈ S (6) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Power Exchange Model of the Customers According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 1, pG cj,s(t) and pCES cj,s (t) denote the real power exchange with the grid and the CES by the customer c at node j at time t in scenario s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' When pG cj,s(t) > 0, it suggests a power import from the grid by a customer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' On the contrary, when that customer exports power back to the grid, it will be pG cj,s(t) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The same sign convention is used for customer power exchanges with the CES, and CES power exchange with the grid pG CES,s(t) (in kW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' When a customer’s SPV generation is insufficient to fulfill its real power consumption, that deficit is attained by importing power from the grid and the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is mathematically represented by (7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Nevertheless, the imported power from each entity should be within the deficit quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is ensured by (7b) and (7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Besides, when that customer has excess SPV generation, it exports its surplus to the grid and the CES, which is mathematically interpreted by (8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Similar to (7b) and (7c), the exported power to the CES and the grid should not exceed the mismatch of the SPV generation and the real power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is demonstrated by (8b) and (8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Considering the ability of the CES to exchange power with the grid and the customers, pG CES,s(t) can be expressed in terms of pCES cj,s , pCES,ch j,s (t) and pCES,dis j,s (t) as in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' If pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ≥ pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t): 0 ≤ pG cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) + pCES cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) = pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) − pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) (7a) 0 ≤ pG cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ≤ pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) − pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) (7b) 0 ≤ pCES cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) − pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) (7c) Otherwise: pG cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) + pCES cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) = pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) − pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ≤ 0 (8a) pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) − pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ≤ pG cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ≤ 0 (8b) pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) − pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ≤ pCES cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ 0 ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (8c) pG CES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) = �N j=1 �� c∈Cj pCES cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) + pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='ch j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) − pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='dis j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) � ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (9) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Community Energy Storage Model In this paper, we assume the CES is owned by a third party, and we designate the owner as the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The planning constraints of the CES are given by (10)-(12), and the operation of the CES is mathematically modeled by (13)-(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The equation (10) finds the optimal CES node, and we use a binary variable Lj, in which Lj = 1 suggests node j as the optimal CES node, and Lj = 0 means there is no CES at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, (10) guarantees only a single CES is installed in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The inequalities in (11) and (12) find the optimal CES capacity Ecap j (in kWh) and its rated power pRate j (in kW) at node j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, Ecap min (in kWh) and Ecap max (in kWh) are the minimum and maximum CES capacity limits, and pRate max is the maximum CES rated power limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' N � j=1 Lj = 1 ∀j ∈ V\\ {0} , Lj ∈ {0, 1} (10) 4 LjECap min ≤ ECap j ≤ LjECap max ∀j ∈ V\\ {0} , Lj ∈ {0, 1} (11) 0 ≤ pRate j ≤ LjpRate max ∀j ∈ V\\ {0} , Lj ∈ {0, 1} (12) The inequality described in (13) avoids simultaneous charg- ing and discharging of the CES, while guaranteeing the CES charging and discharging power does not exceed its optimal rated power pRate j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For this, a binary variable Bj and two additional variables namely, y and z are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, Bj ∈ [0, 1] and y, z ∈ R+ such that these variables also satisfy (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' When the CES is charging, Bj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, z = pCES,dis j,s (t) = 0 ac- cording to (13d) and (13e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, as stated in (13c), y = pRate j and thus, 0 ≤ pCES,ch j,s (t) ≤ y = pRate j ≤ pRate max .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' When the CES discharges, Bj = 0, and hence, y = pCES,ch j,s (t) = 0 according to (13a) and (13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In this instance, z = pRate j , which generates the inequality 0 ≤ pCES,dis j,s (t) ≤ z = pRate j ≤ pRate max .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The equation (14) illustrates how the CES energy level changes with time t, where ECES j,s (t) (in kWh) is the energy level of the CES at node j at time t in scenario s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Additionally, ηch and ηdis are the charging and discharging efficiency of the CES, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Furthermore, the CES energy level should be maintained with in the minimum and maximum state of charge (SoC) levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is regulated by (15), where ηmin and ηmax represent minimum and maximum percentage coefficients of the CES capacity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, it is required to guarantee the continuity of the CES operation over the next day, and thus, the CES energy level at the end of the day should be kept approximately same as the initial energy level at the start of the day .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is managed by (16) [2], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, ε is a small positive number (in kWh), and tn ∈ TN where TN = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='., |T | /24}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 0 ≤ pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='ch j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ y (13a) pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='ch j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ y ≤ pRate max Bj (13b) −pRate max (1 − Bj) ≤ y − pRate j ≤ 0 (13c) 0 ≤ pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='dis j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ z (13d) pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='dis j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ z ≤ pRate max (1 − Bj) (13e) −pRate max Bj ≤ z − pRate j ≤ 0 ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (13f) ECES j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) = ECES j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t − 1) + (ηchpCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='ch j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) − 1 ηdis pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='dis j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t))∆t ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (14) ηminECap j ≤ ECES j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (t) ≤ ηmaxECap j ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (15) ��ECES j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (24tn) − ECES j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s (0) �� ≤ ε ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' tn∈ T N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S (16) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' STOCHASTIC MODELS In this section, the uncertainty modeling of real power consumption and SPV generation are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Similar to [12], [13], the uncertainty of real power consumption and SPV generation are modelled by the probability density functions (PDFs) of normal and beta distributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Uncertainty of the Real Power Consumption The uncertainty of real power consumption of the customers follows the probability density function of normal distribution PDFL(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=') given in (17) [12], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The forecasted real power consumption of the customer c at node j at time t is consid- ered as the mean real power consumption µL,t cj of PDFL(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=') [12], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The standard deviation and a sample real power consumption are denoted by σL,t cj and XL,t cj , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' PDFL(X) = 1 σL,t cj √ 2π e −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='5 � XL,t cj −µL,t cj σL,t cj �2 ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T (17) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Uncertainty of the SPV Generation As mentioned in [12], [13], the uncertainty of SPV gener- ation of the customers mimic the probability density function of beta distribution PDFP V (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=') as given in (18a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, the forecasted SPV generation of the customer c at node j at time t is taken as the mean SPV generation µP V,t cj of PDFP V (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The equations (18a)-(18d) describe the relationship between the shape parameters αt cj, βt cj, a sample SPV generation XP V,t cj , SPV capacity of a customer PVcap,cj, the mean µP V,t cj and the standard deviation σP V,t cj of PDFP V (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=') [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, Γ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=') represents the gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' PDFP V (X) = � � � � � � � � � � � � � Γ(α+β) Γ(α)Γ(β) � XP V,t cj �αt cj−1 � 1 − XP V,t cj �βt cj−1 ∀ 0 < XP V,t cj < 1 αt cj, βt cj ≥ 0 0 Otherwise (18a) µP V,t cj = αt cj αt cj + βt cj (18b) (σP V,t cj )2 = αt cjβt cj � αt cj + βt cj �2 � αt cj + βt cj + 1 � (18c) σP V,t cj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='2µP V,t cj PVcap,cj + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='21 ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T (18d) 5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Scenario-based Stochastic Program Since the normal and beta distributions are continuous PDFs, they represent infinite number of realizations of the random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, a realization refers to a sample real power consumption or SPV generation of a customer at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A large number of realizations can model the uncertainty better at the expense of a large computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' But, continuous PDFs approximated as discrete functions by a finite number of realizations, as given in [15], can be used to eliminate the similar and less probable real power con- sumption or SPV generation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, the discretization of continuous PDFs, reduces the complexity of uncertainty modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, both normal and beta PDFs are approximated as discrete functions by 7 realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, the approximated discrete functions are constructed to have 7 intervals, with every interval having a width of a standard deviation σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The midpoint of an interval is a possible realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For instance, when the forecasted real power consumption is µ, the intervals 1-7 are centered around the 7 realizations µ, µ + σ, µ − σ, µ + 2σ, µ − 2σ, µ + 3σ and µ − 3σ as done in [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The steps of scenario generation and reduction are given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Do Step 1 to Step 5 ∀t ∈ T , j ∈ V\\ {0}, c ∈ Cj Step 1: Find the model parameters in (18) namely, σP V,t cj αt cj and βt cj, by using the forecasted SPV generation µP V,t cj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' It is assumed that PVcap,cj ∀j ∈ V\\ {0} , c ∈ Cj are known prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The standard deviation σL,t cj in (17) is taken as 4% of the mean real power consumption µL,t cj of PDFL(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=') [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Step 2: Discretize the normal and beta distributions de- scribed in (17) and (18) into 7 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For this, 7 possible realizations for each PDF are calculated as µ, µ + σ, µ − σ, µ + 2σ, µ − 2σ, µ + 3σ and µ − 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Then, their respective probability densities are calculated from (17) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Once the probability densities are available, the probability for the occurrence of each SPV generation and real power consump- tion is found by taking the product of probability density and the width of each discrete interval (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Step 3: Normalize the calculated probabilities of real power consumption and SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is done by taking the sum of the probabilities, and dividing each probability by the sum [8], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This should be done for the 7 realizations obtained from each PDF separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since the continuous PDFs are approximated by discretization, sum of the 7 probabilities will only be close to unity but not exactly equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, the normalization guarantees that the sum of the probabilities will be precisely equal to unity [8], [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Step 4: Use the roulette wheel mechanism (RWM) explained in [8], [16], [17] to construct two roulette wheels in the range [0,1], each having 7 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For this, assign the normalized 7 probabilities obtained from each PDF to [0,1] range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, each interval has a width of the normalized probability of the respective real power consumption or SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Step 5: Generate Nr = |R| number of random numbers between 0 and 1, which follow the uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, the random numbers are obtained from a uniform distribution, to guarantee they are generated without any bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Do Step 6 ∀t ∈ T , j ∈ V\\ {0}, c ∈ Cj, r ∈ R Step 6: Assign each random number to the two roulette wheels according to their magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Select φL,t cj,r, φP V,t cj,r , pL cj,r(t) and pP V cj,r(t) from the roulette wheels corresponding to the value of the random number, where φL,t cj,r and φP V,t cj,r are the normalized probabilities of pL cj,r(t) and pP V cj,r(t), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In this way, the initial set of scenarios are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Do Step 7 ∀t ∈ T , j ∈ V\\ {0}, c ∈ Cj Step 7: A scenario reduction approach is essential in sce- nario based stochastic programs to keep the problem tractabil- ity, while sustaining a fair approximation for the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, the initially generated scenarios in R, are then reduced to Ns = |S| number of scenarios to form a new scenario set S, by using the K-Means clustering algorithm [8], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This will generate a new set of values for the probabilities and their realizations as φL,t cj,s, φP V,t cj,s , pL cj,s(t), pP V cj,s(t) ∀s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The K- Means clustering method is illustrated in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Algorithm 1 K-Means Clustering Algorithm 1: Input φL,t cj,r ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T , r∈ R 2: for each t in T do 3: for each j in V\\ {0} do 4: for each c in Cj do 5: Randomly initialize the centroids of K-Means clus- ters as Z = � φL,t cj,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=', φL,t cj,s, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='Ns � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' where |Z| = Ns 6: for each s in S do 7: As ← ∅ 8: end for 9: while centroids of clusters do not change do 10: for each r in R do 11: s∗ ← argmin s ���φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r − φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s ��� 12: As∗ ← As∗ ∪ � φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r � 13: end for 14: for each s in S do 15: φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s ← 1 |As| � φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r∈As φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r 16: end for 17: end while 18: end for 19: end for 20: end for 21: Repeat Step 1 to Step 20 for the inputs φP V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r(t) and pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='r(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' separately ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' r∈ R 22: Return φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' φP V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' pL cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' pP V cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='s(t) ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' t∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s∈ S Do Step 8 ∀t ∈ T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' s ∈ S Step 8: Calculate the overall probability ωs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='t in (19),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' which gives the probability for the occurrence of scenario s at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The numerical values found for ωs,t, pL cj,s(t), and pP V cj,s(t) ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T , s∈ S are then fed into the system models and optimization framework in Section III and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' ωs,t = ��N j=1 �� c∈Cj φL,t cj,sφP V,t cj,s �� �Ns s=1 ��N j=1 �� c∈Cj φL,t cj,sφP V,t cj,s �� (19) 6 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' MULTI-OBJECTIVE OPTIMIZATION FRAMEWORK In this paper, it is aimed to minimize the investment cost of the CES as a planning objective, and minimize the CES opera- tion cost and the social costs of the customers as the operation objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, a multi-objective function is formulated by combining both planning and operation objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Objective Functions 1) Minimizing the Investment Cost of the CES: The in- vestment cost for a CES can be expressed as (20) [8], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The first term of (20) relates the investment cost for the rated power of the CES, and latter for the capacity of the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since minimizing the investment cost is a planning objective, it is not impacted by the uncertainty of the real power consumption and the SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thus, (20) is independent of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' fInv,cost = ρCES(CCES,Inv Rate pRate j + CCES,Inv Cap ECap j ) (20) Here, ρCES = d(1+d)τ (1+d)τ −1, where ρCES, d, and τ are the annual cost of the CES, discount rate and the CES life time (in years), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, CCES,Inv Rate and CCES,Inv Cap are the CES investment cost per kW (in AUD/kW) and the CES investment cost per kWh (in AUD/kWh), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 2) Minimizing the Operation Cost of the CES: The cost for operating the CES is given by (21) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since (21) illustrates an operation objective, it is also a function of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' fop,cost = � t∈T � Ns � s=1 ωs,t � CCES,oppCES j,s (t) � � (21) where pCES j,s (t) = ηchpCES,ch j,s (t) − 1 ηdis pCES,dis j,s (t) and CCES,op is the CES operation cost per kW (in AUD/kW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 3) Minimizing the Social Costs of the Customers: Cus- tomers incur a cost or earn a revenue for trading energy with the CES and the grid, which is jointly named as the social costs as given in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Its first term denotes the energy trading cost with the grid, and latter for trading energy with the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Similar to (21), as the social costs of the customers is also an operation objective, fC,cost is a function of the scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' fC,cost = � t∈T Ns � s=1 ωs,t � λG(t) N � j=1 � c∈Cj pG cj,s(t) + λCES(t) N � j=1 � c∈Cj pCES cj,s (t) � ∆t (22) Here, λG(t) and λCES(t) are the grid energy price and CES provider’s energy price at time t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' We adopt a one- for-one non-dispatchable energy buyback method for λG(t), to value energy imports and exports from/to the grid equally [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Optimization Problem The three objective functions which determine the CES planning and its operation, are combined together to form a multi-objective function as given in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The objective functions are normalized by their corresponding nadir and utopia points to attain a Pareto optimal solution for each objective compatible with the weights assigned for them [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' x∗ = argmin x∈X w1 � fInv,cost−f utopia Inv,cost f Nadir Inv,cost−f utopia Inv,cost � + w2 � fop,cost−f utopia op,cost f Nadir op,cost−f utopia op,cost � +w3 � fC,cost−f utopia C,cost f Nadir C,cost−f utopia C,cost � (23) Here, x = (Lj, pRate j , ECap j , pCES,ch j , pCES,dis j , pCES cj , pG cj) (24) where x is the decision variable vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, Lj, Ecap j and pRate j are the vectors of the optimal CES location, CES capacity and its rated power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Vectors of the CES charging power, CES discharging power, power exchange with the CES and the grid by the customers are given by pCES,ch j , pCES,dis j , pCES cj and pG cj, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The feasible set is given by X, which is constrained by (1)-(16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Furthermore, the calculation of the utopia and nadir values for each objective function are done in line with the techniques mentioned in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, w1, w2 and w3 are the weight coefficients of each objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The implementation of the overall optimization framework is succinctly given in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Algorithm 2 Algorithm to Run the Stochastic Multi-Objective Optimization 1: Input µL,t cj , µP V,t cj ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T 2: Initialize the model parameters used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 3: Execute Step 1 to Step 8 detailed in Section IV-C, in- cluding the Algorithm 1, to model the uncertainty of the real power consumption and the SPV generation of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 4: Return ωs,t, pL cj,s(t), and pP V cj,s(t) ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T , s∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 5: Solve the multi-objective function in (23), subject to the set of constraints (1) - (16), as a MILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' NUMERICAL AND SIMULATION RESULTS In the simulations, a radial distribution network with 7- nodes given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 2 was considered, and its line data can be found in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The forecasted SPV generation and real power consumption data of 30 residential customers in an Australian community, for a period of 1 year, measured in 1-hour time intervals were used for simulations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, all the residential customers generate SPV power and consume real power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Nevertheless, due to the lack of real data on customers’ reactive power consumption, it was not considered for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, we randomly assigned the 30 customers for all the nodes except for the slack 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 2: The 7-Node LV radial feeder with the number of customers marked at each node Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 3: Variation of the grid energy price λG(t) with time node (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Therefore, �N j=1 |Cj| = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Additionally, |V0| = 1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=', |Vmin| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='95p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=', |Vmax| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='05p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=', pRate max = 200kW, ECap min = 50kWh, ECap max = 1000kWh, ηch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='98, ηdis = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='02, ηmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='05, ηmax = 1, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='0001kWh, ∆t = 1h, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='1, τ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='5 years, CCES,Inv Rate = 463AUD/kW, CCES,Inv Cap = 795AUD/kWh and CCES,op = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='69AUD/kW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Moreover, the PV capacities of the customers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' PVcap,cj) were obtained from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, the values for CCES,Inv Rate , CCES,Inv Cap and CCES,op were taken assuming Li-ion as the CES technology [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The three objectives of the multi-objective function in (23), were weighted according to their importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For this, we used the analytic hierarchy process (AHP) illustrated in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' We assigned an equal importance for fInv,cost and fop,cost, and a strong importance for fC,cost compared to fInv,cost and fop,cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Therefore, the values of w1, w2, w3 were calculated as 1/7, 1/7, 5,7, respectively [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' After computing the weight coefficients, the simulations were done over a period of 1 year (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' |T | = 8760) using the CPLEX solver in Python-Pyomo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Case Study I: Comparison of the Proposed Optimization Framework With its Corresponding Deterministic Model To understand the impact of uncertainty on the optimal planning and scheduling decisions of a CES, we compared the results of the proposed stochastic optimization framework and its corresponding deterministic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The simulations were done for our stochastic model by considering 50 initial scenarios, which is then reduced to 10 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Ns = 10) by using the K-Means clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Simulations for the deter- ministic model were obtained by neglecting the uncertainty of real power consumption and SPV generation of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The same set of constraints and the multi-objective function used for the proposed stochastic model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' (1)-(16) and (23) ), were used for the deterministic model as well, while excluding the scenario dependency of (1)-(9), (13)-(16) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In the simulations, a time-of-use (TOU) grid energy price λg(t) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 3 was used [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Due to the lack of accurate real data about the CES provider’s energy price λCES(t), we assumed three different energy price schemes for it as (i) λCES(t) = λG(t), (ii) λCES(t) = 0 and (iii) λCES(t) = λG,avg ,where λG,avg = �24 t=1 λG(t) 24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' A summary of the numerical results obtained for the two types of the optimization models are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For the three energy price schemes of λCES(t), in both deterministic and proposed stochastic models, the optimal CES location is node 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The pRate j and ECap j in the stochastic model are higher than their values in the deterministic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In the stochastic model, due to the impact of the higher values of the realizations with respect to µL,t cj , µP V,t cj ∀j ∈ V\\ {0} , c ∈ Cj, t∈ T , both pRate j and ECap j will be greater than their values in the deterministic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Moreover, in both models, the values of the planning decisions namely, Lj, pRate j and ECap j for their respective models have not changed irrespective of the CES provider’s energy price scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This has happened, as the planning decisions are independent from operation variables, and thus from λCES(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, the investment costs in the deterministic and stochastic models are AUD 32228 and 33695, respectively, irrespective of the CES provider’s energy price scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The operation objectives consider minimizing the CES operation cost fop,cost, and the social costs of the customers fC,cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' According to Table I, fop,cost is same for the deter- ministic model regardless of the CES provider’s energy price scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is resulted as fop,cost is a function independent of λCES(t) (see (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Besides, when λCES(t) = λG(t) and λCES(t) = λG,avg(t), fop,cost in the stochastic model is higher than the corresponding deterministic model values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since the stochastic model takes into account the uncer- tainty of SPV generation and real power consumption of the customers, the costs in the stochastic model are higher than the ones in the deterministic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This behaviour is seen for fC,cost as well when λCES(t) = λG(t) and λCES(t) = λG,avg(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Additionally, as fC,cost < 0 when λCES(t) = 0, it implies that the customers earn a revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The customers minimize their social costs by importing power only from the CES as λCES(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, the customers export power solely to the grid to maximize their social revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is the intuition for fC,cost being negative when λCES(t) = 0 for both deterministic and stochastic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is discussed in detail in Section VI-B-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In the proposed stochastic model, fop,cost for the three energy price schemes of λCES(t) are different from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' As the set of random numbers generated during stochastic modelling are unique and different for every execution of Algorithm 2, a unique set of values for ωs,t, pL cj,s(t), and pP V cj,s(t) ∀j ∈ V\\ {0}, c ∈ Cj, t∈ T , s∈ S are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This results in getting different values for fop,cost, irrespective of fop,cost being independent of λCES(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 2 6 External Transformer IC2l = 4 IC6l = 4 Grid 22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='4 kvV 0 1 3 4 L80 ICil = 3 IC3/ = 5 IC4l = 6 ICzl = 5 5 ICsl = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='25 T2 T3 T4 T5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='20 0 5 10 15 20 25 Time Duration (24 Hours)8 TABLE I: RESULTS OF THE DETERMINISTIC AND PROPOSED STOCHASTIC MODELS Optimal CES Node Lj Optimal CES Rated Power1 pRate j (kW) Optimal CES Capacity1 ECap j (kWh) CES Investment Cost1 (AUD) fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='cost CES Operation Cost1(AUD) fop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='cost Social Costs1 (AUD) fC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='cost Deterministic λCES(t) = λG(t) 7 72 240 32228 24674 124830 Model λCES(t) = 0 7 72 240 32228 24674 94681 λCES(t) = λG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='avg 7 72 240 32228 24674 70810 Proposed Stochastic Model (Ns = 10) λCES(t) = λG(t) 7 78 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='69%) 249 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='61%) 33695 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='35%) 27266 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='51%) 129429 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='55%) λCES(t) = 0 7 78 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='69%) 249 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='61%) 33695 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='35%) 29712 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='96%) 98201 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='58%) λCES(t) = λG,avg 7 78 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='69%) 249 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='61%) 33695 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='35%) 29347 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='92%) 75263 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='92%) 1 Increment percentage values are computed with respect to their corresponding values found in the deterministic model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 4: (a) Total power exchange with the grid and the CES by customers, (b) CES charging/discharging power and temporal variation of CES energy - When Ns = 10, λCES(t) = λG(t) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Analysis of the Results of CES Scheduling and Mutual Power Exchanges Between the CES, the Grid and Customers The results obtained for the proposed stochastic optimiza- tion framework, for different energy price schemes of the CES provider are detailed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Here, we do the analysis for a randomly selected a day, for a duration of 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 1) When λCES(t)=λG(t): Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 4(a) depicts the total power exchange with the grid (brown plot) and the CES (orange plot) by the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' As the customers do not have sufficient SPV generation during T1, T3 T4 and T5, they import power from the grid and the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Besides, the customers export their excess generation to the CES and the grid during T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since λCES(t)=λG(t), the customers do not have any preference whether to exchange power with the grid or the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The charging and discharging pattern of the CES (red plot), and the CES energy level variation (green plot) with time are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' During T1, the CES charges, and by the end of T1, it discharges completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The CES is fully discharged by the end of T1 to exploit its full capacity to charge from the excess SPV generation during T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is evident as the CES energy level has reached its full capacity of 249 kWh during T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' During T3, the CES exports its power to the customers, and at the end of the day, CES reaches its initial energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 5: (a) Total power exchange with the grid and the CES by customers, (b) CES charging/discharging power and temporal variation of CES energy - When Ns = 10, λCES(t) = 0 2) When λCES(t)=0: In this case, as λCES(t)=0, the customers neither incur a cost nor earn a revenue when trading energy with the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 5(a), during T1, the customers import power only from the CES, as the customers do not incur a cost for importing power from the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The same trend is followed by the customers during T3, T4 and T5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' During T2, the customers export the excess SPV generation to the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Also, as λCES(t)=0, the customers do not export power to the CES as they cannot earn a revenue from the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, when λCES(t) = 0, the customers do not incur a cost for all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Instead, they earn a revenue from the grid which is shown in Table I as a negative value for fC,cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 5(b) shows the charging and discharging pattern of the CES, including its energy level variation with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 3) When λCES(t)=λG,avg: In this paper, λG,avg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='34180 AUD/kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, during T3 λCES(t) < λG(t), and during T1, T2 T4, T5 λCES(t) > λG(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 6(a), during T1, the customers import power only from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since λCES(t) > λG(t) during this time period, it is not economically beneficial for the customers to import expensive power from the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' During T2, in which the time period with high SPV generation, the customers export the excess SPV generation to the CES as λCES(t) > λG(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, the customers can earn a higher revenue from the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' (a) 100 Power (kW) 100 0 10 15 oZ 25 Discharging Power (kW) (b) CES Charging and, 50 CES Energy (kWh) 200 0 100 50 0 5 10 15 25 Time Duration (24 hours) pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' (t) /= 1c E G 5=1 pa5(t) /= 1cE G (a) Power (kw) 0 200 0 n 10 15 20 25 Discharging Power (kW) (b) CES Charging and.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 50 Energy (kWh) 200 0 100 50 CES 0 5 10 15 20 25 Time Duration (24 hours) pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' (t) 5 = 1 /= 1c E G 5=1 M Zpas(t) 5 = 1 /= 1cE G 5 = 19 TABLE II: RESULTS OF PROPOSED STOCHASTIC MODEL AND CASE I-IV - (WITH Ns = 10, λCES(t) = λG,avg) Case / CES Node Optimal CES Rated Power1 pRate j (kW) Optimal CES Capacity1 ECap j (kWh) CES Investment Cost1 (AUD) fInv,cost CES Operation Cost1 (AUD) fop,cost Social Costs1 (AUD) fC,cost Cumulative Cost1 (AUD) Proposed Model- 7 (optimal) 78 249 33695 29347 75263 138305 Case I - 3 (chosen) 124 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='10%) 380 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='47%) 51688 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='81%) 44900 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='64%) 77703 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='14%) 174291 (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='65%) Case II - 4 (chosen) 98 (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='41%) 303 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='82%) 41217 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='25%) 36723 (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='09%) 76973 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='22%) 154913 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='72%) Case III - 5 (chosen) 127 (38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='58%) 366 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='97%) 50258 (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='96%) 39093 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='93%) 77024 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='29%) 166375 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='87%) Case IV - 6 (chosen) 96 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='75%) 310 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='68%) 41837 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='46%) 35578 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='51%) 76491 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='61%) 153906 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='14%) 1 Increment percentage values are computed with respect to their corresponding values found in the proposed model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 6: (a) Total power exchange with the grid and the CES by customers, (b) CES charging/discharging power and temporal variation of CES energy - When Ns = 10, λCES(t) = λG,avg Since λCES(t) < λG(t) during T3, the customers import power from the CES, so that they have to pay less for the imported power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' During T4 and T5, the customers import power only from the grid as λCES(t) > λG(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' 6(b) shows the charging and discharging pattern, and the temporal variation of the CES energy level with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Case Study II: Proposed Optimization Framework Vs Mod- els With Arbitrary CES Locations The merits of a CES may be fully exploited if its both planning and scheduling are optimized simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' To test this, we compared our proposed model with four dif- ferent cases that randomly choose the CES location, with λCES(t) = λG,avg(t), Ns = 10, and taking into account the uncertainty of real power consumption and SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' As given in Table II, the CES is allocated for nodes 3, 4, 5 and 6 which represent Case I, II, III and IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For Case I-IV, we considered the constraints (1)-(16), and the objective function in (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Additionally, Lj = 1 where j ∈ {3, 4, 5, 6} in (10)-(12) for Case I-IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In Case I-IV, pRate j and ECap j are significantly higher than in our proposed stochastic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This has resulted in a substantial increase of the CES investment and operation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' But, the social costs show only a minor increase for Case I-IV compared with the increase of the investment and operation costs of the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This can be explained in terms of the weight coefficients used for weighting the objective functions in (20)- (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Since w1, w2 and w3 are 1/7, 1/7 and 5/7, respectively, the highest importance is given for minimizing the social costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Hence, the optimization solver tries to maintain the social costs as much as close to fC,cost obtained in our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' However, this comes at an expense as fInv,cost and fop,cost which have a less significance, increase significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' But, the cumulative cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' sum of fInv,cost, fop,cost and fC,cost ) is the least for the proposed model, while for Case I-IV, it is about 10-21% higher than the cost in our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Case Study III: Impact of Scenario Reduction Approaches In this section, we present a comparison of the results obtained for our optimization model utilizing two scenario re- duction methods namely, backward scenario reduction (BSR) method and K-Means clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In BSR method, the initial number of scenarios are reduced by minimizing the Monge-Kantorovich distance between the scenarios in both initial and reduced scenarios sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Thereby, the initial scenarios are eliminated iteratively one by one, until the desired number of elements in the reduced scenario set is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Further explanation about the BSR method can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' In this case study, we considered λCES(t) = λG,avg, and the initial number of scenarios as 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The proposed opti- mization framework was implemented under the two scenario reduction approaches by taking Ns = 10 and Ns = 30 for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The numerical results obtained for this case study are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Note that, for all the cases, node 7 was recorded as the optimal CES location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The costs for all the three objectives are the highest for the model which did not use a scenario reduction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This is because, the model with Ns = 50 captures more uncertainty of the real power consumption and SPV generation, so that the optimal CES rated power and the capacity are higher than the models with a lesser number of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' For both K-Means and BSR methods, even with a different number of reduced scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Ns = 10 and Ns = 30), the CES planning aspects have not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This occurs as the planning aspects are independent of the number of scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Nevertheless, as the operation decisions are scenario dependent, the operation cost of the CES, and the customers’ social costs have changed according to Ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' This trend is observed in the results obtained for the model which used the BSR method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' (a) Power (kW) 100 0 100 200 0 n 10 15 20 25 Discharging Power (kW) (b) CES Charging and, CES Energy (kWh) 200 0 100 50 0 5 10 15 20 Time Duration (24 hours) pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' (t) 5= 1 /= 1c E G 5=1 MN pa5(t) ws, EE5(t) /=1cE G 10 TABLE III: RESULTS FOR DIFFERENT SCENARIO REDUCTION METHODS - (WITH λCES(t) = λG,avg) No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' of scenarios Computational time1 (min) Optimal CES Rated Power1 pRate j (kW) Optimal CES Capacity1 ECap j (kWh) CES Investment Cost1 (AUD) fInv,cost CES Operation Cost1 (AUD) fop,cost Social Costs1 (AUD) fC,cost Without Scenario Reduction Ns = 50 91 79 257 34573 38434 91321 K-Means Clustering Algorithm Ns = 10 38 (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='24%) 78 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='27%) 249 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='11%) 33695 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='54%) 29347 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='64%) 75263 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='58%) Ns = 30 53 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='76%) 78 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='27%) 249 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='11%) 33695 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='54%) 30019 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='89%) 89425 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='08%) Backward Scenario Reduction Method Ns = 10 41 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='95%) 78 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='27%) 250 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='72%) 33735 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='42%) 29785 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='50%) 75709 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='10%) Ns = 30 57 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='36%) 78 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='27%) 250 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='72%) 33735 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='42%) 30664 (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='22%) 89981 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content='47%) 1 Decrement percentage values are computed with respect to their corresponding values found without scenario reduction The models with reduced number of scenarios have con- verged for a solution in a lesser time than the case with Ns = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Scenario reduction methods like K-Means clustering algorithm and BSR method play a key role in reducing the computational time while maintaining the problem tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' However, according to the results obtained for the models which used the K-Means and the BSR method, it is not conclusive to claim which scenario reduction method is better, as there is no any significant difference between the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' CONCLUSION & FUTURE WORK In this paper, we have explored how the optimization of the planning and scheduling of a community energy storage (CES) benefit the CES provider by minimizing the CES investment and operation costs, and the customers by minimizing their social costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' The uncertainty of real power consumption and solar photovoltaic (SPV) generation of the customers have been accounted to formulate a scenario-based stochastic op- timization program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' To reduce the computational burden of the stochastic model, we have used the K-Means clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' It has been shown that, both the customers and the CES provider can significantly minimize their personal costs by optimizing both the CES planning and its scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Future work includes extending the proposed model for unbalanced distribution networks, developing optimization models for networks with multiple CES, and considering the reactive power regulation capabilities of CES and SPV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQff_yv/content/2301.01462v1.pdf'} +page_content=' Shaw, B.' metadata={'source': 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+Generalized Uncertainty Principle for Entangled +States of Two Identical Particles +K. C. Lemos Filho∗1, B. B. Dilem†2, J. C. Fabris‡1,3, and J. A.Nogueira§1 +1Universidade Federal do Esp´ırito Santo – Ufes, Vit´oria, Esp´ırito Santo, +29.075-910, Brasil +2Instituto Federal do Esp´ırito Santo – Ifes, Alegre, Esp´ırito Santo, +29.520-000, Brasil +3National Research Nuclear University MEPhI, Kashirskoe sh. 31, +Moscow 115409, Russia +Abstract +In this work we determine the consequences of the quantum en- +tanglement of a system of two identical particles when the generalized +uncertainty principle (GUP) is considered. GUP is usually associated +with the existence of a minimal length. We focus on the main formu- +lations of the GUP and then we determine the minimal uncertainties in +position induced by those modified GUP’s. Our results point out that +the minimal uncertainty is reduced by half of its usual value indepen- +dently of the GUP employed. This implies that the minimal length is +also reduced by half. On the other hand, it is generally expected that the +minimal length must not depend on physical system. We overcome this +apparent paradox by realizing that the entangled system is composed by +two particles so that an effective parameter related to the minimal length +must be employed. +PACS numbers: 04.60.-m, 03.65.Ud, 03.65.Ta +Keywords: Minimal length; generalized uncertainty principle; quantum entanglement. +∗kim.vasco@gmail.com +†bernardob@ifes.edu.br +‡julio.fabris@cosmo-ufes.org +§jose.nogueira@ufes.br +1 + +1 +Introduction +Amongst the (many) new concepts introduced by the quantum mechanics, the quantum +entanglement [1, 2] is one that, probably, more contradicts our common sense. Although, +in the beginning, the quantum entanglement were only associated to theoretical aspects +of quantum mechanics, specially those related to the non-locality or the complementarity +(hidden variables) [3], nowadays it is a key component of the applications and experiments +on quantum information, quantum computation and quantum teleportation [4, 5]. +The uncertainty principle is one of the fundamental cornerstones of the quantum me- +chanics. Nevertheless, it is a principle: its correct form can not be proven. That opens +the way to the possibility that its canonical form, described by the Heisenberg’s uncer- +tainty principle (HUP), can be generalized. An example of the possible generalizations +of HUP, whose origin can be traced back to quantum gravity, is given by introducing a +non-zero minimal uncertainty in the measurement of position. That non-zero minimal +uncertainty in position is, then, understood as a minimal-length scale, below which the +necessary amount of energy to probe the position of a particle is so hight that it disturbs +the space-time so that the concept of a length measurement loses its meaning. Hence, the- +ories searching to describe a quantum approach for gravity lead generally to the existence +of a minimal length. In fact, a minimal length actually appears in almost all proposed +theories of the quantum gravity. For this reason theories formulated in a minimal-length +scenario are considered to be effective theories of quantum gravity [6, 7, 8, 9]. +In 1999, Yoon-Ho Kim and Yanhua Shih conducted an experiment whose results ap- +parently suggested a violation of the HUP [10]. However, G. Rigolin pointed out that +in fact there is no violation of the HUP, because the HUP is derived for particles non- +correlated (non-entangled). In the Kim and Shih’s experiment, the photons of the pair +are correlated (entangled) when one of the physical slits is replaced by a virtual slit1: +the canonical HUP is no longer applicable since the quantum entanglement modifies the +canonical HUP [11, 12]. +An immediate question arising from the previous considerations is how the quantum +entanglement modifies a generalized uncertainty principle (GUP), in others words, which +is the effect of the quantum entanglement in the minimal-length scale. The answer for +that question is important in order to know the role of the quantum entanglement at +the minimal-length scale (maybe appearing in the Planck scale) or in the early Universe. +Unfortunately, this issue has been little considered in the literature. In [13], G. Blado, +F. Herrera and J. Erwin have studied the inseparability conditions with the most usual +GUP correction, whereas D. Park has used a coupled harmonic oscillator in order to find +the effects of the quantum entanglement with a linear GUP in [14]. +The purpose of this work is to answer that question considering the main proposals of +generalization for the HUP which take into account the existence of a minimal length in +nature. With this goal in mind, we will analyze the modifications in the HUP arising from +the quantum entanglement of two identical particles determining the minimal uncertainty +1The interaction of the photon with a physical slit destroys the correlation between the photons of the +pair. +2 + +associated to them. +The outline of this paper is as follows. In Section 2 we obtain an expression of the +uncertainty principle for entangled states which is independent of the chosen GUP. In +Section 3 we find the modified uncertainty principle for a pair of identical particles re- +garding the main proposals of GUP’s: Kempf, Mangano and Mann GUP (KMM-GUP), +Ali, Das and Vagenas GUP (ADV-GUP), Pedram GUP and exponential all orders GUP. +In Section 4 we estimate an upper bound for the minimal-length value. We present our +conclusions in the Section 5. +2 +Uncertainty principle for entangled states +The Hilbert space of the state vectors E of a system of N particles is given by the +tensor product of the Hilbert spaces of the state vectors Ei of each particle [15, 16], +E = E1 ⊗ · · · ⊗ EN. +(1) +The position and momentum linear operators of the i-th particle which act on the state +vectors |ψ⟩ ∈ E are the extensions ˜Qi and ˜Pi defined as +˜Qi = I1 ⊗ · · · ⊗ ˆxi ⊗ · · · ⊗ IN, +(2) +˜Pi = I1 ⊗ · · · ⊗ ˆpi ⊗ · · · ⊗ IN, +(3) +where Ii is the identity operator in Ei and ˆxi and ˆpi are the position and the momentum +operators of the i-th particle acting on the state vectors |ψi⟩ ∈ Ei. +The extensions ˜Qi and ˜Pi do not satisfy the canonical uncertainty principle (HUP), +because ˜Qi and ˜Pi are not physical observables [11, 12, 15, 16]. +Physical observables +are operators which commute with every permutation operators of the particles system. +Hence, the operators ˜Q and ˜P, defined as +˜Q := +N +� +i=1 +˜Qi +(4) +and +˜P := +N +� +i=1 +˜Pi, +(5) +are physical observables and they satisfy the relation +(∆Q)2 (∆P)2 ≥ 1 +4 +���⟨[ ˜Q, ˜P]⟩ +��� +2 +. +(6) +The relation (6) is general. It does not depend whether the system of particle is entangled +or not. +3 + +As it was showed by G. Rigolin [11, 12], if the state of the particles system is entangled +then the operators ˜Qi and ˜Pi do not satisfy the canonical Heisenberg uncertainty principle +(HUP) - as previously stated. +We briefly review the Rigolin’s result for a two particles system. From the definitions +of ∆Q and ∆P we have2 +(∆ψQ)2 = +� +ψ| ˜Q2|ψ +� +− +� +ψ| ˜Q|ψ +�2 +. +(7) +From now on we omit the subscript ψ for the sake of simplicity, whenever this does not +cause any confusion. Thus, +(∆Q)2 = (∆Q1)2 + (∆Q2)2 + 2 +�� +˜Q1 ˜Q2 +� +− +� +˜Q1 +� � +˜Q2 +�� +. +(8) +In the same way, +(∆P)2 = (∆P1)2 + (∆P2)2 + 2 +�� +˜P1 ˜P2 +� +− +� +˜P1 +� � +˜P2 +�� +. +(9) +Using the results (8) and (9) into Eq. (6) we obtain +� +(∆Q1)2 + (∆Q2)2 + 2 +�� +˜Q1 ˜Q2 +� +− +� +˜Q1 +� � +˜Q2 +��� +× +� +(∆P1)2 + (∆P2)2 + 2 +�� +˜P1 ˜P2 +� +− +� +˜P1 +� � +˜P2 +��� +≥ 1 +4 +���⟨[ ˜Q, ˜P]⟩ +��� +2 +. +(10) +We now use the functions +CQ(1, 2) +:= +� +˜Q1 ˜Q2 +� +− +� +˜Q1 +� � +˜Q2 +� +, +(11) +CP(1, 2) +:= +� +˜P1 ˜P2 +� +− +� +˜P1 +� � +˜P2 +� +, +(12) +which are called quantum covariance functions (QCF). By definition, QCF’s vanish if and +only if the system is separable [17]. Therefore (11) and (12) are zero for any not entangled +quantum system. +Using the QCF’s (11) and (12) we have +� +(∆Q1)2 + (∆Q2)2 + 2CQ(1, 2) +� � +(∆P1)2 + (∆P2)2 + 2CP(1, 2) +� +≥ 1 +4 +���⟨[ ˜Q, ˜P]⟩ +��� +2 +, +(13) +or +2 +� +i,j=1 +CQ(i, j) +2 +� +k,l=1 +CP(k, l) ≥ 1 +4 +���⟨[ ˜Q, ˜P]⟩ +��� +2 +, +(14) +since CQ(i, i) = (∆Qi)2, CP(i, i) = (∆Pi)2, CQ(i, j) = CQ(j, i) and CP(i, j) = CP(j, i). +2Note that [ ˜Q1, ˜Q2] = 0 and [ ˜P1, ˜P2] = 0. +4 + +In this work, we concern with the case of an entangled system of two identical particles, +so we are going to handle Eq. (13) in order to express it in a more appropriate way. For +this end, we define +|ψ′⟩ := +� +˜Q1 − ˜Q2 +� +|ψ⟩, +(15) +with |ψ′⟩, |ψ⟩ ∈ E and ⟨ψ | ψ⟩ = 1. Therefore, +⟨ψ′ | ψ′⟩ = (∆ψQ1)2 + (∆ψQ2)2 − 2 +� +˜Q1 ˜Q2 +� +ψ + +� +˜Q1 +�2 +ψ + +� +˜Q2 +�2 +ψ . +(16) +Now, using the Schwarz inequality, ⟨ψ | ψ⟩ ⟨ψ′ | ψ′⟩ ≥ ⟨ψ | ψ′⟩ ⟨ψ′ | ψ⟩, we have +(∆ψQ1)2 + (∆ψQ2)2 ≥ 2 +�� +˜Q1 ˜Q2 +� +ψ − +� +˜Q1 +� +ψ +� +˜Q2 +� +ψ +� +. +(17) +In the same way +(∆ψP1)2 + (∆ψP2)2 ≥ 2 +�� +˜P1 ˜P2 +� +ψ − +� +˜P1 +� +ψ +� +˜P2 +� +ψ +� +. +(18) +Finally, from inequalities (17), (18) and (13) we obtain +� +(∆Q1)2 + (∆Q2)2� � +(∆P1)2 + (∆P2)2� +≥ 1 +16 +���⟨[ ˜Q, ˜P]⟩ +��� +2 +. +(19) +In the case where (∆Q1)2 = (∆Q2)2 and (∆P1)2 = (∆P2)2 the inequality (19) becomes +∆Qi∆Pi ≥ 1 +8 +���⟨[ ˜Q, ˜P]⟩ +��� . +(20) +It is worth noting that the expression of the inequality (20) is independent of the +chosen uncertainty principle that does not take into account the quantum correlation. +This uncertainty principle is related to the commutation relation [ ˜Q, ˜P]. +3 +Uncertainty principle for entangled states in different minimal- +length scenarios +In this section we consider a system of two entangled identical particles whose momenta +have the same value but opposite directions, that is, ⃗p1 = −⃗p2, just as in the Kim +and Shih’s experiment [10]. Therefore, in this case ⟨ˆp1⟩ + ⟨ˆp2⟩ = 0. Moreover, such a +consideration also allows us to estimate, in the next section, an upper bound for the value +of the minimal length based on the experimental results obtained by Kim and Shih. +5 + +3.1 +Heisenberg uncertainty principle +Before we consider a minimal-length scenario it is appropriate to determine the change +in the canonical HUP, that is, in a scenario in which effects of quantum gravity are not +present. The canonical HUP for states of one simple-particle is +∆x∆p ≥ ℏ +2. +(21) +The commutation relation related to the HUP is +[ˆx, ˆp] = iℏ. +(22) +Hence +[ ˜Q, ˜P] = [ ˜Q1 + ˜Q2, ˜P1 + ˜P2] = 2iℏ. +(23) +Substituting Eq. (23) into Eq. (20) we get +∆Qi∆Pi ≥ ℏ +4. +(24) +The result (24) shows that for a system of two entangled identical particles the HUP is +modified. Such an outcome is not new, it was already obtained by G. Rigolin in 2002 [11] +and then in 2016 [12]. +From a quick glance at the result (24) and recalling that dimensionally (∆Q)min ∝ ℏ, +we expect the minimal uncertainty in the position will be reduced by half for all GUP’s. +3.2 +KMM GUP +The GUP +∆xi∆pi ≥ ℏ +2 +� +1 + β (∆pi)2 + β ⟨ˆpi⟩2� +, +(25) +where β is a parameter related to the minimal length, has been proposed by A. Kempf, +G. Mangano an R. B. Mann (KMM-GUP) [18] and it is the most used in the literature. +The commutation relation related to it is given by +[ˆxi, ˆpi] = iℏ +� +1 + βˆp2 +i +� +. +(26) +Hence +[ ˜Q, ˜P] = iℏ +� +1 + β +� +˜P 2 +1 + ˜P 2 +2 +�� += iℏ +� +1 + 2β ˜P 2 +i +� +. +(27) +Substituting Eq. (27) into Eq. (20) we get +∆Qi∆Pi ≥ ℏ +4 +� +1 + β (∆Pi)2 + γ +� +, +(28) +where γ := β +� +˜Pi +�2 +. +6 + +The modified KMM-GUP (28) induces the existence of a minimal uncertainty given +by +(∆Qi)min = ℏ +2 +� +β. +(29) +The result above shows that the non-zero minimal uncertainty in position induced by +the KMM-GUP for two entangled identical particles is twice smaller than for a separable +system of two identical particles (non-entangled). +3.3 +ADV GUP +A. Farag Ali, S. Das and E. C. Vagenas have proposed a GUP related to a commutation +relation which has a linear and a quadratic term in the momentum operator [19], +[ˆxi, ˆpi] = iℏ +� +1 − 2αˆpi + 4α2ˆp2 +i +� +, +(30) +where α is a parameter related to the minimal length. Besides the existence of a minimal +length this linear approach induces a maximal uncertainty in the momentum, too. Then, +from Eq. (30) we get +[ ˜Q, ˜P] = 2iℏ +� +1 − α +� +˜P1 + ˜P2 +� ++ 2α2 � +˜P 2 +1 + ˜P 2 +2 +�� +. +(31) +Therefore, +⟨[ ˜Q, ˜P]⟩ = 2iℏ +� +1 − α +�� +˜P1 +� ++ +� +˜P2 +�� ++ 2α2 �� +˜P 2 +1 +� ++ +� +˜P 2 +2 +��� +, +(32) +⟨[ ˜Q, ˜P]⟩ = 2iℏ +� +1 + 4α2 � +˜P 2 +i +�� +, +(33) +Substituting Eq. (33) into Eq. (20) we obtain +∆Qi∆Pi ≥ ℏ +4 +� +1 + 4α2 (∆Pi)2 + γ′ +� +, +(34) +where γ′ := 4α2 � +˜Pi +�2 +. +The modified ADV-GUP (34) induces the existence of a non-zero minimal uncertainty +given by +(∆Qi)min = ℏα, +(35) +which once again is twice smaller than for a non-entangled system of two particles. +It is important to note that the linear GUP (ADV-GUP) becomes non-linear in this +case and consequently a maximal uncertainty in the momentum is no longer induced. +7 + +3.4 +Pedram GUP +In order to overcome some problems arising from KMM-GUP and ADV-GUP - such +as incorporation of a maximal momentum required in doubly special relativity (DSR) +theories, commutative geometry and an approach valid for all order in the parameter +related to the minimal length - P. Pedram has proposed a GUP [20, 21] based on the +commutation relation given by +[ˆxi, ˆpi] = +iℏ +1 − βˆp2 +i +. +(36) +Hence +[ ˜Q, ˜P] = +iℏ +1 − β ˜P 2 +1 ++ +iℏ +1 − β ˜P 2 +2 +. +(37) +Using the so-called Jensen’s Inequality [22] we have +⟨[ ˜Q, ˜P]⟩ ≥ +iℏ +1 − β +� +˜P 2 +1 +� + +iℏ +1 − β +� +˜P 2 +2 +� +. +(38) +Substituting Eq. (38) into Eq. (20) we obtain +∆Qi∆Pi ≥ ℏ +4 +1 +� +1 − β (∆Pi)2 − γ +�. +(39) +Therefore, the modified Pedram-GUP (39) introduces a non-zero minimal uncertainty +given by +(∆Qi)min = 3ℏ +8 +� +3β, +(40) +which is also twice smaller than for a non-entangled system of a pair of identical particles. +3.5 +Exponential all orders GUP +In the canonical field theory in the context of non-commutative coherent states repre- +sentation and field theory on non-anticommutative superspace the Feynman propagator +displays an ultra-violet (UV) cut-off of the form e−βp2 [23, 24, 25, 26]. In consequence, K. +Nouicer has proposed an exponential all orders GUP [27, 28] based on the commutation +relation given by +[ˆxi, ˆpi] = iℏeβˆp2 +i . +(41) +Hence +[ ˜Q, ˜P] = iℏ +� +eβ ˜P 2 +1 + eβ ˜P 2 +2 +� +. +(42) +Again, using the so-called Jensen’s Inequality [22] we have +⟨[ ˜Q, ˜P]⟩ ≥ iℏ +� +eβ⟨ ˜P 2 +1⟩ + eβ⟨ ˜P 2 +2⟩� +. +(43) +8 + +Substituting Eq. (42) into Eq. (20) we get +∆Qi∆Pi ≥ ℏ +4eβ(∆Pi)2+γ. +(44) +Therefore, the modified exponential all order GUP (44) introduces a minimal uncer- +tainty given by +(∆Qi)min = ℏ +2 +� +eβ +2 . +(45) +Finally, we note that (∆Qi)min is also twice smaller than for a non-entangled system of a +pair of identical particles, as we have already said. +3.6 +Minimal length +If a non-zero minimal uncertainty in position can be interpreted as a minimal length then +the previous results show that the minimal length for two entangled identical particles +can be twice smaller than for a separable system. It is clear this statement must be taken +with care because a minimal length should be a constant, that is, it must not depend on +the physical system, it is a quantum gravitation effect. In fact, the minimal length should +be an invariant as well as the light speed is. The answer for that apparent contradiction +is possibly because the system is made up of two particles. In references [29], C. Quesne +and V. M. Tkachuck have claimed that for a system composed by N particles the effective +parameter related to the minimal length, β, is reduced by a factor +1 +N2. Hence, ℏ +2 +√β is not +the correct minimal uncertainty in position (∆Qi)min for the modified KMM-GUP, but +ℏ +2 +√βi, where [29, 30] +β = βi +22. +(46) +Consequently, we find that (∆Qi)min = ℏ√β, therefore lmin = ℏ√β, as we expected3. In +the same way for others GUP’s. +The reader might want to claim that the minimal length is actually described by the +minimal uncertainty of the entangled system. However, according to reference [12] we +can presume that the minimal uncertainty for a system of N entangled identical particles +is reduced by +1 +N . Since, in principle, there is no limit for the number of particles for a +entangled system, there would also be no limit for the minimal length. +4 +Upper bound for the minimal-length value +In the Kim and Shih’s experiment entangled identical pairs of photons were produced +by spontaneous parametric down conversion (SPDC) with momentum conservation. A +narrow physical slit was placed along the trajectory of one of the photons, whereas the +3It is worth noting that if the minimal uncertainty was greater for the entangled system, we could sup- +pose that the quantum entanglement decreases the accuracy of a position measurement thereby increasing +the minimal uncertainty. +9 + +other photon of the pair (called 2) passed through a virtual slit. The ghost image exper- +imental technique [31] ensured that the quantum correlation between the pair of photons +was not destroyed. Then simultaneous detection of photons of the pairs were performed +and data just for coincidence events were obtained in case when the photon 2 passed +through a virtual slit (non-slit case) and in case when the photon 2 passed through a +physical slit (slit case). +From the experimental data obtained by Kim and Shih one gets that [12] +∆P ns +2 +∆P s +2 += 1.25 +2.15, +(47) +where ∆P ns +2 +is uncertainty in momentum of the photon 2 in the non-slit case and ∆P s +2 is +uncertainty in momentum of the photon 2 in slit case. Since the width of the slit was 0.16 +mm (∆Q2 = 0.16 mm), the uncertainty in momentum in the slit case for KMM-GUP can +be find from4 +0.16∆P s +2 = ℏ +2 +� +1 + 4β (∆P s +i )2 + γ +� +. +(48) +Eq. (48) has real roots only if β ≤ 0.162 +ℏ2η , where η := 1 + γ. Thus, +∆P s +2+ = 0.32 +ℏβ − ℏη +0.32 − β ℏ3η2 +0.323 +(49) +and +∆P s +2− = ℏη +0.32 + β ℏ3η2 +0.323, +(50) +Now, using the above results we can estimate an upper bound for the minimal-length +value induced by KMM-GUP. Hence, substituting the root (49) into Eq. (28) we have +that +β ≤ 3.58 × 10−2 +ℏ2η +. +(51) +Therefore, +lmin ≤ ℏ +� +3.58 × 10−2 +ℏ2η +≤ ℏ +� +3.58 × 10−2 +ℏ2 += 1.9 × 10−4m. +(52) +The substitution of the root (50) hold the inequality (28) for all β > 0. +Consequently, the upper bound for the minimal-length value is order 10−4 m. There- +fore, using the experiment described above a result with less restrictions than those re- +ported in the literature is obtained [30, 32, 33, 34, 35, 36, 37, 38]. +4Note that in with-slit case the correlation between the photons of the pair was destroyed, therefore +the photons were not entangled and the usual KMM-GUP, Eq. (25), is held. +10 + +5 +Conclusion +In this work we find the non-zero minimal uncertainties induced by the main proposals +of GUP’s (KMM, ADV, Pedram and exponential) which are modified due to the quantum +entanglement of a system of two identical particles. In principle, our results have pointed +out that the minimal uncertainties are reduced at half for a system of two entangled +identical particles independently of the GUP. Hence, if a non-zero minimal uncertainty in +position can be interpreted as a minimal length then the quantum entanglement reduces +by half the minimal length. However, the minimal length must not depend on the phys- +ical system. We overcome this apparent paradox by using the Quesne and Tkachuck’s +proposal for a system composed. Consequently, despite the quantum entanglement to +change the GUP, the minimal length does not change. Based on our results and using +the reference [12] we can expect that the apparent minimal uncertainty for a system of +N entangled identical particles is reduced by +1 +N , nonetheless the minimal length does not +change because the effective parameter β is also reduced by a factor +1 +N2. +Finally, we have estimated from the data obtained from the Kim and Shih’s experiment +an upper bound value for the minimal length of the order of 10−4 m. Consequently, it is +rather an inexpressive value (in the sense of leading to poor predictive power) as compared +to ones have been found in the literature. This is due to the high imprecision of the +experiment on the entangled system described above. However, we may expect that more +refined version of the experiment may lead to more stringent bounds on the minimum +length. +Acknowledgements +We would like to thank FAPES, CAPES and CNPq (Brazil) for financial support. +References +[1] R. Horodecki, P. Horodecki, M. Horodecki and K. 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C 73, 2495 (2013). doi.org/10.1140/epjc/s10052-013-2495-6 +14 + diff --git a/7tFLT4oBgHgl3EQfAi4b/content/tmp_files/load_file.txt b/7tFLT4oBgHgl3EQfAi4b/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2db1fea2e21430899f82ed33f7f9f8dd2e96fc6e --- /dev/null +++ b/7tFLT4oBgHgl3EQfAi4b/content/tmp_files/load_file.txt @@ -0,0 +1,565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf,len=564 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='11966v1 [quant-ph] 27 Jan 2023 Generalized Uncertainty Principle for Entangled States of Two Identical Particles K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Lemos Filho∗1, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Dilem†2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Fabris‡1,3, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='Nogueira§1 1Universidade Federal do Esp´ırito Santo – Ufes, Vit´oria, Esp´ırito Santo, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='075-910, Brasil 2Instituto Federal do Esp´ırito Santo – Ifes, Alegre, Esp´ırito Santo, 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='520-000, Brasil 3National Research Nuclear University MEPhI, Kashirskoe sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 31, Moscow 115409, Russia Abstract In this work we determine the consequences of the quantum en- tanglement of a system of two identical particles when the generalized uncertainty principle (GUP) is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' GUP is usually associated with the existence of a minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' We focus on the main formu- lations of the GUP and then we determine the minimal uncertainties in position induced by those modified GUP’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Our results point out that the minimal uncertainty is reduced by half of its usual value indepen- dently of the GUP employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' This implies that the minimal length is also reduced by half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' On the other hand, it is generally expected that the minimal length must not depend on physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' We overcome this apparent paradox by realizing that the entangled system is composed by two particles so that an effective parameter related to the minimal length must be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='-m, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='Ud, 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='Ta Keywords: Minimal length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' generalized uncertainty principle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' ∗kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='vasco@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='com †bernardob@ifes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='br ‡julio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='fabris@cosmo-ufes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='org §jose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='nogueira@ufes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='br 1 1 Introduction Amongst the (many) new concepts introduced by the quantum mechanics, the quantum entanglement [1, 2] is one that, probably, more contradicts our common sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Although, in the beginning, the quantum entanglement were only associated to theoretical aspects of quantum mechanics, specially those related to the non-locality or the complementarity (hidden variables) [3], nowadays it is a key component of the applications and experiments on quantum information, quantum computation and quantum teleportation [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The uncertainty principle is one of the fundamental cornerstones of the quantum me- chanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Nevertheless, it is a principle: its correct form can not be proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' That opens the way to the possibility that its canonical form, described by the Heisenberg’s uncer- tainty principle (HUP), can be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' An example of the possible generalizations of HUP, whose origin can be traced back to quantum gravity, is given by introducing a non-zero minimal uncertainty in the measurement of position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' That non-zero minimal uncertainty in position is, then, understood as a minimal-length scale, below which the necessary amount of energy to probe the position of a particle is so hight that it disturbs the space-time so that the concept of a length measurement loses its meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Hence, the- ories searching to describe a quantum approach for gravity lead generally to the existence of a minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In fact, a minimal length actually appears in almost all proposed theories of the quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' For this reason theories formulated in a minimal-length scenario are considered to be effective theories of quantum gravity [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In 1999, Yoon-Ho Kim and Yanhua Shih conducted an experiment whose results ap- parently suggested a violation of the HUP [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' However, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Rigolin pointed out that in fact there is no violation of the HUP, because the HUP is derived for particles non- correlated (non-entangled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In the Kim and Shih’s experiment, the photons of the pair are correlated (entangled) when one of the physical slits is replaced by a virtual slit1: the canonical HUP is no longer applicable since the quantum entanglement modifies the canonical HUP [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' An immediate question arising from the previous considerations is how the quantum entanglement modifies a generalized uncertainty principle (GUP), in others words, which is the effect of the quantum entanglement in the minimal-length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The answer for that question is important in order to know the role of the quantum entanglement at the minimal-length scale (maybe appearing in the Planck scale) or in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Unfortunately, this issue has been little considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In [13], G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Blado, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Herrera and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Erwin have studied the inseparability conditions with the most usual GUP correction, whereas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Park has used a coupled harmonic oscillator in order to find the effects of the quantum entanglement with a linear GUP in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The purpose of this work is to answer that question considering the main proposals of generalization for the HUP which take into account the existence of a minimal length in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' With this goal in mind, we will analyze the modifications in the HUP arising from the quantum entanglement of two identical particles determining the minimal uncertainty 1The interaction of the photon with a physical slit destroys the correlation between the photons of the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 2 associated to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In Section 2 we obtain an expression of the uncertainty principle for entangled states which is independent of the chosen GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In Section 3 we find the modified uncertainty principle for a pair of identical particles re- garding the main proposals of GUP’s: Kempf, Mangano and Mann GUP (KMM-GUP), Ali, Das and Vagenas GUP (ADV-GUP), Pedram GUP and exponential all orders GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In Section 4 we estimate an upper bound for the minimal-length value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' We present our conclusions in the Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 2 Uncertainty principle for entangled states The Hilbert space of the state vectors E of a system of N particles is given by the tensor product of the Hilbert spaces of the state vectors Ei of each particle [15, 16], E = E1 ⊗ · · · ⊗ EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (1) The position and momentum linear operators of the i-th particle which act on the state vectors |ψ⟩ ∈ E are the extensions ˜Qi and ˜Pi defined as ˜Qi = I1 ⊗ · · · ⊗ ˆxi ⊗ · · · ⊗ IN, (2) ˜Pi = I1 ⊗ · · · ⊗ ˆpi ⊗ · · · ⊗ IN, (3) where Ii is the identity operator in Ei and ˆxi and ˆpi are the position and the momentum operators of the i-th particle acting on the state vectors |ψi⟩ ∈ Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The extensions ˜Qi and ˜Pi do not satisfy the canonical uncertainty principle (HUP), because ˜Qi and ˜Pi are not physical observables [11, 12, 15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Physical observables are operators which commute with every permutation operators of the particles system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Hence, the operators ˜Q and ˜P, defined as ˜Q := N � i=1 ˜Qi (4) and ˜P := N � i=1 ˜Pi, (5) are physical observables and they satisfy the relation (∆Q)2 (∆P)2 ≥ 1 4 ���⟨[ ˜Q, ˜P]⟩ ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (6) The relation (6) is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' It does not depend whether the system of particle is entangled or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 3 As it was showed by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Rigolin [11, 12], if the state of the particles system is entangled then the operators ˜Qi and ˜Pi do not satisfy the canonical Heisenberg uncertainty principle (HUP) - as previously stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' We briefly review the Rigolin’s result for a two particles system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' From the definitions of ∆Q and ∆P we have2 (∆ψQ)2 = � ψ| ˜Q2|ψ � − � ψ| ˜Q|ψ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (7) From now on we omit the subscript ψ for the sake of simplicity, whenever this does not cause any confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Thus, (∆Q)2 = (∆Q1)2 + (∆Q2)2 + 2 �� ˜Q1 ˜Q2 � − � ˜Q1 � � ˜Q2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (8) In the same way, (∆P)2 = (∆P1)2 + (∆P2)2 + 2 �� ˜P1 ˜P2 � − � ˜P1 � � ˜P2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (9) Using the results (8) and (9) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (6) we obtain � (∆Q1)2 + (∆Q2)2 + 2 �� ˜Q1 ˜Q2 � − � ˜Q1 � � ˜Q2 ��� × � (∆P1)2 + (∆P2)2 + 2 �� ˜P1 ˜P2 � − � ˜P1 � � ˜P2 ��� ≥ 1 4 ���⟨[ ˜Q, ˜P]⟩ ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (10) We now use the functions CQ(1, 2) := � ˜Q1 ˜Q2 � − � ˜Q1 � � ˜Q2 � , (11) CP(1, 2) := � ˜P1 ˜P2 � − � ˜P1 � � ˜P2 � , (12) which are called quantum covariance functions (QCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' By definition, QCF’s vanish if and only if the system is separable [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Therefore (11) and (12) are zero for any not entangled quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Using the QCF’s (11) and (12) we have � (∆Q1)2 + (∆Q2)2 + 2CQ(1, 2) � � (∆P1)2 + (∆P2)2 + 2CP(1, 2) � ≥ 1 4 ���⟨[ ˜Q, ˜P]⟩ ��� 2 , (13) or 2 � i,j=1 CQ(i, j) 2 � k,l=1 CP(k, l) ≥ 1 4 ���⟨[ ˜Q, ˜P]⟩ ��� 2 , (14) since CQ(i, i) = (∆Qi)2, CP(i, i) = (∆Pi)2, CQ(i, j) = CQ(j, i) and CP(i, j) = CP(j, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 2Note that [ ˜Q1, ˜Q2] = 0 and [ ˜P1, ˜P2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 4 In this work, we concern with the case of an entangled system of two identical particles, so we are going to handle Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (13) in order to express it in a more appropriate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' For this end, we define |ψ′⟩ := � ˜Q1 − ˜Q2 � |ψ⟩, (15) with |ψ′⟩, |ψ⟩ ∈ E and ⟨ψ | ψ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Therefore, ⟨ψ′ | ψ′⟩ = (∆ψQ1)2 + (∆ψQ2)2 − 2 � ˜Q1 ˜Q2 � ψ + � ˜Q1 �2 ψ + � ˜Q2 �2 ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (16) Now, using the Schwarz inequality, ⟨ψ | ψ⟩ ⟨ψ′ | ψ′⟩ ≥ ⟨ψ | ψ′⟩ ⟨ψ′ | ψ⟩, we have (∆ψQ1)2 + (∆ψQ2)2 ≥ 2 �� ˜Q1 ˜Q2 � ψ − � ˜Q1 � ψ � ˜Q2 � ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (17) In the same way (∆ψP1)2 + (∆ψP2)2 ≥ 2 �� ˜P1 ˜P2 � ψ − � ˜P1 � ψ � ˜P2 � ψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (18) Finally, from inequalities (17), (18) and (13) we obtain � (∆Q1)2 + (∆Q2)2� � (∆P1)2 + (∆P2)2� ≥ 1 16 ���⟨[ ˜Q, ˜P]⟩ ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (19) In the case where (∆Q1)2 = (∆Q2)2 and (∆P1)2 = (∆P2)2 the inequality (19) becomes ∆Qi∆Pi ≥ 1 8 ���⟨[ ˜Q, ˜P]⟩ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (20) It is worth noting that the expression of the inequality (20) is independent of the chosen uncertainty principle that does not take into account the quantum correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' This uncertainty principle is related to the commutation relation [ ˜Q, ˜P].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 3 Uncertainty principle for entangled states in different minimal- length scenarios In this section we consider a system of two entangled identical particles whose momenta have the same value but opposite directions, that is, ⃗p1 = −⃗p2, just as in the Kim and Shih’s experiment [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Therefore, in this case ⟨ˆp1⟩ + ⟨ˆp2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Moreover, such a consideration also allows us to estimate, in the next section, an upper bound for the value of the minimal length based on the experimental results obtained by Kim and Shih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='1 Heisenberg uncertainty principle Before we consider a minimal-length scenario it is appropriate to determine the change in the canonical HUP, that is, in a scenario in which effects of quantum gravity are not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The canonical HUP for states of one simple-particle is ∆x∆p ≥ ℏ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (21) The commutation relation related to the HUP is [ˆx, ˆp] = iℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (22) Hence [ ˜Q, ˜P] = [ ˜Q1 + ˜Q2, ˜P1 + ˜P2] = 2iℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (23) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (23) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (20) we get ∆Qi∆Pi ≥ ℏ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (24) The result (24) shows that for a system of two entangled identical particles the HUP is modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Such an outcome is not new, it was already obtained by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Rigolin in 2002 [11] and then in 2016 [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' From a quick glance at the result (24) and recalling that dimensionally (∆Q)min ∝ ℏ, we expect the minimal uncertainty in the position will be reduced by half for all GUP’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='2 KMM GUP The GUP ∆xi∆pi ≥ ℏ 2 � 1 + β (∆pi)2 + β ⟨ˆpi⟩2� , (25) where β is a parameter related to the minimal length, has been proposed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Kempf, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Mangano an R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Mann (KMM-GUP) [18] and it is the most used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The commutation relation related to it is given by [ˆxi, ˆpi] = iℏ � 1 + βˆp2 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (26) Hence [ ˜Q, ˜P] = iℏ � 1 + β � ˜P 2 1 + ˜P 2 2 �� = iℏ � 1 + 2β ˜P 2 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (27) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (27) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (20) we get ∆Qi∆Pi ≥ ℏ 4 � 1 + β (∆Pi)2 + γ � , (28) where γ := β � ˜Pi �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 6 The modified KMM-GUP (28) induces the existence of a minimal uncertainty given by (∆Qi)min = ℏ 2 � β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (29) The result above shows that the non-zero minimal uncertainty in position induced by the KMM-GUP for two entangled identical particles is twice smaller than for a separable system of two identical particles (non-entangled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='3 ADV GUP A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Farag Ali, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Das and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Vagenas have proposed a GUP related to a commutation relation which has a linear and a quadratic term in the momentum operator [19], [ˆxi, ˆpi] = iℏ � 1 − 2αˆpi + 4α2ˆp2 i � , (30) where α is a parameter related to the minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Besides the existence of a minimal length this linear approach induces a maximal uncertainty in the momentum, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Then, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (30) we get [ ˜Q, ˜P] = 2iℏ � 1 − α � ˜P1 + ˜P2 � + 2α2 � ˜P 2 1 + ˜P 2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (31) Therefore, ⟨[ ˜Q, ˜P]⟩ = 2iℏ � 1 − α �� ˜P1 � + � ˜P2 �� + 2α2 �� ˜P 2 1 � + � ˜P 2 2 ��� , (32) ⟨[ ˜Q, ˜P]⟩ = 2iℏ � 1 + 4α2 � ˜P 2 i �� , (33) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (33) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (20) we obtain ∆Qi∆Pi ≥ ℏ 4 � 1 + 4α2 (∆Pi)2 + γ′ � , (34) where γ′ := 4α2 � ˜Pi �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The modified ADV-GUP (34) induces the existence of a non-zero minimal uncertainty given by (∆Qi)min = ℏα, (35) which once again is twice smaller than for a non-entangled system of two particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' It is important to note that the linear GUP (ADV-GUP) becomes non-linear in this case and consequently a maximal uncertainty in the momentum is no longer induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='4 Pedram GUP In order to overcome some problems arising from KMM-GUP and ADV-GUP - such as incorporation of a maximal momentum required in doubly special relativity (DSR) theories, commutative geometry and an approach valid for all order in the parameter related to the minimal length - P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Pedram has proposed a GUP [20, 21] based on the commutation relation given by [ˆxi, ˆpi] = iℏ 1 − βˆp2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (36) Hence [ ˜Q, ˜P] = iℏ 1 − β ˜P 2 1 + iℏ 1 − β ˜P 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (37) Using the so-called Jensen’s Inequality [22] we have ⟨[ ˜Q, ˜P]⟩ ≥ iℏ 1 − β � ˜P 2 1 � + iℏ 1 − β � ˜P 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (38) Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (38) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (20) we obtain ∆Qi∆Pi ≥ ℏ 4 1 � 1 − β (∆Pi)2 − γ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (39) Therefore, the modified Pedram-GUP (39) introduces a non-zero minimal uncertainty given by (∆Qi)min = 3ℏ 8 � 3β, (40) which is also twice smaller than for a non-entangled system of a pair of identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='5 Exponential all orders GUP In the canonical field theory in the context of non-commutative coherent states repre- sentation and field theory on non-anticommutative superspace the Feynman propagator displays an ultra-violet (UV) cut-off of the form e−βp2 [23, 24, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In consequence, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Nouicer has proposed an exponential all orders GUP [27, 28] based on the commutation relation given by [ˆxi, ˆpi] = iℏeβˆp2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (41) Hence [ ˜Q, ˜P] = iℏ � eβ ˜P 2 1 + eβ ˜P 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (42) Again, using the so-called Jensen’s Inequality [22] we have ⟨[ ˜Q, ˜P]⟩ ≥ iℏ � eβ⟨ ˜P 2 1⟩ + eβ⟨ ˜P 2 2⟩� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (43) 8 Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (42) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (20) we get ∆Qi∆Pi ≥ ℏ 4eβ(∆Pi)2+γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (44) Therefore, the modified exponential all order GUP (44) introduces a minimal uncer- tainty given by (∆Qi)min = ℏ 2 � eβ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (45) Finally, we note that (∆Qi)min is also twice smaller than for a non-entangled system of a pair of identical particles, as we have already said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='6 Minimal length If a non-zero minimal uncertainty in position can be interpreted as a minimal length then the previous results show that the minimal length for two entangled identical particles can be twice smaller than for a separable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' It is clear this statement must be taken with care because a minimal length should be a constant, that is, it must not depend on the physical system, it is a quantum gravitation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In fact, the minimal length should be an invariant as well as the light speed is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The answer for that apparent contradiction is possibly because the system is made up of two particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In references [29], C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Quesne and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Tkachuck have claimed that for a system composed by N particles the effective parameter related to the minimal length, β, is reduced by a factor 1 N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Hence, ℏ 2 √β is not the correct minimal uncertainty in position (∆Qi)min for the modified KMM-GUP, but ℏ 2 √βi, where [29, 30] β = βi 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (46) Consequently, we find that (∆Qi)min = ℏ√β, therefore lmin = ℏ√β, as we expected3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In the same way for others GUP’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The reader might want to claim that the minimal length is actually described by the minimal uncertainty of the entangled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' However, according to reference [12] we can presume that the minimal uncertainty for a system of N entangled identical particles is reduced by 1 N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Since, in principle, there is no limit for the number of particles for a entangled system, there would also be no limit for the minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 4 Upper bound for the minimal-length value In the Kim and Shih’s experiment entangled identical pairs of photons were produced by spontaneous parametric down conversion (SPDC) with momentum conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' A narrow physical slit was placed along the trajectory of one of the photons, whereas the 3It is worth noting that if the minimal uncertainty was greater for the entangled system, we could sup- pose that the quantum entanglement decreases the accuracy of a position measurement thereby increasing the minimal uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 9 other photon of the pair (called 2) passed through a virtual slit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' The ghost image exper- imental technique [31] ensured that the quantum correlation between the pair of photons was not destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Then simultaneous detection of photons of the pairs were performed and data just for coincidence events were obtained in case when the photon 2 passed through a virtual slit (non-slit case) and in case when the photon 2 passed through a physical slit (slit case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' From the experimental data obtained by Kim and Shih one gets that [12] ∆P ns 2 ∆P s 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='15, (47) where ∆P ns 2 is uncertainty in momentum of the photon 2 in the non-slit case and ∆P s 2 is uncertainty in momentum of the photon 2 in slit case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Since the width of the slit was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='16 mm (∆Q2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='16 mm), the uncertainty in momentum in the slit case for KMM-GUP can be find from4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='16∆P s 2 = ℏ 2 � 1 + 4β (∆P s i )2 + γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (48) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (48) has real roots only if β ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='162 ℏ2η , where η := 1 + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Thus, ∆P s 2+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='32 ℏβ − ℏη 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='32 − β ℏ3η2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='323 (49) and ∆P s 2− = ℏη 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='32 + β ℏ3η2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='323, (50) Now, using the above results we can estimate an upper bound for the minimal-length value induced by KMM-GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Hence, substituting the root (49) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (28) we have that β ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='58 × 10−2 ℏ2η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (51) Therefore, lmin ≤ ℏ � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='58 × 10−2 ℏ2η ≤ ℏ � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='58 × 10−2 ℏ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content='9 × 10−4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (52) The substitution of the root (50) hold the inequality (28) for all β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Consequently, the upper bound for the minimal-length value is order 10−4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' There- fore, using the experiment described above a result with less restrictions than those re- ported in the literature is obtained [30, 32, 33, 34, 35, 36, 37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 4Note that in with-slit case the correlation between the photons of the pair was destroyed, therefore the photons were not entangled and the usual KMM-GUP, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' (25), is held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' 10 5 Conclusion In this work we find the non-zero minimal uncertainties induced by the main proposals of GUP’s (KMM, ADV, Pedram and exponential) which are modified due to the quantum entanglement of a system of two identical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' In principle, our results have pointed out that the minimal uncertainties are reduced at half for a system of two entangled identical particles independently of the GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Hence, if a non-zero minimal uncertainty in position can be interpreted as a minimal length then the quantum entanglement reduces by half the minimal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' However, the minimal length must not depend on the phys- ical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' We overcome this apparent paradox by using the Quesne and Tkachuck’s proposal for a system composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Consequently, despite the quantum entanglement to change the GUP, the minimal length does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Based on our results and using the reference [12] we can expect that the apparent minimal uncertainty for a system of N entangled identical particles is reduced by 1 N , nonetheless the minimal length does not change because the effective parameter β is also reduced by a factor 1 N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Finally, we have estimated from the data obtained from the Kim and Shih’s experiment an upper bound value for the minimal length of the order of 10−4 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Consequently, it is rather an inexpressive value (in the sense of leading to poor predictive power) as compared to ones have been found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' This is due to the high imprecision of the experiment on the entangled system described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' However, we may expect that more refined version of the experiment may lead to more stringent bounds on the minimum length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tFLT4oBgHgl3EQfAi4b/content/2301.11966v1.pdf'} +page_content=' Acknowledgements We would like to thank FAPES, CAPES and CNPq (Brazil) for financial support.' 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to deceive a Face Recognition (FR) system by presenting a morphed image consisting of the +biometric qualities from two different identities with the aim of triggering a false acceptance with one of the two identities, thereby +presenting a significant threat to biometric systems. The success of a morphing attack is dependent on the ability of the morphed +image to represent the biometric characteristics of both identities that were used to create the image. We present a novel morphing +attack that uses a Diffusion-based architecture to improve the visual fidelity of the image and improve the ability of the morphing attack +to represent characteristics from both identities. We demonstrate the high fidelity of the proposed attack by evaluating its visual fidelity +via the Fr´echet Inception Distance. Extensive experiments are conducted to measure the vulnerability of FR systems to the proposed +attack. The proposed attack is compared to two state-of-the-art GAN-based morphing attacks along with two Landmark-based attacks. +The ability of a morphing attack detector to detect the proposed attack is measured and compared against the other attacks. +Additionally, a novel metric to measure the relative strength between morphing attacks is introduced and evaluated. +Index Terms—Biometrics, Morphing Attack, GAN, Vulnerability Analysis, Face Recognition, Diffusion Models +! +1 +INTRODUCTION +F +ACE recognition (FR) systems have become one of the most +common biometric modalities used for identity verification +across a wide range of modern-day applications, from trivial tasks +such as unlocking a smart phone to official businesses such as +banking, e-commerce, and law enforcement. Unfortunately, while +FR systems can often reach low false rejection and acceptance +rates [1], they are especially vulnerable to a new class of emerging +attacks, known as the face morphing attack [2], [3], [4], [5]. The +face morphing attack aims to compromise a fundamental property +of biometric security, i.e., the one-to-one mapping from biometric +data to the associated identity. This compromise is achieved +by creating a morphed face which contains biometric data of +both identities in such a manner that presenting one morphed +image triggers a match with two disjoint identities, violating the +fundamental principle. +This poses a significant threat towards FR systems, especially +the applications of e-passports and border access. Notably the e- +passport scenario, wherein the applicant submits a passport photo +either in digital or printed form, is especially vulnerable to face +morphing attack. Moreover, digital morphs can be easily generated +hence offer a low-cost attack in the digital domain. It is not un- +common for a digital image submitted to a document submission +portal to not have its authenticity verified by a human agent [6]. +Critically, an adversary who is blacklisted from accessing a certain +system can create a morph with a non-blacklisted individual to +gain access. This is especially relevant for countries such as New +Zealand, Estonia, and Ireland, where e-passports are used for both +issuance and renewal of documents [7]. In 2018 a German activist +was reported to have received a German passport with a photo of +his faced morphed with an Italian politician [8]. +Due to the severity of face morphing attacks, an abundance of +algorithms have been developed to identify these attacks [2], [9], +[10], [11]. Methods for Morphing Attack Detection (MAD) can +be broadly characterized into two classes based on the manner in +which they obtain the features used for detection, i.e., handcrafted +features or deep features. Handcrafted features are used in the +so-called classical algorithms which seek to find evidence of the +morphing attack in the pixel domain, whether that is evidence +of degradation in image quality [10], residual noise left from +the morphing attack [12], or local geometric features such as +Local Binary Patterns (LBP) [13], Binarized Statisical Image +Feature (BSIF) [14], and Local Phase Quantitization (LPQ) [4]. +Conversely, deep features are used with deep learning based algo- +rithms. These features are often extracted by a deep Convolutional +Neural Network (CNN) often from a pre-trained network such as +ResNet150 [6], [15]. Generally, the best success has been found +using deep CNN-based features in comparison to handcrafted +features on both digital and print-scan face data [16]. +Comparatively, there has been less research on face mor- +phing attack algorithms. Similar to the two classes of MAD +algorithms, there exist two broad classes of face morphing at- +tacks: Landmark-based attacks and deep learning-based attacks. +Landmark-based morphing attacks use local features to create the +morphed image by warping and aligning the landmarks within +each face to then create a morphed face by pixel-wise compositing. +Landmark-based attacks have been shown to be effective against +FR systems [17]. Recent work has enhanced the effectiveness of +Landmark-based attacks by using adversarial perturbation [18]. +In contrast, deep learning-based morphing attacks use a machine +learning model to embed the original bona fide faces into a +semantic representation which are then combined to produce a +new representation which should contain information from both +identities. This is new representation is then used to generate a +morphed face. +However, nearly all state-of-the-art deep learning based morph +methods were based on the Generative Adversarial Network +arXiv:2301.04218v1 [cs.CV] 10 Jan 2023 + +2 +(a) Identity a +(b) Morphed image +(c) Identity b +Fig. 1: Diffusion-based morphing attack. Samples are from FRLL dataset. +(GAN) framework [19], [20] with the primary difference being +architectural improvements such as using the StyleGAN2 [21] +architecture over the vanilla GAN architecture or changes to +the morph generation pipeline as seen in the Morphing through +Identity Driven Prior GAN (MIPGAN) [22]. At the same time, +there exists a handful of alternative state-of-the-art deep generative +models which offer their own advantages in terms of visual +fidelity, semantic representation capabilities, and inference speed. +In particular, a class of generative models collectively known +as “Diffusion models” haven been shown to possess high visual +fidelity, beating even state-of-the-art GANs in visual fidelity [23], +at the cost of increased inference time. As visual fidelity and +semantic representation abilities are far more important for the +potency of a potential morphing attack than inference speed, we +present a novel methodology for generating strong face morphing +attacks by leveraging Diffusion-based methods. Figure 1 shows a +morphed image generated by the proposed attack constructed from +two identities from the FRLL dataset [24]. We present a summary +of contributions of the proposed work as shown below: +• +We propose a novel method for generating morphed faces +by using a Diffusion-based model which calculates twin +embeddings, one for the semantic details and one for +the stochastic details, to generate images of high visual +fidelity. +• +The proposed morphing attack is evaluated against four +other morphing attacks by the Fr´echet Inception Distance, +a quantitative measure of visual fidelity. +• +We evaluated our proposed attack against four other mor- +phing attacks with extensive experiments assessing the +vulnerability of two FR systems across three different +datasets. +• +The proposed attack is further evaluated its ability to +evade detection from MAD algorithms trained against +other morphing attacks. +• +We introduce and evaluate a novel metric to measure the +relative strength from one morphing attack to another. +• +A small study on the impact of the pre-processing in +the FR pipeline on the vulnerability of FR systems to +morphing attacks is presented. +• +Four variants of the proposed morphing attack are com- +pared against each other, resulting in the creation of over +10,000 new morphed images between them. +2 +PRIOR WORK +Several face morphing attacks have been developed by researchers +with the morph generating process generally using face land- +marks or deep learning. In particular we compare our proposed +Diffusion-based attack against four state-of-the-art morphing at- +tacks, two deep learning-based and two Landmark-based; namely, +the OpenCV, FaceMorpher, StyleGAN2, and MIPGAN-II face +morphing attacks. These were chosen as they represent several +types of morphing attacks and provide a substantial baseline to +measure the performance of the proposed Diffusion-based attack. +To the best of our knowledge all deep learning-based attacks +generate the morphed images via a type of GAN architecture [17], +[19], [22], [25]. +2.1 +Landmark-Based Morphing Attacks +The FaceMorpher and OpenCV attacks were chosen to represent +the Landmark-based attacks as they are commonly used to repre- +sent this class of attack [11], [17], [26]. +FaceMorpher is an open-source algorithm that uses the +STASM landmark detector [27], [28]. From the landmarks on the +images Delaunay triangles are formed, which are then warped and +blended together. The areas outside the landmarks are averaged, +typically introducing strong artifacts in the neck and hair regions +of the image [29]. +OpenCV is a morphing attack which uses the open-source +OpenCV library with a 68-point annotator from the Dlib li- +brary [30]. The images and associated landmarks are used to form +Delaunay triangles. Then, in a similar manner to FaceMorpher, +the landmarks are warped and blended. In contrast to the approach +of FaceMorpher, the areas outside the landmarks do not consist +of an averaged image, but rather additional Delaunay triangles. +However, these morphs also exhibit strong artifacts outside the +facial area due to the missing landmarks [29]. +2.2 +GAN-Based Morphing Attacks +As mentioned earlier, prior deep learning-based attacks have used +a GAN architecture for the morph generation process [26], [31]. +GANs are a type of deep generative model which seeks to learn +the sampling process for some data distribution Pdata on X , i.e., +given some simple distribution Z ∼ p(z) on Z, the generator G : +Z → X is to learn G(Z) ∼ Pdata. A discriminator, sometimes + +3 +Fig. 2: Proposed architecture for Diffusion-based morphs, where the green traces indicate variables associated with identity a, likewise +red traces denote identity b, and blue traces for the morphed identity ab. +called the critic, D : X → [0, 1] is trained adversarially against +the generator in a minimax game described by +min +G max +D +E +x∼Pdata log D(x) + +E +z∼p(z) log(1 − (D ◦ G)(z)) (1) +where the discriminator attempts to get better at distinguishing +synthetic samples from genuine samples, while the generator tries +to get better at deceiving the discriminator. For a GAN-based mor- +phing attack it is necessary for an encoding algorithm E : X → Z +to exist, which can embed images in the latent space such that the +inversion has low distortion (G ◦ E)(x) ≈ x. The latent codes +for two identities are then averaged to produce a new latent code +representing the morphed face which is passed to the generator. +Notably, there exists a trade-off between the inversion distortion +and editability of the latent embeddings [32] Damer et al. proposed +to use the GAN architecture for generating morphs by combining +two latent codes encoded from two real identities to create a +morphed code [19]. This proposed attack, known as MorGAN, +was based on a modification to the vanilla GAN architecture +by the addition of an explicitly defined encoder architecture that +was trained jointly with the generator via a modified adversarial +loss formulation. Since then the StyleGAN2 [17] and MIPGAN- +II [22] attacks have improved upon the MorGAN formulation by +improving the GAN architecture, loss formulation, or encoding +algorithm. +StyleGAN2 morphs offer promising results as shown in Damer +et al. [19]. StyleGAN2 offers a host of improvements over the +standard GAN implementation that enables the architecture to +achieve state of the art image quality when generating high resolu- +tion images. The StyleGAN2 model was pre-trained on the Flickr- +Faces-HQ (FFHQ) dataset [33]. The faces were then cropped to +possess the same landmark alignment as in the FFHQ dataset. +Following the approach in [21], the images, xa, xb, are embedded +by optimizing an initial latent code through stochastic gradient +descent, minimizing the perceptual loss between the generated +image and target image. After each embedded latent code, za, zb, +is found, a morphed latent code is created by linearly interpolating +between the two, zab = lerp(za, zb; 0.5). Lastly, the interpolated +latent code is passed to the StyleGAN2 synthesis network to get +the morphed image xab. The StyleGAN2 morphs are strong when +used with images containing a uniform background, which makes +them especially powerful when used in conjunction with the Face +Research Lab London (FRLL) dataset [24]. +Fig. 3: The forward and reverse Diffusion processes. +MIPGAN-II proposes an extension on StyleGAN2 by adding +an optimization procedure for the latent vector used in creating +the morphed image [22]. The StyleGAN2 portion of MIPGAN- +II was pre-trained on the FFHQ dataset. The two bona fide +images are embedded into the latent space using the StyleGAN2 +optimization procedure. The latent code is initially constructed +as z0 = lerp(za, zb; 0.5). For n epochs the latent code is +optimized to minimize a combination of perceptual loss, identity +loss, identity difference loss, and Multi-Scale Structural Similarity +loss, finding a fully optimized latent zn. The latent code zn is then +passed to the StyleGAN2 synthesis network to create the morphed +image. As MIPGAN-II presents a refinement on StyleGAN2 for +the application of morphing attack, it possesses similar advantages +and disadvantages that StyleGAN2 morphs offer. +3 +DIFFUSION-BASED MORPHING ATTACK +Unlike GANs which learn the sampling process for the data +distribution through adversarial training between the generator +and the critic, diffusion-based and score-based generative models +learn the data distribution by learning a denoising process for +varying noise levels. Diffusion-based models can achieve image +fidelity superior to state-of-the-art generative models, matching +even the acclaimed BigGAN-deep [34] model, while maintaining +better coverage of the data distribution [23]. For these reasons we +propose a morphing attack that uses Diffusion-based methods as +the generative process. +3.1 +Diffusion Models +Given data distribution q(x0) on data space X , the goal is to +learn a model pθ(x0) approximating q(x0) which can be easily +sampled. Denoising Diffusion Probabilistic Models (DDPMs) [35] +are latent variable models of the form +pθ(x0) = +� +pθ(x0:T ) dx1:T +(2) + +(x, x) +lz(za, Zb) +E(xo) +q(XtXt-1, z) +lx(x(), x) +pe(Xt-1|xt, z)XT +Xt +Xt-1 +X04 +Algorithm 1 Diffusion Morphing Algorithm. +Require: Bona fide images, x(a) +0 , x(b) +0 +∈ X , image space preprocessing function, ξ : X ×X → X , image space interpolation function, +ℓX : X × X × [0, 1] → X , and latent space interpolation function, ℓZ : Z × Z × [0, 1] → Z. +procedure DIFFUSIONMORPH(x(a) +0 , x(b) +0 ) +za ← E(x(a) +0 ) +▷ Calculate semantic latent codes +zb ← E(x(b) +0 ) +x(a) +0 +← ξ(x(a) +0 , x(b) +0 ) +▷ Preprocess images passed to stochastic encoder +x(b) +0 +← ξ(x(b) +0 , x(a) +0 ) +t ← 0 +while t < T do +x(a) +t+1 ← √αt+1f (t) +θ (x(a) +t +, za) + √1 − αt+1ϵ(t) +θ (x(a) +t +, za) +▷ Forward pass of diffusion algorithm +x(b) +t+1 ← √αt+1f (t) +θ (x(b) +t , zb) + √1 − αt+1ϵ(t) +θ (x(b) +t , zb) +t ← t + 1 +end while +x(ab) +T +← ℓX (x(a) +T , x(b) +T ; 0.5) +▷ Stochastic code interpolation +zab ← ℓZ(za, zb; 0.5) +▷ Semantic code interpolation +t ← T +while t > 0 do +x(ab) +t−1 ← √αt−1f (t) +θ (x(ab) +t +, zab) + √1 − αt−1ϵ(t) +θ (x(ab) +t +, zab) +▷ Diffusion generative process +t ← t − 1 +end while +return x(ab) +0 +end procedure +where {xt}T +t=1 ∈ X are latent variables and T ∈ N. The reverse +process is a Markov chain starting at pθ(xT ) = N(0, I), i.e., a +normal distribution with mean vector 0 and variance I (the identity +matrix), with Gaussian transitions +pθ(x0:T ) = pθ(x0) +T +� +t=1 +pθ(xt−1|xt) +(3) +The diffusion (forward) process is fixed to a Markov chain that +gradually adds Gaussian noise to the original sample x0 according +to variance schedule {βt}T +t=1, such that +q(x1:T , x0) = +T +� +t=1 +q(xt|xt−1) +(4) +and q(xt|xt−1) = N(√1 − βtxt−1, βtI). See Figure 3 for an +illustration of this process. The transition probability p(xt−1|xt) +is likely to be very complex, unless the gap between t and t − 1 is +very small, i.e., T → ∞. In this case, p(xt−1|xt) can be modelled +as N(µ(t) +θ (xt), σt), where µ(t) +θ +: X → X is an estimator at step +t parameterized by θ. Ho et al. [35] proposes to use the following +form: +µ(t) +θ (xt) = +1 +√αt +� +xt − +βt +√1 − ¯αt +ϵ(t) +θ (xt) +� +(5) +where αt = 1 − βt, ¯αt = �t +s=1 αs, and ϵ(t) +θ +: X → X is a +function approximator parameterized by θ, which learns to predict +the noise added to x0 to get xt. This is achieved by using a U-Net, +a type of CNN consisting of several skip connections formed at +each resolution size in the architecture to model ϵ(t) +θ +[36]. +Like with Variational Autoencoders, the model is trained by +optimizing the variational bound on the negative log likelihood: +E[− log pθ(x0)] ≤ E +� +− log pθ(xT ) − +T +� +t=1 +log pθ(xt−1|xt) +q(xt|xt−1) +� +Relaxing the constraint that the inference process has to be +Markovian leads to another kind of diffusion model known as the +Denoising Diffusion Implicit Model (DDIM) as proposed by Song +et al. [37]. This model has the benefit of a deterministic generative +process such that: +xt−1 = √αt−1f (t) +θ (xt) + +� +1 − αt−1ϵ(t) +θ (xt) +(6) +where +f (t) +θ (xt) = +1 +√αt +(xt − +√ +1 − αtϵ(t) +θ (xt)) +(7) +which allows for deterministic embedding of the image x0 into a +latent representation xT . However, xT does not contain high-level +semantics, which is a necessary property for deep learning-based +morphing attacks. +3.2 +Diffusion Autoencoders +Preechakul et al. [38] proposed a Diffusion Autoencoder by +employing a conditional DDIM. The DDIM is conditioned on a +semantic representation z ∈ Z by modifying ϵ(t) +θ +to take z as an +additional input. Training is done via a simplified loss [35] defined +as: +L = +T +� +t=1 +E +x0∼q(x0) +ϵt∼N (0,I) +||ϵ(t) +θ (xt, z) − ϵt||2 +2 +(8) +Therefore, Equation (6) functions as a decoder network from latent +code (xT , z). Let E : X → Z be the semantic encoder network +such that +z = E(x0) +which +provides +the +information +necessary +for +decoder +pθ(xt−1|xt, z) to properly denoise images. Likewise, the stochas- +tic encoder can be thought of as a reversal of Equation (6) such +that +xt+1 = √αt+1f (t) +θ (xt, z) + +� +1 − αt+1ϵ(t) +θ (xt, z) +(9) + +5 +where xT is encouraged to encode only stochastic details left out +from the semantic representation z. +3.3 +Proposed Morphing Algorithm +We propose a novel process for the creation of morphed images +by employing both the stochastic and semantic encoders. In +particular, let xa, xb ∈ X be two bona fide images of identities +a, b, and let x(a) +0 += xa and x(b) +0 += b. Algorithm 1 outlines the +structure of the proposed Diffusion-based attack, hereafter called +the Diffusion attack for simplicity, with additional illustration +provided in Figure 2 +Beyond the core components of the DDIM and semantic +encoder, three additional functions are added to the architec- +ture. Namely, these are the image space preprocessing func- +tion, ξ : X × X → X , image space interpolation function, +ℓX : X ×X ×[0, 1] → X , and latent space interpolation function, +ℓZ : Z × Z × [0, 1] → Z. +The interpolation functions are used to interpolate between +the semantic and stochastic values by some factor γ ∈ [0, 1]. +The image space preprocessing function is used to prepare the +image passed to the semantic encoder. The simplest form of +the interpolation function is the linear interpolation function, +lerp(a, b; γ) = γa+(1−γ)b, where linear interpolation was found +to be the best choice for ℓZ. However, for ℓX Song et al. [37] +suggests the usage of spherical linear interpolation. For a vector +space V and two vectors u, v ∈ V the spherical interpolation by +a factor of γ is given as +slerp(u, v; γ) = sin((1 − γ)θ) +sin θ +u + sin(γθ) +sin θ v +(10) +where θ = arccos(u·v) +||u|| ||v|| . +While the semantic code provides most of the fundamental +information, such as positioning of facial features, the stochastic +code is used to provide information on the details not explicitly +associated with the identity, but are necessary for realism of the +generated image. By altering the stochastic code, details such as +direction of strands of hair, clothing, etc., are altered whilst the +identity of the image is preserved [38]. Unlike the rather straight- +forward nature of linearly interpolating between the semantic +codes to produce an image with key identifying characteristics of +both identities, the nature of the stochastic code can lead to images +of low visual fidelity if the interpolation is not done carefully. +In particular, linear interpolation between two stochastic codes +is does guarantee a smooth interpolation between the stochastic +details in the images. For this reason the preprocessing function, +ξ, is used to prepare the image passed to the stochastic encoder. +One strategy is to “pre-morph” the image when extracting the +stochastic details, i.e., ξ performs an image space morph of the +image with the goal of reducing the artifacts induced by stochastic +interpolation. +4 +EXPERIMENTAL SETUP +To evaluate the effectiveness of the proposed morphing attack, the +attack is evaluated on three datasets against two different state-of- +the-art FR systems. All training, optimization, and evaluation was +conducted on a system with dual Intel Xeon Silver 4114 CPUs +and a NVIDIA Tesla V100 32GB GPU with CUDA version 10.1 +and CUDNN version 8.4. The proposed morphing attack, MAD +algorithm, and the FR systems are implemented in PyTorch [39]. +4.1 +Face Recognition Systems +In order to evaluate the strength of the proposed morphing attack, +two publicly available FR systems are used, specifically, the +FaceNet and VGGFace2 models. These models are widely used +and representative recognition systems with state-of-the-art face +verification performance [40], [41]. For both models the last fully +connected layer is used to provid a rich feature representation +of the input image. Then for a presented face, its feature vector +is compared with that of the feature vector belonging to the +target face. If the distance between these two representations is +sufficiently “small”, the presented face is then said to have the +same identity as the target face. The VGGFace2 model improves +upon acclaimed VGGFace [42] by using an improved training +dataset, also called VGGFace2. Cao et al. [40] present the SENet +architecture—the Squeeze and Excitation Network (SENet) was +introduced by Hu et al. [43]—as the optimal choice when used +with the VGGFace2 dataset. Google’s FaceNet model consists of +an Inception-ResNet V1 architecture which is pre-trained on the +VGGFace2 dataset [41]. +Additionally, the two FR systems use a different pre- +processing pipeline. As all datasets the images and generated +morphs are cropped as to be appropriate for passport photos, a +face extractor such as MTCNN [44], is omitted in the verification +pipeline. The FaceNet model resizes the image such that the short +side of the image is 180 pixels long and then the image is cropped +to a 160 × 160 resolution. Lastly, the images are normalized to +[−1, 1]. The VGGFace2 model resizes the image such that the +short side of the image is 256 pixels long and then crops the +image to 224 × 224 pixels. The mean RGB vector1 is subtracted +from the cropped image to normalize the image. +4.2 +Datasets +In this work, the FERET [45], FRLL [24], and FRGC v2.0 [46] +datasets were used to evaluate the proposed technique, as they +are commonly used in MAD with a large number of different +identities [17], [26]. Notably, the FRLL dataset consists of high +quality close-up frontal images at a 1350 × 1350 resolution +with 189 facial landmarks—a large number of landmarks. The +StyleGAN2, MIPGAN-II, and diffusion models were all trained on +the FFHQ dataset, which contains 70,000 images at a 1024×1024 +resolution [33]. Morphs using OpenCV, FaceMorpher, and Style- +GAN2 were created by Sarkar et al. [17] on the FRLL, FERET, +and FRGC datasets. Additionally, Zhang et al. [22] created morphs +via MIPGAN-II on the three datasets. +In order to create a morphed face, two component identities are +needed. Naturally, if the two component identities are disparate, +the resulting morph is likely to be very weak. To rectify this and +for evaluation purposes the component identity pairs were selected +by following the existing protocol offered by Sarkar et al. [17]. +These pairings resulted in 1222 unique morphs on FRLL, 964 on +FRGC, and 529 on FERET. +5 +RESULTS +The proposed morphing attack is compared to state-of-the-art +techniques drawing from both GAN-based and Landmark-based +methods. The effectiveness of the proposed method is quantita- +tively measured on three fronts, these are: the visual fidelity of +1. The mean vector is specifically ⟨131.0912, 103.8827, 91.4953⟩ for the +red, green, and blue channels. + +6 +(a) Identity a +(b) OpenCV +(c) StyleGAN2 +(d) Diffusion +(e) MIPGAN-II +(f) FaceMorpher +(g) Identity b +Fig. 4: Different generated morphs from two identities from the FRLL dataset. +(a) Diffusion +(b) MIPGAN-II +Fig. 5: Comparison of Diffusion and MIPGAN-II morphed faces +on FRGC. Images are resized to 256 × 256 and cropped to 224 × +224 to match VGGFace2 pre-processing pipeline. +the generated morphed images, the vulnerability of state-of-the-art +FR systems to the morphing attack, and the detection potential of +the morphing attack, respectively. Furthermore, an exploration of +interpolation techniques for the stochastic latent code is provided. +5.1 +Evaluation of Visual Fidelity +The visual fidelity of the Diffusion attack is compared against +other morphing attacks. Whilst on first glance it may appear that +the ability to deceive a FR system should imply a high level of +visual fidelity, this is not a simple assertion. We posit two reasons +for this discrepancy: +1) +The image pre-processing pipeline for an FR system may +crop out a significant portion of artifacts in the original +morphed image. +2) +The FR system is vulnerable to forms of adversarial +attacks which can degrade the performance of the FR sys- +tem while injecting noticeable artifacts into the morphed +image. +This can lead to a situation wherein the FR system is fooled by a +morphed image; however, it would be trivial for a human agent to +TABLE 1: FID across different morphing attacks. Lower is better. +Morphing Attack +FRLL +FRGC +FERET +StyleGAN2 +45.19 +86.41 +41.91 +FaceMorpher +91.97 +88.14 +79.58 +OpenCV +85.71 +100.02 +91.94 +MIPGAN-II +66.41 +115.96 +70.88 +Diffusion +42.63 +64.16 +50.45 +notice the artifacts present in the image. Moreover, a deep learning +system could be specifically trained to notice such artifacts, greatly +reducing the potential of such an attack to go undetected. +To quantitatively assess the visual fidelity of the generated +images the Fr´echet Inception Distance (FID) is employed, as it has +shown to correlate well with human assessment of fidelity [47]. +The FID is a measure of distance between the generated and +target distributions, therefore, the lower the FID metric the more +similar the generated distribution is to the target distribution which +correlates well with visual fidelity. The metric is defined as the +Fr´echet distance, or 2-Wasserstein metric2, between two Gaussian +distributions each representing the activations the deepest layer of +an Inception v3 network induced by images from the generated +and target distributions. +Table 1 shows the FID metric between the generated images +from different morphing attacks and bona fide samples from +the dataset the morphing attack is drawn from. The FID met- +ric is calculated using pytorch-fid [48] Morphed images +generated from the Diffusion attack generally had the lowest +FID with the StyleGAN2-based attack following closely behind. +Both Landmark-based morphs—OpenCV and FaceMorpher—had +noticeably higher FIDs than the deep learning-based morphs. +These results correlate well with visual inspection of the morphs as +both Figure 4b and Figure 4f exhibit prominent artifacts outside +the central face region. Likewise, the MIPGAN-II attack seems +to struggle with some distortion outside the central face region, +see Figure 4e. Interestingly, on the FRLL dataset the StyleGAN2 +morphing pipeline consistently darkens morphs relative to its com- +ponent images; however, the visual fidelity is relatively high albeit +with noticeable darkening. Importantly, stochastic details, such as +hair, seem to be modelled well by the Diffusion attack whereas +other attacks distort such details, the OpenCV, FaceMorpher, and +MIPGAN-II attacks, or present details that have little similarity +to both identities, the StyleGAN2 attack. While exhibiting the far +less visual artifacts than other morphing techniques, the Diffusion +attack does tend to slightly smooth out the skin texture. Overall, +the Diffusion attack exhibits the highest consistent visual fidelity +among all the presented attacks. +Strangely, MIPGAN-II exhibited a much higher FID than the +StyleGAN2 morphs. This is a surprising result as the MIPGAN- +II attack was positioned as an improvement over the StyleGAN2 +attack. Figure 5 compares two morphed faces generated by the +2. The 2-Wasserstein metric between two probability measures µ, ν with +finite moments on Rn is defined as +W2(µ, ν) = +� +inf +π∈Π(µ,ν) +� +Rn×Rn ||x − y||2 +2 dπ(x, y) +� 1 +2 +where Π(µ, ν) is the set of all distributions with marginals µ and ν. + +二二7 +TABLE 2: The APCER at specific BPCER values. Higher is better. +Dataset +FR System +Morphing Attack +APCER @ BPCER = 0.1% +APCER @ BPCER = 1% +APCER @ BPCER = 5% +FRLL +FaceNet +StyleGAN2 +0.99 +0.05 +0 +FRLL +FaceNet +FaceMorpher +2.25 +0.14 +0.05 +FRLL +FaceNet +OpenCV +3.24 +0.33 +0 +FRLL +FaceNet +MIPGAN-II +8.87 +0.47 +0.09 +FRLL +FaceNet +Diffusion +8.83 +0.99 +0.23 +FRLL +VGGFace2 +StyleGAN2 +0.05 +0.05 +0 +FRLL +VGGFace2 +FaceMorpher +1.36 +1.08 +0.23 +FRLL +VGGFace2 +OpenCV +2.35 +2.11 +0.28 +FRLL +VGGFace2 +MIPGAN-II +1.31 +0.99 +0.23 +FRLL +VGGFace2 +Diffusion +2.68 +2.07 +0.52 +FRGC +FaceNet +StyleGAN2 +74.04 +36.69 +17.73 +FRGC +FaceNet +FaceMorpher +87.9 +38.85 +14.9 +FRGC +FaceNet +OpenCV +84.1 +31.43 +11.36 +FRGC +FaceNet +MIPGAN-II +96.54 +61.9 +33.48 +FRGC +FaceNet +Diffusion +91.73 +48.86 +24.95 +FRGC +VGGFace2 +StyleGAN2 +81.42 +46.22 +26.7 +FRGC +VGGFace2 +FaceMorpher +95.42 +63.65 +38.18 +FRGC +VGGFace2 +OpenCV +95.31 +64.62 +39.4 +FRGC +VGGFace2 +MIPGAN-II +91.92 +57.8 +30.84 +FRGC +VGGFace2 +Diffusion +93.71 +58.25 +32.18 +FERET +FaceNet +StyleGAN2 +15.65 +9.35 +2.49 +FERET +FaceNet +FaceMorpher +10.71 +5.1 +0.91 +FERET +FaceNet +OpenCV +8.79 +3.06 +0.17 +FERET +FaceNet +MIPGAN-II +21.03 +10.77 +2.21 +FERET +FaceNet +Diffusion +24.04 +13.95 +4.99 +FERET +VGGFace2 +StyleGAN2 +54.08 +18.42 +5.73 +FERET +VGGFace2 +FaceMorpher +80.5 +32.65 +12.7 +FERET +VGGFace2 +OpenCV +81.01 +32.6 +12.87 +FERET +VGGFace2 +MIPGAN-II +66.5 +18.14 +5.84 +FERET +VGGFace2 +Diffusion +80.9 +35.2 +14.34 +Diffusion and MIPGAN-II attacks on the FRGC dataset. Numer- +ous high frequency artifacts are present in Figure 5b, particularly, +near the hairline and transition between hair and the background. +Comparing Figure 5a and Figure 5b the hair generated by the +MIPGAN-II attack looks unnatural with a strange texture as +though an image sharpening filter has been applied to the image, +greatly enhancing the magnitude of high frequency content, which +aligns with the observation in Figure 4. Moreover, the MIPGAN- +II images seem to be desaturated when compared to images +produced by other attacks leading to a washed out appearance. +Perhaps the low visual fidelity can be explained by the identity +loss overpowering the perceptual quality loss leading to morphed +images with low visual fidelity but high effectiveness against FR +systems. +5.2 +Vulnerability of FR Systems +The strength of the proposed face morphing algorithm is further +evaluated by measuring the ability of the morph to fool a FR +system. The attack success is quantitatively verified against two +state-of-the-art FR systems. To ensure a valid comparison between +the five different morphing attacks the same pairs of component +identities were used in evaluating every morphing attack, i.e., for +every pair of component identities a morphed image was created +for each of the five attacks. For both the FaceNet and VGGFace2 +FR systems the False Match Rate (FMR) is set at 0.1% following +the guidelines of Frontex [49]. Additionally, the distance between +faces is measured using the L2 distance between the outputs of +the FR model. +The vulnerability of FR systems to morphing attacks is as- +sessed by comparing the error rates in detection, specifically, +the Attack Presentation Classification Error Rate (APCER)3 is +measured at specific Bona fide Presentation Classification Error +Rate (BPCER)4 values. In Table 2 the APCER values for the five +different morphing attacks is presented across all three datasets +evaluated on three different BPCER values of 0.1%, 1%, and 5%. +Due to a variety of factors—such as image quality and number +of bona fide images per identity—the results vary between the +different datasets; while there is some variance between both +FR systems, they tend to agree more closely. Noticeably, all +attacks performed rather poorly on the FRLL dataset, although +the Diffusion-based morphing attack performed the best among +them, which could be attributed to the limited number of bona +fide images per identity; for in the FRLL dataset there are only +two bona fide images per identity: a neutral face (used to create +the morph) and a smiling face. Both FR systems on the FRLL +dataset tend to give close relative rankings between the morphing +attacks with the StyleGAN2-based attack being noticeably weaker +than the rest. Both FR systems were much more vulnerable when +evaluated on the FRGC dataset. The MIPGAN-II attack performed +very well against FaceNet which makes sense as this technique +was refined on the FRGC dataset in particular [22]. The attack was +not as strong on VGGFace2 and instead that FR system was more +vulnerable to OpenCV and FaceMorpher, this could possibly be +attributed to the different pre-processing pipelines. The Diffusion- +based generally performs close to the top performer on either FR +system. As with FRGC, on the FERET dataset VGGFace2 is more +3. APCER is the proportion of attack presentations incorrectly classified as +bona fide presentations. +4. BPCER is the proportion of bona fide presentations incorrectly classified +as attack presentations. + +8 +TABLE 3: MMPMR at FMR = 0.1% across diffrent morphing attacks. Higher is better. +FRLL +FRGC +FERET +Morphing Attack +FaceNet +VGGFace2 +FaceNet +VGGFace2 +FaceNet +VGGFace2 +Geometric Mean +StyleGAN2 +4.69 +6.05 +0.18 +0.85 +0.54 +0.76 +1.10 +FaceMorpher +11.26 +36.4 +0.51 +9.15 +2.3 +10.78 +6.02 +OpenCV +17.34 +40.93 +0.14 +12.16 +1.69 +11.12 +5.32 +MIPGAN-II +30.96 +26.74 +3.12 +7.94 +6 +5.39 +9.34 +Diffusion +28.14 +35.37 +2.68 +8.47 +6.47 +13.03 +11.13 +vulnerable to landmark-based attacks, OpenCV and FaceMorpher, +than FaceNet. Diffusion-based morphs pose the greatest threat on +FERET consistently having high APCER values. In general the +following observations can be drawn from Table 2: +• +Among the five different attacks, FR systems are most vul- +nerable to Diffusion attacks. Moreover, Diffusion attacks +always rank in the top three in terms of performance. +• +FR systems are the least vulnerable to the StyleGAN2 at- +tack. The StyleGAN2 attack is always outperformed by its +successor, MIPGAN-II, and the other deep learning-based +attack, Diffusion, while often falling behind landmark- +based attacks. +In addition to using the error rates to assess the vulnerability +of FR systems the Mated Morphed Presentation Match Rate +(MMPMR) [50] is used as a measure of vulnerability. Scherhag et +al. [50] propose two variants of the MMPMR metric for the +scenario in which there multiple bona fide images of an identity +used in morph process, excluding the image used to create the +morph, called the MinMax-MMPMR and ProdAvg-MMPMR. +The MinMax-MMPMR metric is especially likely to increase the +number of accepted morphs as the number of bona fide images +per identity increases. Therefore, the ProdAvg-MMPMR is the +specific MMPMR variant used to assess the vulnerability of FR +systems, any mention hereafter to MMPMR refers specifically to +ProdAvg-MMPMR unless stated otherwise. +Let PM ∈ P(X) be the distribution of morphed images such +that for some xab ∼ PM, xab denotes a morphed image made +from identities a, b, where P(X) denotes the set of all probability +measures on X . Let Pk ∈ P(X) denote the distribution of bona +fide images of identity k. Then with abuse of notation Pk\xab is +the distribution of bona fide images of identity k excluding those +images used in creating the morph xab. The MMPMR metric for +a particular threshold, γ > 0, equipped with FR system F : X → +V is then defined as +M(γ) = +E +xab∼PM +� +� +k∈{a,b} +E +x∼Pk\xab +� +||F(xab)−F(x)||2 < γ +�� +i.e., the expected success rate of the morphing attack to fool the +FR system. The product term is the joint probability of successful +verification of both identities +Table 3 presents the MMPMR metric when the FMR is set +at 0.1% for all datasets and FR systems. Interestingly, the FRLL +dataset had the highest overall MMPMR metrics in contrast to +the results from Table 2. This can likely be attributed to limited +number of bona fide images per identity and FRLL in contrast +with other datasets as the particular choice of MMPMR metric +heavily punishes failed verifications in either identity, thus having +only one possible image per identity could boost the metric for +FRLL. On average the Diffusion attack greatly outperforms the +TABLE 4: APCER at FMR = 0.1% across different margin sizes +on the FaceNet FR system. Higher is Better. +Margin Size +Dataset +Morphing Attack +0 +20 +40 +80 +FRLL +MIPGAN-II +54.84 +56.53 +57.18 +58.03 +FRLL +StyleGAN2 +15.12 +15.57 +17.14 +25.11 +FRLL +FaceMorpher +74.48 +75.26 +73.86 +47.91 +FRLL +OpenCV +76.21 +75.82 +74.96 +48.4 +FRLL +Diffusion +51.25 +54.47 +57.07 +59.02 +FERET +MIPGAN-II +19.33 +21.71 +22.11 +26.36 +FERET +StyleGAN2 +14.12 +17.57 +18.14 +22.17 +FERET +FaceMorpher +36.11 +36.45 +36.28 +17.8 +FERET +OpenCV +36.11 +38.44 +37.64 +13.89 +FERET +Diffusion +23.24 +26.02 +25.91 +30.39 +FRGC +MIPGAN-II +12.33 +14.3 +16.2 +20.97 +FRGC +StyleGAN2 +7.18 +8.71 +9.74 +14.42 +FRGC +FaceMorpher +17.5 +18.87 +20.39 +9.07 +FRGC +OpenCV +17.02 +18.47 +19.91 +6.6 +FRGC +Diffusion +9.7 +10.71 +12.27 +15.62 +other attacks; conversely, the Landmark-based attacks, on average, +exhibit mediocre performance. In agreement with Table 2 the +StyleGAN2 attack shows abysmal performance in comparison +with the other attacks. +5.2.1 +The Effect of Pre-processing on a FR System +The impact of the pre-processing pipeline on the vulnerability of +a FR system is examined, in particular the cropping process is +further explored. To study this an additional margin size is added +to the image after a initial face extraction and cropping performed +by MTCNN, such that a margin size N adds back at most N pixels +to the cropped image in both dimensions. Therefore, the larger N +is the less tightly cropped the image passed to the FR system +is. Table 4 shows the illustrates the impact of the margin size +on the APCER metric on the FaceNet FR system. Generally, as +the margin size increases the performance of the Landmark-based +attacks decreases and the performance of the deep learning-based +attacks increases. As illustrated in Figure 4 the Landmark-based +attacks have noticeable artifacts outside the central face region; +conversely, the deep learning-based morphs have less artifacts in +the outside regions and generally look more realistic to a human +observer. This observation aligns the visual fidelity results from +Table 1. Therefore, a MAD algorithm or FR system which uses +less tightly cropped faces would be more resilient against attacks +with visual artifacts outside the core face region. +5.2.2 +General Remarks on the Vulnerability Study +The poor performance of the StyleGAN2 attack could be at- +tributed to the darkening of images with light backgrounds, see +Figure 4, and due to aliasing effects latent to the StyleGAN2 +generation pipeline which is addressed by Karras et al. [51]. + +9 +TABLE 5: Ablation study on the impact morphing attack on validation accuracy. +Training Attack +Validation Attack +Dataset +Diffusion +FaceMorpher +MIPGAN-II +OpenCV +StyleGAN2 +Diffusion +FaceMorpher +MIPGAN-II +OpenCV +StyleGAN2 +FERET + + + + + +72.73 +99.23 +100 +99.95 +99.33 +FERET + + + + + +99.9 +76.39 +100 +99.85 +99.64 +FERET + + + + + +99.69 +99.38 +100 +99.95 +99.54 +FERET + + + + + +99.74 +99.48 +100 +99.74 +99.43 +FERET + + + + + +99.74 +98.56 +99.9 +99.74 +87.89 +FRGC + + + + + +75.89 +99.98 +99.97 +99.9 +99.93 +FRGC + + + + + +99.95 +99.48 +100 +99.9 +99.95 +FRGC + + + + + +99.83 +99.85 +99.82 +99.8 +99.85 +FRGC + + + + + +99.93 +100 +100 +99.23 +99.93 +FRGC + + + + + +99.93 +99.93 +99.94 +99.88 +97.83 +FRLL + + + + + +13.96 +99.58 +99.32 +99.65 +99.65 +FRLL + + + + + +99.23 +99.09 +98.91 +99.37 +99.44 +FRLL + + + + + +99.09 +98.95 +98.24 +99.02 +99.09 +FRLL + + + + + +99.51 +99.44 +99.19 +99.16 +99.58 +FRLL + + + + + +99.93 +99.86 +99.86 +99.93 +95.02 +Moreover, the structure of the StyleGAN2 latent space can make +exploration in the space difficult which could possibly explain +the poor performance in attacking the FR system compared to +other attacks. MIPGAN-II, on the other hand, likely avoids these +pitfalls due its explicit latent optimization process for fooling a FR +system. The Diffusion attack utilizes an entirely different latent +representation scheme which seems to yield an advantage in the +task of generating morphed faces. The pre-processing pipeline of +the FR system seems to mostly mitigate the artifacts latent to the +Landmark-based attacks; however, such artifacts could easily be +detected by a human observer. +5.3 +Detectability of Morphing Attacks +The performance of the proposed attack is further evaluated by the +ability of Morphing Attack Detection (MAD) algorithms trained +against other attacks to detect an unseen attack. To quantita- +tively assess the detectability of a particular morphing attack a +SE-ResNeXt101-32x4d model pre-trained on ImageNet [52] by +NVIDIA is trained to detect morphing attacks. SE-ResNeXt101- +32x4d is a state-of-the-art image recognition model based on the +ResNeXt101-32x4d model [53] with the addition of the Squeeze- +and-Excitation architecture [43]. For all experiments a 5-fold strat- +ified k-fold cross validation strategy is employed, thus preserving +the class balance between morphed and bona fide images in each +fold. The model is fine-tuned on a collection of morphing attacks +for 5 training epochs using exponential learning rate scheduler +with differential learning rates in order to mitigate overfitting of +the model. +5.3.1 +Ablation Study +To study the impact of a particular morphing attack on the ability +of a MAD algorithm to detect morphing attacks an ablation +study was conducted where the SE-ResNeXt101-32x4d model +was trained on all the morphing attacks except for one holdout. +Table 5 shows the validation accuracy of each morphing attack +when different morphing attacks were withheld from the training +process. Due to the similar natures between the OpenCV and +FaceMorpher attacks the absence of one of these attacks does not +greatly impact the validation accuracy. Interestingly, the absence +of MIPGAN-II does not significantly change the validation accu- +racy of the attacks; however, the omission of StyleGAN2 during +training does decrease the performance of the StyleGAN2 during +validation despite the presence of the MIPGAN-II. Notably, the +Diffusion attack is very difficult to detect as a novel attack, which +can be partially attributed to its unique morph generation process +in contrast with the other morphing attacks. +5.3.2 +A Metric For Relative Strength +In this section we introduce a metric to measure the relative +strength from one morph to another. We say a morph α is “strong” +relative to a morph β if the following conditions are satisfied: +1) +It is easy to detect β when a detector is trained on α, i.e., +high transferability. +2) +It is hard to detect α when a detector is trained on β, i.e., +low detectability. +Additionally, the relative strength metric, ∆(α||β), should be +positive when α is stronger than β and negative when α is weaker. +A relative strength of 0 would denote that the two morphing +attacks are equally strong. +As some of the morphing attacks are not deterministic but +probabilistic, we chose to represent a morphing attack α by +the random variable Xα : Ω → X such that P(Xα|xa, xb) +denotes the distribution of morphs generated from images xa, xb. +Moreover, we suppose there exists a detector f α : X → {0, 1} +trained to distinguish between bona fide presentations and mor- +phed presentations generated by α; wherein 0 denotes a bona fide +presentation and 1 denotes a morphed presentation. The transfer- +ability of a morphing attack α to β is defined as the probability the +detector f α is able to detect the attack β given the probability f α +detects α, i.e., T(α, β) = P(f α(Xβ) = 1|f α(Xα) = 1). This +metric can be represented as a ratio of expectations taken over the +pairs of component bona fide images: +T(α, β) = P(f α(Xβ) = 1, f α(Xα) = 1) +P(f α(Xα) = 1) += Exa,xb[P(f α(Xβ) = 1, f α(Xα) = 1|xa, xb)] +Exa,xb[P(f α(Xα) = 1|xa, xb)] +(11) +Let {xα +i }N +i=1 denote a collection of N samples drawn from +P(Xα|xa, xb) such that xα +i denotes the morph generated from +i-th pair of bona fide identities (ai, bi), and likewise for β. Then +the metric in Equation (11) can be closely approximated by +T(α, β) ≈ +�N +i=1 +� +f α(xβ +i ) = 1 ∧ f α(xα +i ) = 1 +� +�N +i=1 +� +f α(xα +i ) = 1 +� +(12) + +10 +(a) RSM on FRGC +(b) RSM on FERET +(c) RSM on FRLL +Fig. 6: Blue indicates higher strength and red indicates weak +strength. +i.e., the number of morphs from both α and β detected over the +number of morphs detected from α. +The relative strength metric (RSM) from α to β is defined +as the log ratio of the transferability metrics between the two +morphing attacks: +∆(α||β) = log +�T(α, β) +T(β, α) +� +(13) +The log of the ratio is chosen such that ∆(α||β) the RSM takes +(a) Variant A +(b) Variant B +(c) Variant C +(d) Variant D +Fig. 7: Morphed image generated by different Diffusion attack +variants on FRLL. +positive values when α is “stronger” than β and negative values +when weaker—with a value of zero denoting equal strength. +Additionally, there is an antisymmetry such that ∆(α||β) = +−∆(β||α). +In contrast to the ablation study, the SE-ResNeXt101-32x4d +model is only trained on a single attack per k-fold. The RSM is +calculated between all attacks with the results shown in Figure 6. +From Figure 6 it is observed that the RSM between the Landmark- +based morphs and the RSM between the StyleGAN-based morphs +is very small. As these attacks have similar morph generation +pipelines it makes sense that the transferability between the attacks +is near identical. In general, the Landmark-based attacks seem to +be stronger than the StyleGAN-based attacks, in particular the +FaceMorpher attack. The MIPGAN-II attack is generally weaker +than the other attacks. Overall, the Diffusion attack is the least +detectable among the attacks along with generally being the +strongest attack across the three datasets. +The results from Figure 6 corroborate with the results from +Table 5 demonstrating the difficulty in detecting Diffusion attacks. +From the perspective of training a MAD system including samples +from the FaceMorpher, StyleGAN, and Diffusion attacks would +greatly increase the ability for the system to detect unknown +attacks. Additionally, Table 5 and Figure 6 demonstrates a par- +ticular vulnerability existing MAD systems may have to the new +Diffusion attack. +5.4 +Study of the Diffusion-based Morphing Process +The diffusion morphing algorithm leverages both a stochastic +and semantic representation of an image. While the semantic +representation contains many of the key “identifying” features, +the stochastic representation represents many of the details nec- +essary for high visual fidelity. Due to the importance of the +stochastic code for high fidelity, we investigated several methods +for finding the morphed stochastic latent code, x(ab) +T +. The first + +3.517 +StyleGAN2 +OpenCv +Attack +Training A +MIPGAN-II +0.000 +FaceMorpher +Diffusion +3.517 +OP +eGAN2 +ValidationAttack2.734 +StyleGAN2 +OpenCv +Attack +Training A +MIPGAN-II +0.000 +FaceMorpher +Diffusion +2.734 +OR +MIPGAN-II +ValidationAttack18.2 +StyleGAN2 +Opencv +Training Attack +MIPGAN-II +0.0 +FaceMorpher +Diffusion +18.2 +MIPGAN-II +OR +GAN +Validation Attack11 +TABLE 6: MMPMR at FMR = 0.1% across different configurations. Higher is better. † indicates our default choices. +FRLL +FRGC +FERET +Variant +ℓX +ξ(x, y) +FaceNet +VGGFace2 +FaceNet +VGGFace2 +FaceNet +VGGFace2 +Geometric Mean +A +slerp +x, y �→ x +32.97 +34.71 +3.2 +9.59 +7.17 +11.54 +11.95 +B +lerp +x, y �→ x +10.81 +11 +1.17 +2.17 +2.33 +4.69 +3.86 +C† +slerp +x, y �→ 1 +2 (x + y) +28.14 +35.37 +2.68 +8.47 +6.47 +13.03 +11.13 +D +slerp +x, y �→ OpenCV(x, y) +9.14 +9.34 +0 +1.37 +0.14 +1.42 +0 +TABLE 7: FID across different configurations. Lower is better. +† indicates our default choices. +Variant +ℓX +ξ(x, y) +FRLL +FRGC +FERET +A +slerp +x, y �→ x +48.13 +52.97 +55.66 +B +lerp +x, y �→ x +82.05 +119.33 +97.75 +C† +slerp +x, y �→ 1 +2 (x + y) +42.63 +64.16 +50.45 +D +slerp +x, y �→ OpenCV(x, y) +93.85 +84.51 +108.49 +variant, variant A, is the baseline implementation with ℓZ using +linear interpolation, ℓX using spherical linear interpolation, and +ξ does not perform any “pre-morphing”. Conversely, in variant B +the stochastic codes are interpolated via linear interpolation. In +variants C and D, instead of using the original image to calculate +the stochastic code, the function ξ is used to construct the “pre- +morph” passed to the stochastic encoder. Specifically, in variant +C the two images are averaged pixel-wise and presented to the +stochastic encoder; in contrast, in variant D the OpenCV morph is +presented to the stochastic encoder. +In Table 7 the FID is calculated between the generated morphs +and the bona fide samples for each particular dataset. Variant +C generally presents the lowest FID score closely followed by +variant A. Both variants B and D exhibit clear degradation in +performance when compared to variants A and C. Furthermore, +the FID scores seems to correlate well to human assessment +of the generated samples, see Figure 7. Noticeably, the linear +interpolation in variant B results in an overly smoothed face and +generally darker image, greatly degrading visual fidelity. Variant +D has prominent visual artifacts, similar to the artifacts found in +the OpenCV morphs. Moreover, the poor performance seems to +be aided by an issue of differing alignment strategies between the +OpenCV and diffusion pipeline. +Notably, variant C often removes many of the high frequency +artifacts found in variant A. This is likely due to the difficulty +in smoothly interpolating between points in the stochastic latent +space in contrast with the semantic latent space. As such, variant +C which performs a pixel-wise average of the two source images +before using the stochastic encoder seems to greatly improve +the ability to smoothly interpolate between different stochastic +representations. This appears to be the primary reason variant C +has a generally lower FID when compared to variant A. Both +Figure 7 and Table 7 demonstrate the large importance that the +stochastic code plays in creating high fidelity morphed images. +Due to the high fidelity exhibited by variant C, this particular +diffusion process was used in evaluation against other morphing +attacks. +The MMPMR metric is calculated for each variant, see Ta- +ble 6. Variant A is slightly stronger than variant C, with variants +B and D falling far behind likely due to the high number of visual +distortions. These results stand in contrast to the assessment of vi- +sual fidelity wherein variant C outperforms variant A. This, again, +illustrates a trade-off between visual fidelity and ability to fool +the FR system; however, in this case the trade-off effectiveness +against the FR system is relatively small in comparison to the +gains in visual fidelity. Due to its excellent visual fidelity and +strong MMPMR results variant C was chosen to be the default +configuration for the Diffusion attack. +6 +CONCLUSION +By addressing some of the key limitations of prior deep-learning +based morphing attacks, namely, the trade-off between visual +fidelity and effectiveness against FR systems, we have proposed +a novel morphing attack using Diffusion-based methods for the +generative process. The proposed attack consistently generates +realistic morphed images with high visual fidelity while also +being able to strongly threaten FR systems. To evaluate the attack +potential of the proposed attack, we evaluated the vulnerability +of two FR systems over three distinct datasets and created over +10,000 new morphs between the four variants, with the strongest +variant achieving state-of-the-art performance. We conducted an +additional study on the impact the pre-processing pipeline has on +the vulnerability of an FR system to morphing attacks. 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Gelly, “Are +gans created equal? a large-scale study,” in Proceedings of the 32nd +International Conference on Neural Information Processing Systems, +ser. NIPS’18. +Red Hook, NY, USA: Curran Associates Inc., 2018, +p. 698–707. +[48] M. Seitzer, “pytorch-fid: FID Score for PyTorch,” https://github.com/ +mseitzer/pytorch-fid, August 2020, version 0.2.1. +[49] “Best practice technical guidelines for automated border control abc +systems,” 2015. +[50] U. Scherhag, A. Nautsch, C. Rathgeb, M. Gomez-Barrero, R. N. J. +Veldhuis, L. Spreeuwers, M. Schils, D. Maltoni, P. Grother, S. Marcel, +R. Breithaupt, R. Ramachandra, and C. Busch, “Biometric systems under +morphing attacks: Assessment of morphing techniques and vulnerability +reporting,” in 2017 International Conference of the Biometrics Special +Interest Group (BIOSIG), 2017, pp. 1–7. + +13 +[51] T. Karras, M. Aittala, S. Laine, E. H¨ark¨onen, J. Hellsten, J. Lehtinen, and +T. Aila, “Alias-free generative adversarial networks,” in Proc. NeurIPS, +2021. +[52] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei, “Imagenet: +A large-scale hierarchical image database,” in IEEE conf. on computer +vision and pattern recognition, 2009, pp. 248–255. +[53] S. Xie, R. B. Girshick, P. Doll´ar, Z. Tu, and K. He, “Aggregated residual +transformations for deep neural networks,” 2017 IEEE Conference on +Computer Vision and Pattern Recognition (CVPR), pp. 5987–5995, 2017. + diff --git a/A9E2T4oBgHgl3EQf8Qn_/content/tmp_files/load_file.txt b/A9E2T4oBgHgl3EQf8Qn_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b4c53be625bf897a0d444ae707aaa0150a971726 --- /dev/null +++ b/A9E2T4oBgHgl3EQf8Qn_/content/tmp_files/load_file.txt @@ -0,0 +1,1172 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf,len=1171 +page_content='1 Diffusion Models For Stronger Face Morphing Attacks Zander Blasingame and Chen Liu Department of Electrical and Computer Engineering Clarkson University, Potsdam, New York, USA {blasinzw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' cliu}@clarkson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='edu Abstract—Face morphing attacks seek to deceive a Face Recognition (FR) system by presenting a morphed image consisting of the biometric qualities from two different identities with the aim of triggering a false acceptance with one of the two identities, thereby presenting a significant threat to biometric systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The success of a morphing attack is dependent on the ability of the morphed image to represent the biometric characteristics of both identities that were used to create the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We present a novel morphing attack that uses a Diffusion-based architecture to improve the visual fidelity of the image and improve the ability of the morphing attack to represent characteristics from both identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We demonstrate the high fidelity of the proposed attack by evaluating its visual fidelity via the Fr´echet Inception Distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Extensive experiments are conducted to measure the vulnerability of FR systems to the proposed attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The proposed attack is compared to two state-of-the-art GAN-based morphing attacks along with two Landmark-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The ability of a morphing attack detector to detect the proposed attack is measured and compared against the other attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, a novel metric to measure the relative strength between morphing attacks is introduced and evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Index Terms—Biometrics, Morphing Attack, GAN, Vulnerability Analysis, Face Recognition, Diffusion Models !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 1 INTRODUCTION F ACE recognition (FR) systems have become one of the most common biometric modalities used for identity verification across a wide range of modern-day applications, from trivial tasks such as unlocking a smart phone to official businesses such as banking, e-commerce, and law enforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Unfortunately, while FR systems can often reach low false rejection and acceptance rates [1], they are especially vulnerable to a new class of emerging attacks, known as the face morphing attack [2], [3], [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The face morphing attack aims to compromise a fundamental property of biometric security, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', the one-to-one mapping from biometric data to the associated identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This compromise is achieved by creating a morphed face which contains biometric data of both identities in such a manner that presenting one morphed image triggers a match with two disjoint identities, violating the fundamental principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This poses a significant threat towards FR systems, especially the applications of e-passports and border access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Notably the e- passport scenario, wherein the applicant submits a passport photo either in digital or printed form, is especially vulnerable to face morphing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, digital morphs can be easily generated hence offer a low-cost attack in the digital domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' It is not un- common for a digital image submitted to a document submission portal to not have its authenticity verified by a human agent [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Critically, an adversary who is blacklisted from accessing a certain system can create a morph with a non-blacklisted individual to gain access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This is especially relevant for countries such as New Zealand, Estonia, and Ireland, where e-passports are used for both issuance and renewal of documents [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In 2018 a German activist was reported to have received a German passport with a photo of his faced morphed with an Italian politician [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Due to the severity of face morphing attacks, an abundance of algorithms have been developed to identify these attacks [2], [9], [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Methods for Morphing Attack Detection (MAD) can be broadly characterized into two classes based on the manner in which they obtain the features used for detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', handcrafted features or deep features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Handcrafted features are used in the so-called classical algorithms which seek to find evidence of the morphing attack in the pixel domain, whether that is evidence of degradation in image quality [10], residual noise left from the morphing attack [12], or local geometric features such as Local Binary Patterns (LBP) [13], Binarized Statisical Image Feature (BSIF) [14], and Local Phase Quantitization (LPQ) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Conversely, deep features are used with deep learning based algo- rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' These features are often extracted by a deep Convolutional Neural Network (CNN) often from a pre-trained network such as ResNet150 [6], [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Generally, the best success has been found using deep CNN-based features in comparison to handcrafted features on both digital and print-scan face data [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Comparatively, there has been less research on face mor- phing attack algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Similar to the two classes of MAD algorithms, there exist two broad classes of face morphing at- tacks: Landmark-based attacks and deep learning-based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Landmark-based morphing attacks use local features to create the morphed image by warping and aligning the landmarks within each face to then create a morphed face by pixel-wise compositing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Landmark-based attacks have been shown to be effective against FR systems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Recent work has enhanced the effectiveness of Landmark-based attacks by using adversarial perturbation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In contrast, deep learning-based morphing attacks use a machine learning model to embed the original bona fide faces into a semantic representation which are then combined to produce a new representation which should contain information from both identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This is new representation is then used to generate a morphed face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' However, nearly all state-of-the-art deep learning based morph methods were based on the Generative Adversarial Network arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='04218v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='CV] 10 Jan 2023 2 (a) Identity a (b) Morphed image (c) Identity b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 1: Diffusion-based morphing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Samples are from FRLL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' (GAN) framework [19], [20] with the primary difference being architectural improvements such as using the StyleGAN2 [21] architecture over the vanilla GAN architecture or changes to the morph generation pipeline as seen in the Morphing through Identity Driven Prior GAN (MIPGAN) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' At the same time, there exists a handful of alternative state-of-the-art deep generative models which offer their own advantages in terms of visual fidelity, semantic representation capabilities, and inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In particular, a class of generative models collectively known as “Diffusion models” haven been shown to possess high visual fidelity, beating even state-of-the-art GANs in visual fidelity [23], at the cost of increased inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As visual fidelity and semantic representation abilities are far more important for the potency of a potential morphing attack than inference speed, we present a novel methodology for generating strong face morphing attacks by leveraging Diffusion-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Figure 1 shows a morphed image generated by the proposed attack constructed from two identities from the FRLL dataset [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We present a summary of contributions of the proposed work as shown below: We propose a novel method for generating morphed faces by using a Diffusion-based model which calculates twin embeddings, one for the semantic details and one for the stochastic details, to generate images of high visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The proposed morphing attack is evaluated against four other morphing attacks by the Fr´echet Inception Distance, a quantitative measure of visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We evaluated our proposed attack against four other mor- phing attacks with extensive experiments assessing the vulnerability of two FR systems across three different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The proposed attack is further evaluated its ability to evade detection from MAD algorithms trained against other morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We introduce and evaluate a novel metric to measure the relative strength from one morphing attack to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' A small study on the impact of the pre-processing in the FR pipeline on the vulnerability of FR systems to morphing attacks is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Four variants of the proposed morphing attack are com- pared against each other, resulting in the creation of over 10,000 new morphed images between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 2 PRIOR WORK Several face morphing attacks have been developed by researchers with the morph generating process generally using face land- marks or deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In particular we compare our proposed Diffusion-based attack against four state-of-the-art morphing at- tacks, two deep learning-based and two Landmark-based;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' namely, the OpenCV, FaceMorpher, StyleGAN2, and MIPGAN-II face morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' These were chosen as they represent several types of morphing attacks and provide a substantial baseline to measure the performance of the proposed Diffusion-based attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To the best of our knowledge all deep learning-based attacks generate the morphed images via a type of GAN architecture [17], [19], [22], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 Landmark-Based Morphing Attacks The FaceMorpher and OpenCV attacks were chosen to represent the Landmark-based attacks as they are commonly used to repre- sent this class of attack [11], [17], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' FaceMorpher is an open-source algorithm that uses the STASM landmark detector [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' From the landmarks on the images Delaunay triangles are formed, which are then warped and blended together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The areas outside the landmarks are averaged, typically introducing strong artifacts in the neck and hair regions of the image [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' OpenCV is a morphing attack which uses the open-source OpenCV library with a 68-point annotator from the Dlib li- brary [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The images and associated landmarks are used to form Delaunay triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Then, in a similar manner to FaceMorpher, the landmarks are warped and blended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In contrast to the approach of FaceMorpher, the areas outside the landmarks do not consist of an averaged image, but rather additional Delaunay triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' However, these morphs also exhibit strong artifacts outside the facial area due to the missing landmarks [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 GAN-Based Morphing Attacks As mentioned earlier, prior deep learning-based attacks have used a GAN architecture for the morph generation process [26], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' GANs are a type of deep generative model which seeks to learn the sampling process for some data distribution Pdata on X , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', given some simple distribution Z ∼ p(z) on Z, the generator G : Z → X is to learn G(Z) ∼ Pdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' A discriminator, sometimes 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 2: Proposed architecture for Diffusion-based morphs, where the green traces indicate variables associated with identity a, likewise red traces denote identity b, and blue traces for the morphed identity ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' called the critic, D : X → [0, 1] is trained adversarially against the generator in a minimax game described by min G max D E x∼Pdata log D(x) + E z∼p(z) log(1 − (D ◦ G)(z)) (1) where the discriminator attempts to get better at distinguishing synthetic samples from genuine samples, while the generator tries to get better at deceiving the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For a GAN-based mor- phing attack it is necessary for an encoding algorithm E : X → Z to exist, which can embed images in the latent space such that the inversion has low distortion (G ◦ E)(x) ≈ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The latent codes for two identities are then averaged to produce a new latent code representing the morphed face which is passed to the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Notably, there exists a trade-off between the inversion distortion and editability of the latent embeddings [32] Damer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' proposed to use the GAN architecture for generating morphs by combining two latent codes encoded from two real identities to create a morphed code [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This proposed attack, known as MorGAN, was based on a modification to the vanilla GAN architecture by the addition of an explicitly defined encoder architecture that was trained jointly with the generator via a modified adversarial loss formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Since then the StyleGAN2 [17] and MIPGAN- II [22] attacks have improved upon the MorGAN formulation by improving the GAN architecture, loss formulation, or encoding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' StyleGAN2 morphs offer promising results as shown in Damer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' StyleGAN2 offers a host of improvements over the standard GAN implementation that enables the architecture to achieve state of the art image quality when generating high resolu- tion images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The StyleGAN2 model was pre-trained on the Flickr- Faces-HQ (FFHQ) dataset [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The faces were then cropped to possess the same landmark alignment as in the FFHQ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Following the approach in [21], the images, xa, xb, are embedded by optimizing an initial latent code through stochastic gradient descent, minimizing the perceptual loss between the generated image and target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' After each embedded latent code, za, zb, is found, a morphed latent code is created by linearly interpolating between the two, zab = lerp(za, zb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Lastly, the interpolated latent code is passed to the StyleGAN2 synthesis network to get the morphed image xab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The StyleGAN2 morphs are strong when used with images containing a uniform background, which makes them especially powerful when used in conjunction with the Face Research Lab London (FRLL) dataset [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 3: The forward and reverse Diffusion processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' MIPGAN-II proposes an extension on StyleGAN2 by adding an optimization procedure for the latent vector used in creating the morphed image [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The StyleGAN2 portion of MIPGAN- II was pre-trained on the FFHQ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The two bona fide images are embedded into the latent space using the StyleGAN2 optimization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The latent code is initially constructed as z0 = lerp(za, zb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For n epochs the latent code is optimized to minimize a combination of perceptual loss, identity loss, identity difference loss, and Multi-Scale Structural Similarity loss, finding a fully optimized latent zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The latent code zn is then passed to the StyleGAN2 synthesis network to create the morphed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As MIPGAN-II presents a refinement on StyleGAN2 for the application of morphing attack, it possesses similar advantages and disadvantages that StyleGAN2 morphs offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 3 DIFFUSION-BASED MORPHING ATTACK Unlike GANs which learn the sampling process for the data distribution through adversarial training between the generator and the critic, diffusion-based and score-based generative models learn the data distribution by learning a denoising process for varying noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Diffusion-based models can achieve image fidelity superior to state-of-the-art generative models, matching even the acclaimed BigGAN-deep [34] model, while maintaining better coverage of the data distribution [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For these reasons we propose a morphing attack that uses Diffusion-based methods as the generative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 Diffusion Models Given data distribution q(x0) on data space X , the goal is to learn a model pθ(x0) approximating q(x0) which can be easily sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Denoising Diffusion Probabilistic Models (DDPMs) [35] are latent variable models of the form pθ(x0) = � pθ(x0:T ) dx1:T (2) (x, x) lz(za, Zb) E(xo) q(XtXt-1, z) lx(x(), x) pe(Xt-1|xt, z)XT Xt Xt-1 X04 Algorithm 1 Diffusion Morphing Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Require: Bona fide images, x(a) 0 , x(b) 0 ∈ X , image space preprocessing function, ξ : X ×X → X , image space interpolation function, ℓX : X × X × [0, 1] → X , and latent space interpolation function, ℓZ : Z × Z × [0, 1] → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' procedure DIFFUSIONMORPH(x(a) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' x(b) 0 ) za ← E(x(a) 0 ) ▷ Calculate semantic latent codes zb ← E(x(b) 0 ) x(a) 0 ← ξ(x(a) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' x(b) 0 ) ▷ Preprocess images passed to stochastic encoder x(b) 0 ← ξ(x(b) 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' x(a) 0 ) t ← 0 while t < T do x(a) t+1 ← √αt+1f (t) θ (x(a) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' za) + √1 − αt+1ϵ(t) θ (x(a) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' za) ▷ Forward pass of diffusion algorithm x(b) t+1 ← √αt+1f (t) θ (x(b) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' zb) + √1 − αt+1ϵ(t) θ (x(b) t ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' zb) t ← t + 1 end while x(ab) T ← ℓX (x(a) T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' x(b) T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5) ▷ Stochastic code interpolation zab ← ℓZ(za, zb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5) ▷ Semantic code interpolation t ← T while t > 0 do x(ab) t−1 ← √αt−1f (t) θ (x(ab) t , zab) + √1 − αt−1ϵ(t) θ (x(ab) t , zab) ▷ Diffusion generative process t ← t − 1 end while return x(ab) 0 end procedure where {xt}T t=1 ∈ X are latent variables and T ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The reverse process is a Markov chain starting at pθ(xT ) = N(0, I), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', a normal distribution with mean vector 0 and variance I (the identity matrix), with Gaussian transitions pθ(x0:T ) = pθ(x0) T � t=1 pθ(xt−1|xt) (3) The diffusion (forward) process is fixed to a Markov chain that gradually adds Gaussian noise to the original sample x0 according to variance schedule {βt}T t=1, such that q(x1:T , x0) = T � t=1 q(xt|xt−1) (4) and q(xt|xt−1) = N(√1 − βtxt−1, βtI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' See Figure 3 for an illustration of this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The transition probability p(xt−1|xt) is likely to be very complex, unless the gap between t and t − 1 is very small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In this case, p(xt−1|xt) can be modelled as N(µ(t) θ (xt), σt), where µ(t) θ : X → X is an estimator at step t parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [35] proposes to use the following form: µ(t) θ (xt) = 1 √αt � xt − βt √1 − ¯αt ϵ(t) θ (xt) � (5) where αt = 1 − βt, ¯αt = �t s=1 αs, and ϵ(t) θ : X → X is a function approximator parameterized by θ, which learns to predict the noise added to x0 to get xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This is achieved by using a U-Net, a type of CNN consisting of several skip connections formed at each resolution size in the architecture to model ϵ(t) θ [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Like with Variational Autoencoders, the model is trained by optimizing the variational bound on the negative log likelihood: E[− log pθ(x0)] ≤ E � − log pθ(xT ) − T � t=1 log pθ(xt−1|xt) q(xt|xt−1) � Relaxing the constraint that the inference process has to be Markovian leads to another kind of diffusion model known as the Denoising Diffusion Implicit Model (DDIM) as proposed by Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This model has the benefit of a deterministic generative process such that: xt−1 = √αt−1f (t) θ (xt) + � 1 − αt−1ϵ(t) θ (xt) (6) where f (t) θ (xt) = 1 √αt (xt − √ 1 − αtϵ(t) θ (xt)) (7) which allows for deterministic embedding of the image x0 into a latent representation xT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' However, xT does not contain high-level semantics, which is a necessary property for deep learning-based morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 Diffusion Autoencoders Preechakul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [38] proposed a Diffusion Autoencoder by employing a conditional DDIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The DDIM is conditioned on a semantic representation z ∈ Z by modifying ϵ(t) θ to take z as an additional input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Training is done via a simplified loss [35] defined as: L = T � t=1 E x0∼q(x0) ϵt∼N (0,I) ||ϵ(t) θ (xt, z) − ϵt||2 2 (8) Therefore, Equation (6) functions as a decoder network from latent code (xT , z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Let E : X → Z be the semantic encoder network such that z = E(x0) which provides the information necessary for decoder pθ(xt−1|xt, z) to properly denoise images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Likewise, the stochas- tic encoder can be thought of as a reversal of Equation (6) such that xt+1 = √αt+1f (t) θ (xt, z) + � 1 − αt+1ϵ(t) θ (xt, z) (9) 5 where xT is encouraged to encode only stochastic details left out from the semantic representation z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='3 Proposed Morphing Algorithm We propose a novel process for the creation of morphed images by employing both the stochastic and semantic encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In particular, let xa, xb ∈ X be two bona fide images of identities a, b, and let x(a) 0 = xa and x(b) 0 = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Algorithm 1 outlines the structure of the proposed Diffusion-based attack, hereafter called the Diffusion attack for simplicity, with additional illustration provided in Figure 2 Beyond the core components of the DDIM and semantic encoder, three additional functions are added to the architec- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Namely, these are the image space preprocessing func- tion, ξ : X × X → X , image space interpolation function, ℓX : X ×X ×[0, 1] → X , and latent space interpolation function, ℓZ : Z × Z × [0, 1] → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The interpolation functions are used to interpolate between the semantic and stochastic values by some factor γ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The image space preprocessing function is used to prepare the image passed to the semantic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The simplest form of the interpolation function is the linear interpolation function, lerp(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' γ) = γa+(1−γ)b, where linear interpolation was found to be the best choice for ℓZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' However, for ℓX Song et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [37] suggests the usage of spherical linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For a vector space V and two vectors u, v ∈ V the spherical interpolation by a factor of γ is given as slerp(u, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' γ) = sin((1 − γ)θ) sin θ u + sin(γθ) sin θ v (10) where θ = arccos(u·v) ||u|| ||v|| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' While the semantic code provides most of the fundamental information, such as positioning of facial features, the stochastic code is used to provide information on the details not explicitly associated with the identity, but are necessary for realism of the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' By altering the stochastic code, details such as direction of strands of hair, clothing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', are altered whilst the identity of the image is preserved [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Unlike the rather straight- forward nature of linearly interpolating between the semantic codes to produce an image with key identifying characteristics of both identities, the nature of the stochastic code can lead to images of low visual fidelity if the interpolation is not done carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In particular, linear interpolation between two stochastic codes is does guarantee a smooth interpolation between the stochastic details in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For this reason the preprocessing function, ξ, is used to prepare the image passed to the stochastic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' One strategy is to “pre-morph” the image when extracting the stochastic details, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', ξ performs an image space morph of the image with the goal of reducing the artifacts induced by stochastic interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 4 EXPERIMENTAL SETUP To evaluate the effectiveness of the proposed morphing attack, the attack is evaluated on three datasets against two different state-of- the-art FR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' All training, optimization, and evaluation was conducted on a system with dual Intel Xeon Silver 4114 CPUs and a NVIDIA Tesla V100 32GB GPU with CUDA version 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 and CUDNN version 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The proposed morphing attack, MAD algorithm, and the FR systems are implemented in PyTorch [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 Face Recognition Systems In order to evaluate the strength of the proposed morphing attack, two publicly available FR systems are used, specifically, the FaceNet and VGGFace2 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' These models are widely used and representative recognition systems with state-of-the-art face verification performance [40], [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For both models the last fully connected layer is used to provid a rich feature representation of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Then for a presented face, its feature vector is compared with that of the feature vector belonging to the target face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' If the distance between these two representations is sufficiently “small”, the presented face is then said to have the same identity as the target face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The VGGFace2 model improves upon acclaimed VGGFace [42] by using an improved training dataset, also called VGGFace2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [40] present the SENet architecture—the Squeeze and Excitation Network (SENet) was introduced by Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [43]—as the optimal choice when used with the VGGFace2 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Google’s FaceNet model consists of an Inception-ResNet V1 architecture which is pre-trained on the VGGFace2 dataset [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, the two FR systems use a different pre- processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As all datasets the images and generated morphs are cropped as to be appropriate for passport photos, a face extractor such as MTCNN [44], is omitted in the verification pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The FaceNet model resizes the image such that the short side of the image is 180 pixels long and then the image is cropped to a 160 × 160 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Lastly, the images are normalized to [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The VGGFace2 model resizes the image such that the short side of the image is 256 pixels long and then crops the image to 224 × 224 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The mean RGB vector1 is subtracted from the cropped image to normalize the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 Datasets In this work, the FERET [45], FRLL [24], and FRGC v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='0 [46] datasets were used to evaluate the proposed technique, as they are commonly used in MAD with a large number of different identities [17], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Notably, the FRLL dataset consists of high quality close-up frontal images at a 1350 × 1350 resolution with 189 facial landmarks—a large number of landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The StyleGAN2, MIPGAN-II, and diffusion models were all trained on the FFHQ dataset, which contains 70,000 images at a 1024×1024 resolution [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Morphs using OpenCV, FaceMorpher, and Style- GAN2 were created by Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [17] on the FRLL, FERET, and FRGC datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [22] created morphs via MIPGAN-II on the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In order to create a morphed face, two component identities are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Naturally, if the two component identities are disparate, the resulting morph is likely to be very weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To rectify this and for evaluation purposes the component identity pairs were selected by following the existing protocol offered by Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' These pairings resulted in 1222 unique morphs on FRLL, 964 on FRGC, and 529 on FERET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5 RESULTS The proposed morphing attack is compared to state-of-the-art techniques drawing from both GAN-based and Landmark-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The effectiveness of the proposed method is quantita- tively measured on three fronts, these are: the visual fidelity of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The mean vector is specifically ⟨131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='0912, 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='8827, 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='4953⟩ for the red, green, and blue channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 6 (a) Identity a (b) OpenCV (c) StyleGAN2 (d) Diffusion (e) MIPGAN-II (f) FaceMorpher (g) Identity b Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 4: Different generated morphs from two identities from the FRLL dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' (a) Diffusion (b) MIPGAN-II Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5: Comparison of Diffusion and MIPGAN-II morphed faces on FRGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Images are resized to 256 × 256 and cropped to 224 × 224 to match VGGFace2 pre-processing pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' the generated morphed images, the vulnerability of state-of-the-art FR systems to the morphing attack, and the detection potential of the morphing attack, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Furthermore, an exploration of interpolation techniques for the stochastic latent code is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 Evaluation of Visual Fidelity The visual fidelity of the Diffusion attack is compared against other morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Whilst on first glance it may appear that the ability to deceive a FR system should imply a high level of visual fidelity, this is not a simple assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We posit two reasons for this discrepancy: 1) The image pre-processing pipeline for an FR system may crop out a significant portion of artifacts in the original morphed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 2) The FR system is vulnerable to forms of adversarial attacks which can degrade the performance of the FR sys- tem while injecting noticeable artifacts into the morphed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This can lead to a situation wherein the FR system is fooled by a morphed image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' however, it would be trivial for a human agent to TABLE 1: FID across different morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Morphing Attack FRLL FRGC FERET StyleGAN2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='19 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='41 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='91 FaceMorpher 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='97 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='58 OpenCV 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='94 MIPGAN-II 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='41 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='96 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='88 Diffusion 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='63 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='16 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='45 notice the artifacts present in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, a deep learning system could be specifically trained to notice such artifacts, greatly reducing the potential of such an attack to go undetected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To quantitatively assess the visual fidelity of the generated images the Fr´echet Inception Distance (FID) is employed, as it has shown to correlate well with human assessment of fidelity [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The FID is a measure of distance between the generated and target distributions, therefore, the lower the FID metric the more similar the generated distribution is to the target distribution which correlates well with visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The metric is defined as the Fr´echet distance, or 2-Wasserstein metric2, between two Gaussian distributions each representing the activations the deepest layer of an Inception v3 network induced by images from the generated and target distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Table 1 shows the FID metric between the generated images from different morphing attacks and bona fide samples from the dataset the morphing attack is drawn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The FID met- ric is calculated using pytorch-fid [48] Morphed images generated from the Diffusion attack generally had the lowest FID with the StyleGAN2-based attack following closely behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Both Landmark-based morphs—OpenCV and FaceMorpher—had noticeably higher FIDs than the deep learning-based morphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' These results correlate well with visual inspection of the morphs as both Figure 4b and Figure 4f exhibit prominent artifacts outside the central face region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Likewise, the MIPGAN-II attack seems to struggle with some distortion outside the central face region, see Figure 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Interestingly, on the FRLL dataset the StyleGAN2 morphing pipeline consistently darkens morphs relative to its com- ponent images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' however, the visual fidelity is relatively high albeit with noticeable darkening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Importantly, stochastic details, such as hair, seem to be modelled well by the Diffusion attack whereas other attacks distort such details, the OpenCV, FaceMorpher, and MIPGAN-II attacks, or present details that have little similarity to both identities, the StyleGAN2 attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' While exhibiting the far less visual artifacts than other morphing techniques, the Diffusion attack does tend to slightly smooth out the skin texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Overall, the Diffusion attack exhibits the highest consistent visual fidelity among all the presented attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Strangely, MIPGAN-II exhibited a much higher FID than the StyleGAN2 morphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This is a surprising result as the MIPGAN- II attack was positioned as an improvement over the StyleGAN2 attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Figure 5 compares two morphed faces generated by the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The 2-Wasserstein metric between two probability measures µ, ν with finite moments on Rn is defined as W2(µ, ν) = � inf π∈Π(µ,ν) � Rn×Rn ||x − y||2 2 dπ(x, y) � 1 2 where Π(µ, ν) is the set of all distributions with marginals µ and ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 二二7 TABLE 2: The APCER at specific BPCER values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Dataset FR System Morphing Attack APCER @ BPCER = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1% APCER @ BPCER = 1% APCER @ BPCER = 5% FRLL FaceNet StyleGAN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='05 0 FRLL FaceNet FaceMorpher 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='05 FRLL FaceNet OpenCV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='33 0 FRLL FaceNet MIPGAN-II 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='09 FRLL FaceNet Diffusion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='23 FRLL VGGFace2 StyleGAN2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='05 0 FRLL VGGFace2 FaceMorpher 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='23 FRLL VGGFace2 OpenCV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='28 FRLL VGGFace2 MIPGAN-II 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='23 FRLL VGGFace2 Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='68 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='52 FRGC FaceNet StyleGAN2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='04 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='69 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='73 FRGC FaceNet FaceMorpher 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='85 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 FRGC FaceNet OpenCV 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='43 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='36 FRGC FaceNet MIPGAN-II 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='54 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='48 FRGC FaceNet Diffusion 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='73 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='86 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 FRGC VGGFace2 StyleGAN2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='42 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='22 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='7 FRGC VGGFace2 FaceMorpher 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='42 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='65 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='18 FRGC VGGFace2 OpenCV 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='31 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='62 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='4 FRGC VGGFace2 MIPGAN-II 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='92 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='84 FRGC VGGFace2 Diffusion 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='25 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='18 FERET FaceNet StyleGAN2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='65 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='49 FERET FaceNet FaceMorpher 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='91 FERET FaceNet OpenCV 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='17 FERET FaceNet MIPGAN-II 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='03 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='21 FERET FaceNet Diffusion 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='04 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='99 FERET VGGFace2 StyleGAN2 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='08 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='73 FERET VGGFace2 FaceMorpher 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='65 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='7 FERET VGGFace2 OpenCV 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='01 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='87 FERET VGGFace2 MIPGAN-II 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='84 FERET VGGFace2 Diffusion 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='34 Diffusion and MIPGAN-II attacks on the FRGC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Numer- ous high frequency artifacts are present in Figure 5b, particularly, near the hairline and transition between hair and the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Comparing Figure 5a and Figure 5b the hair generated by the MIPGAN-II attack looks unnatural with a strange texture as though an image sharpening filter has been applied to the image, greatly enhancing the magnitude of high frequency content, which aligns with the observation in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, the MIPGAN- II images seem to be desaturated when compared to images produced by other attacks leading to a washed out appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Perhaps the low visual fidelity can be explained by the identity loss overpowering the perceptual quality loss leading to morphed images with low visual fidelity but high effectiveness against FR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 Vulnerability of FR Systems The strength of the proposed face morphing algorithm is further evaluated by measuring the ability of the morph to fool a FR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The attack success is quantitatively verified against two state-of-the-art FR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To ensure a valid comparison between the five different morphing attacks the same pairs of component identities were used in evaluating every morphing attack, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', for every pair of component identities a morphed image was created for each of the five attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For both the FaceNet and VGGFace2 FR systems the False Match Rate (FMR) is set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1% following the guidelines of Frontex [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, the distance between faces is measured using the L2 distance between the outputs of the FR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The vulnerability of FR systems to morphing attacks is as- sessed by comparing the error rates in detection, specifically, the Attack Presentation Classification Error Rate (APCER)3 is measured at specific Bona fide Presentation Classification Error Rate (BPCER)4 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In Table 2 the APCER values for the five different morphing attacks is presented across all three datasets evaluated on three different BPCER values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1%, 1%, and 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Due to a variety of factors—such as image quality and number of bona fide images per identity—the results vary between the different datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' while there is some variance between both FR systems, they tend to agree more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Noticeably, all attacks performed rather poorly on the FRLL dataset, although the Diffusion-based morphing attack performed the best among them, which could be attributed to the limited number of bona fide images per identity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' for in the FRLL dataset there are only two bona fide images per identity: a neutral face (used to create the morph) and a smiling face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Both FR systems on the FRLL dataset tend to give close relative rankings between the morphing attacks with the StyleGAN2-based attack being noticeably weaker than the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Both FR systems were much more vulnerable when evaluated on the FRGC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The MIPGAN-II attack performed very well against FaceNet which makes sense as this technique was refined on the FRGC dataset in particular [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The attack was not as strong on VGGFace2 and instead that FR system was more vulnerable to OpenCV and FaceMorpher, this could possibly be attributed to the different pre-processing pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The Diffusion- based generally performs close to the top performer on either FR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As with FRGC, on the FERET dataset VGGFace2 is more 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' APCER is the proportion of attack presentations incorrectly classified as bona fide presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' BPCER is the proportion of bona fide presentations incorrectly classified as attack presentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 8 TABLE 3: MMPMR at FMR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1% across diffrent morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' FRLL FRGC FERET Morphing Attack FaceNet VGGFace2 FaceNet VGGFace2 FaceNet VGGFace2 Geometric Mean StyleGAN2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='69 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='10 FaceMorpher 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='26 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='51 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='78 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 OpenCV 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='34 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='69 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='32 MIPGAN-II 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='96 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='94 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='34 Diffusion 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='68 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='13 vulnerable to landmark-based attacks, OpenCV and FaceMorpher, than FaceNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Diffusion-based morphs pose the greatest threat on FERET consistently having high APCER values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In general the following observations can be drawn from Table 2: Among the five different attacks, FR systems are most vul- nerable to Diffusion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, Diffusion attacks always rank in the top three in terms of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' FR systems are the least vulnerable to the StyleGAN2 at- tack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The StyleGAN2 attack is always outperformed by its successor, MIPGAN-II, and the other deep learning-based attack, Diffusion, while often falling behind landmark- based attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In addition to using the error rates to assess the vulnerability of FR systems the Mated Morphed Presentation Match Rate (MMPMR) [50] is used as a measure of vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Scherhag et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [50] propose two variants of the MMPMR metric for the scenario in which there multiple bona fide images of an identity used in morph process, excluding the image used to create the morph, called the MinMax-MMPMR and ProdAvg-MMPMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The MinMax-MMPMR metric is especially likely to increase the number of accepted morphs as the number of bona fide images per identity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Therefore, the ProdAvg-MMPMR is the specific MMPMR variant used to assess the vulnerability of FR systems, any mention hereafter to MMPMR refers specifically to ProdAvg-MMPMR unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Let PM ∈ P(X) be the distribution of morphed images such that for some xab ∼ PM, xab denotes a morphed image made from identities a, b, where P(X) denotes the set of all probability measures on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Let Pk ∈ P(X) denote the distribution of bona fide images of identity k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Then with abuse of notation Pk\\xab is the distribution of bona fide images of identity k excluding those images used in creating the morph xab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The MMPMR metric for a particular threshold, γ > 0, equipped with FR system F : X → V is then defined as M(γ) = E xab∼PM � � k∈{a,b} E x∼Pk\\xab � ||F(xab)−F(x)||2 < γ �� i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', the expected success rate of the morphing attack to fool the FR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The product term is the joint probability of successful verification of both identities Table 3 presents the MMPMR metric when the FMR is set at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1% for all datasets and FR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Interestingly, the FRLL dataset had the highest overall MMPMR metrics in contrast to the results from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This can likely be attributed to limited number of bona fide images per identity and FRLL in contrast with other datasets as the particular choice of MMPMR metric heavily punishes failed verifications in either identity, thus having only one possible image per identity could boost the metric for FRLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' On average the Diffusion attack greatly outperforms the TABLE 4: APCER at FMR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1% across different margin sizes on the FaceNet FR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Higher is Better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Margin Size Dataset Morphing Attack 0 20 40 80 FRLL MIPGAN-II 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='84 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='53 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='18 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='03 FRLL StyleGAN2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='12 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='57 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='11 FRLL FaceMorpher 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='48 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='26 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='86 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='91 FRLL OpenCV 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='21 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='82 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='96 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='4 FRLL Diffusion 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='25 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='07 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 FERET MIPGAN-II 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='33 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='11 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='36 FERET StyleGAN2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='12 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='57 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='17 FERET FaceMorpher 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='11 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='45 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='28 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='8 FERET OpenCV 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='11 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='44 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='64 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='89 FERET Diffusion 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='24 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='91 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='39 FRGC MIPGAN-II 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='33 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='97 FRGC StyleGAN2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='18 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='74 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='42 FRGC FaceMorpher 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='87 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='39 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='07 FRGC OpenCV 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='91 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='6 FRGC Diffusion 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='27 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='62 other attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' conversely, the Landmark-based attacks, on average, exhibit mediocre performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In agreement with Table 2 the StyleGAN2 attack shows abysmal performance in comparison with the other attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 The Effect of Pre-processing on a FR System The impact of the pre-processing pipeline on the vulnerability of a FR system is examined, in particular the cropping process is further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To study this an additional margin size is added to the image after a initial face extraction and cropping performed by MTCNN, such that a margin size N adds back at most N pixels to the cropped image in both dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Therefore, the larger N is the less tightly cropped the image passed to the FR system is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Table 4 shows the illustrates the impact of the margin size on the APCER metric on the FaceNet FR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Generally, as the margin size increases the performance of the Landmark-based attacks decreases and the performance of the deep learning-based attacks increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As illustrated in Figure 4 the Landmark-based attacks have noticeable artifacts outside the central face region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' conversely, the deep learning-based morphs have less artifacts in the outside regions and generally look more realistic to a human observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This observation aligns the visual fidelity results from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Therefore, a MAD algorithm or FR system which uses less tightly cropped faces would be more resilient against attacks with visual artifacts outside the core face region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 General Remarks on the Vulnerability Study The poor performance of the StyleGAN2 attack could be at- tributed to the darkening of images with light backgrounds, see Figure 4, and due to aliasing effects latent to the StyleGAN2 generation pipeline which is addressed by Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 9 TABLE 5: Ablation study on the impact morphing attack on validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Training Attack Validation Attack Dataset Diffusion FaceMorpher MIPGAN-II OpenCV StyleGAN2 Diffusion FaceMorpher MIPGAN-II OpenCV StyleGAN2 FERET \x17 \x13 \x13 \x13 \x13 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='73 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='23 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='33 FERET \x13 \x17 \x13 \x13 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='39 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='85 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='64 FERET \x13 \x13 \x17 \x13 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='69 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='38 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='54 FERET \x13 \x13 \x13 \x17 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='74 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='48 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='74 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='43 FERET \x13 \x13 \x13 \x13 \x17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='74 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='56 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='74 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='89 FRGC \x17 \x13 \x13 \x13 \x13 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='89 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='98 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='97 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 FRGC \x13 \x17 \x13 \x13 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='48 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 FRGC \x13 \x13 \x17 \x13 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='83 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='85 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='82 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='8 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='85 FRGC \x13 \x13 \x13 \x17 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 100 100 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='23 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 FRGC \x13 \x13 \x13 \x13 \x17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='94 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='88 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='83 FRLL \x17 \x13 \x13 \x13 \x13 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='96 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='58 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='32 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='65 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='65 FRLL \x13 \x17 \x13 \x13 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='23 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='09 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='91 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='37 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='44 FRLL \x13 \x13 \x17 \x13 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='09 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='24 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='09 FRLL \x13 \x13 \x13 \x17 \x13 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='51 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='44 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='19 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='16 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='58 FRLL \x13 \x13 \x13 \x13 \x17 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='86 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='86 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='93 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='02 Moreover, the structure of the StyleGAN2 latent space can make exploration in the space difficult which could possibly explain the poor performance in attacking the FR system compared to other attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' MIPGAN-II, on the other hand, likely avoids these pitfalls due its explicit latent optimization process for fooling a FR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The Diffusion attack utilizes an entirely different latent representation scheme which seems to yield an advantage in the task of generating morphed faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The pre-processing pipeline of the FR system seems to mostly mitigate the artifacts latent to the Landmark-based attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' however, such artifacts could easily be detected by a human observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='3 Detectability of Morphing Attacks The performance of the proposed attack is further evaluated by the ability of Morphing Attack Detection (MAD) algorithms trained against other attacks to detect an unseen attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To quantita- tively assess the detectability of a particular morphing attack a SE-ResNeXt101-32x4d model pre-trained on ImageNet [52] by NVIDIA is trained to detect morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' SE-ResNeXt101- 32x4d is a state-of-the-art image recognition model based on the ResNeXt101-32x4d model [53] with the addition of the Squeeze- and-Excitation architecture [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' For all experiments a 5-fold strat- ified k-fold cross validation strategy is employed, thus preserving the class balance between morphed and bona fide images in each fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The model is fine-tuned on a collection of morphing attacks for 5 training epochs using exponential learning rate scheduler with differential learning rates in order to mitigate overfitting of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1 Ablation Study To study the impact of a particular morphing attack on the ability of a MAD algorithm to detect morphing attacks an ablation study was conducted where the SE-ResNeXt101-32x4d model was trained on all the morphing attacks except for one holdout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Table 5 shows the validation accuracy of each morphing attack when different morphing attacks were withheld from the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Due to the similar natures between the OpenCV and FaceMorpher attacks the absence of one of these attacks does not greatly impact the validation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Interestingly, the absence of MIPGAN-II does not significantly change the validation accu- racy of the attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' however, the omission of StyleGAN2 during training does decrease the performance of the StyleGAN2 during validation despite the presence of the MIPGAN-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Notably, the Diffusion attack is very difficult to detect as a novel attack, which can be partially attributed to its unique morph generation process in contrast with the other morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 A Metric For Relative Strength In this section we introduce a metric to measure the relative strength from one morph to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We say a morph α is “strong” relative to a morph β if the following conditions are satisfied: 1) It is easy to detect β when a detector is trained on α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', high transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 2) It is hard to detect α when a detector is trained on β, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', low detectability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, the relative strength metric, ∆(α||β), should be positive when α is stronger than β and negative when α is weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' A relative strength of 0 would denote that the two morphing attacks are equally strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As some of the morphing attacks are not deterministic but probabilistic, we chose to represent a morphing attack α by the random variable Xα : Ω → X such that P(Xα|xa, xb) denotes the distribution of morphs generated from images xa, xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, we suppose there exists a detector f α : X → {0, 1} trained to distinguish between bona fide presentations and mor- phed presentations generated by α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' wherein 0 denotes a bona fide presentation and 1 denotes a morphed presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The transfer- ability of a morphing attack α to β is defined as the probability the detector f α is able to detect the attack β given the probability f α detects α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', T(α, β) = P(f α(Xβ) = 1|f α(Xα) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This metric can be represented as a ratio of expectations taken over the pairs of component bona fide images: T(α, β) = P(f α(Xβ) = 1, f α(Xα) = 1) P(f α(Xα) = 1) = Exa,xb[P(f α(Xβ) = 1, f α(Xα) = 1|xa, xb)] Exa,xb[P(f α(Xα) = 1|xa, xb)] (11) Let {xα i }N i=1 denote a collection of N samples drawn from P(Xα|xa, xb) such that xα i denotes the morph generated from i-th pair of bona fide identities (ai, bi), and likewise for β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Then the metric in Equation (11) can be closely approximated by T(α, β) ≈ �N i=1 � f α(xβ i ) = 1 ∧ f α(xα i ) = 1 � �N i=1 � f α(xα i ) = 1 � (12) 10 (a) RSM on FRGC (b) RSM on FERET (c) RSM on FRLL Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 6: Blue indicates higher strength and red indicates weak strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=', the number of morphs from both α and β detected over the number of morphs detected from α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The relative strength metric (RSM) from α to β is defined as the log ratio of the transferability metrics between the two morphing attacks: ∆(α||β) = log �T(α, β) T(β, α) � (13) The log of the ratio is chosen such that ∆(α||β) the RSM takes (a) Variant A (b) Variant B (c) Variant C (d) Variant D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 7: Morphed image generated by different Diffusion attack variants on FRLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' positive values when α is “stronger” than β and negative values when weaker—with a value of zero denoting equal strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, there is an antisymmetry such that ∆(α||β) = −∆(β||α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In contrast to the ablation study, the SE-ResNeXt101-32x4d model is only trained on a single attack per k-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The RSM is calculated between all attacks with the results shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' From Figure 6 it is observed that the RSM between the Landmark- based morphs and the RSM between the StyleGAN-based morphs is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As these attacks have similar morph generation pipelines it makes sense that the transferability between the attacks is near identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In general, the Landmark-based attacks seem to be stronger than the StyleGAN-based attacks, in particular the FaceMorpher attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The MIPGAN-II attack is generally weaker than the other attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Overall, the Diffusion attack is the least detectable among the attacks along with generally being the strongest attack across the three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The results from Figure 6 corroborate with the results from Table 5 demonstrating the difficulty in detecting Diffusion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' From the perspective of training a MAD system including samples from the FaceMorpher, StyleGAN, and Diffusion attacks would greatly increase the ability for the system to detect unknown attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Additionally, Table 5 and Figure 6 demonstrates a par- ticular vulnerability existing MAD systems may have to the new Diffusion attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='4 Study of the Diffusion-based Morphing Process The diffusion morphing algorithm leverages both a stochastic and semantic representation of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' While the semantic representation contains many of the key “identifying” features, the stochastic representation represents many of the details nec- essary for high visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Due to the importance of the stochastic code for high fidelity, we investigated several methods for finding the morphed stochastic latent code, x(ab) T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The first 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='517 StyleGAN2 OpenCv Attack Training A MIPGAN-II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='000 FaceMorpher Diffusion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='517 OP eGAN2 ValidationAttack2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='734 StyleGAN2 OpenCv Attack Training A MIPGAN-II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='000 FaceMorpher Diffusion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='734 OR MIPGAN-II ValidationAttack18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 StyleGAN2 Opencv Training Attack MIPGAN-II 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='0 FaceMorpher Diffusion 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 MIPGAN-II OR GAN Validation Attack11 TABLE 6: MMPMR at FMR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='1% across different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Higher is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' † indicates our default choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' FRLL FRGC FERET Variant ℓX ξ(x, y) FaceNet VGGFace2 FaceNet VGGFace2 FaceNet VGGFace2 Geometric Mean A slerp x, y �→ x 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='97 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='17 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='54 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='95 B lerp x, y �→ x 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='81 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='86 C† slerp x, y �→ 1 2 (x + y) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='68 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='47 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='03 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='13 D slerp x, y �→ OpenCV(x, y) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='34 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='42 0 TABLE 7: FID across different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Lower is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' † indicates our default choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Variant ℓX ξ(x, y) FRLL FRGC FERET A slerp x, y �→ x 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='13 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='97 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='66 B lerp x, y �→ x 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='05 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='33 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='75 C† slerp x, y �→ 1 2 (x + y) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='63 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='16 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='45 D slerp x, y �→ OpenCV(x, y) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='85 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='51 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content='49 variant, variant A, is the baseline implementation with ℓZ using linear interpolation, ℓX using spherical linear interpolation, and ξ does not perform any “pre-morphing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Conversely, in variant B the stochastic codes are interpolated via linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In variants C and D, instead of using the original image to calculate the stochastic code, the function ξ is used to construct the “pre- morph” passed to the stochastic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Specifically, in variant C the two images are averaged pixel-wise and presented to the stochastic encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' in contrast, in variant D the OpenCV morph is presented to the stochastic encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' In Table 7 the FID is calculated between the generated morphs and the bona fide samples for each particular dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Variant C generally presents the lowest FID score closely followed by variant A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Both variants B and D exhibit clear degradation in performance when compared to variants A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Furthermore, the FID scores seems to correlate well to human assessment of the generated samples, see Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Noticeably, the linear interpolation in variant B results in an overly smoothed face and generally darker image, greatly degrading visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Variant D has prominent visual artifacts, similar to the artifacts found in the OpenCV morphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, the poor performance seems to be aided by an issue of differing alignment strategies between the OpenCV and diffusion pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Notably, variant C often removes many of the high frequency artifacts found in variant A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This is likely due to the difficulty in smoothly interpolating between points in the stochastic latent space in contrast with the semantic latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' As such, variant C which performs a pixel-wise average of the two source images before using the stochastic encoder seems to greatly improve the ability to smoothly interpolate between different stochastic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This appears to be the primary reason variant C has a generally lower FID when compared to variant A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Both Figure 7 and Table 7 demonstrate the large importance that the stochastic code plays in creating high fidelity morphed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Due to the high fidelity exhibited by variant C, this particular diffusion process was used in evaluation against other morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The MMPMR metric is calculated for each variant, see Ta- ble 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Variant A is slightly stronger than variant C, with variants B and D falling far behind likely due to the high number of visual distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' These results stand in contrast to the assessment of vi- sual fidelity wherein variant C outperforms variant A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' This, again, illustrates a trade-off between visual fidelity and ability to fool the FR system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' however, in this case the trade-off effectiveness against the FR system is relatively small in comparison to the gains in visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Due to its excellent visual fidelity and strong MMPMR results variant C was chosen to be the default configuration for the Diffusion attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' 6 CONCLUSION By addressing some of the key limitations of prior deep-learning based morphing attacks, namely, the trade-off between visual fidelity and effectiveness against FR systems, we have proposed a novel morphing attack using Diffusion-based methods for the generative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The proposed attack consistently generates realistic morphed images with high visual fidelity while also being able to strongly threaten FR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' To evaluate the attack potential of the proposed attack, we evaluated the vulnerability of two FR systems over three distinct datasets and created over 10,000 new morphs between the four variants, with the strongest variant achieving state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' We conducted an additional study on the impact the pre-processing pipeline has on the vulnerability of an FR system to morphing attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' A novel metric to assess the strength of morphing attacks relative to each other has been introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' Moreover, the proposed attack was evaluated by its detection performance against a state-of-the-art MAD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The Diffusion attack was shown to be very difficult to detect if not specifically trained against presenting showing the proposed attack can greatly threaten preexisting FR systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' The images generated by the Diffusion attack possess high visual fidelity, can fool state-of-the-art FR systems, and are difficult for MAD mechanisms to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9E2T4oBgHgl3EQf8Qn_/content/2301.04218v1.pdf'} +page_content=' REFERENCES [1] L.' 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826004 , INDIA +E-MAIL: RANJIT.CHAOS@GMAIL.COM +( Accepted: 02 November 2021 ) +Abstract. +The concept of superfluous prey, or an excess of prey in certain areas within +a patchy ecosystem, has significant implications for the synchronization of the predator +population. +These areas, known as ”hotspots,” have a higher density of prey compared +to other areas and attract a higher concentration of predators. +As a result, the preda- +tor population becomes more stable and predictable, as they are less likely to migrate to +other areas in search of food. This phenomenon can have important consequences for both +the predators and their prey, as well as the overall functioning of the ecosystem. +This +work investigates the synchronization between two chaotic food webs using the general- +ized Lotka-Volterra (GLV) model consisting of one prey and two predator populations. +We, first, examine the impact of three functional responses (linear, Holling type II, and +Holling type III) on system dynamics For the study, we consider the model with a linear +functional response consisting of chaotic oscillations and apply controllers to stabilize its +unstable fixed points. +This research contributes to the understanding of how to apply +chaotic ecological models to predict the population of competing species in one habitat +using information about similar populations in another system. To do this, we configure a +drive-response system where prey acts as the driving variable and both predators depend +only on the prey. We use active and adaptive control methods to synchronize two coupled +GLV models and verify the analytical results through numerical simulations. +AMS Classification: — +Keywords: +Lyapunov exponents, slow manifold equation, chaos control, complete +replacement synchronization, active and adaptive control techniques +JOURNAL OF DYNAMICAL SYSTEMS & GEOMETRIC THEORIES +©TARU PUBLICATIONS +1 + +2 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +1. Introduction +In ancient philosophy and mythology, the word chaos meant the disordered state +of unformed matter supposed to have existed before the ordered universe. In the +course of its journey to truth, science has met with startling phenomena called chaos. +Since the 1960s, with the discovery of chaotic systems, chaos has set a nonlinear +dynamics research boom. Chaos theory is attributed to the work of Edward Lorenz. +His 1963 paper, “Deterministic Non-periodic Flow” [1], is credited for laying the +foundation for chaos theory. Hunt and Ott [2] reviewed the problem and proposed a +computationally feasible entropy-based good definition of chaos. They define chaos +as “ the existence of positive Expansion Entropy (EE) (equal to topological entropy +for infinitely differentiable maps) on a given restraining region (bounded positive +volume subset),” which confirms both the notions (COS as well as OS). Since EE +enjoys the properties of simplicity, computability, and generality, so they call it a +“ good ” definition of chaos. Chaotic systems are sensitive to initial conditions, +topologically mixing and with dense periodic orbits [3], [4]. Because of slightest dif- +ference, chaotic dynamical systems can lead to entirely different trajectories. The +main characteristic of chaos is that the system does not repeat its past behaviour. +Mathematically, chaotic dynamical systems are classified as non-linear dynamical +systems having at least one positive Lyapunov exponent[5]. +Chaos theory is not just about chaos - it has two sides to it: chaos control and chaos +synchronization. The study of chaos control and understanding chaotic model be- +haviour has gained significant interest, with applications in various fields such as +secure communications, biology, neural networks, finance, and more. Whereas chaos +synchronization refers to the alignment in time of different chaotic processes. At +first glance, chaotic systems may seem to defy synchronization, but it has been +observed and studied in various contexts. +In 1665, Dutch physicist Huygens observed the adjustment of rhythms via a cou- +pling. +He noticed that pendulum clocks suspended from the same beam would +slowly adjust their phases. In 1984, Kuramoto set theory for the onset of sync, and +Pecora and Carroll [6] reviewed the area of synchronization in chaotic systems and +presented a more geometric view using synchronization manifold. In 1999, Blasius +explained the theoretical analysis of seasonally synchronized chaotic population cy- +cles [7]. All these contributions help the researcher think that synchronization is + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +3 +an essential phenomenon in physical and biological systems. In literature, this phe- +nomenon has been nominated by various types, such as phase locking, frequency +pulling, generalized synchrony, and complete locking, depending on the degree or +type of synchronization. +Synchronization of chaotic systems can be achieved by configuring drive and re- +sponse systems, with the goal of using the output of the drive system to control +the response system so that the output tracks the drive system asymptotically. +However, creating identical chaotic synchronized ecological systems is questionable +as it has a potential threat to biodiversity. As discovered in Pecora and Carroll’s +pioneering work, another way of achieving complete synchronization between two +systems is what is now called the technique of complete replacement. The complete +replacement synchronization is helpful in a network of patchy ecosystems as it can +help in identifying the underlying mechanism that derive the group co-ordination +in the present of severely chaotic oscillations. In population biology, the chaotic +dynamics may synchronize if populations are coupled through environmental or bi- +ological interactions. +From the ecological aspect, it is crucial to figure out the ecologically feasible cou- +pling scheme that guarantee the permanence and global attractiveness of all species +in a multi-patch ecosystem. Upadhyay and Rai [8] have demonstrated that the +two non-identical chaotic ecological systems having different kinds of top-predators +can be synchronized using an algorithm proposed by Lu and Cao[9]. The idea of +this approach is that it takes care of the uncertainties involved in the parameter +estimation. There are many methods in control theory for synchronizing chaotic +systems, including the Adaptive Control Method, Back-stepping Method, Active +Control Method, Time-Delay feedback approach, and others. +Among above-listed methods, the linear active control technique for chaos syn- +chronization is popular and effective for synchronizing identical and non-identical +chaotic systems. In this work, we will use the active and adaptive control meth- +ods for achieving synchronization of chaos within either identical or non-identical +systems. The Active control method was first used for chaos synchronization by +E.W. Bai and K.E. Lonngren [10], [11]. In this method, non-linear controllers are +designed based on the Lyapunov stability theory to achieve synchronization in cou- +pled systems using the known parameters of the drive and response systems. + +4 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +Since we will be dealing with an ecological model, the system’s parameters cannot be +precisely known. The adaptive control is one of the popular technique for controlling +and synchronizing non-linear systems with uncertain parameters [12]. The method +allows the model to adapt data assimilation along the way that may be useful for +predicting the real system’s future behaviour [13], [14]. In this method, control law +and parameter update law are designed in such a way that the chaotic response +system to behave like chaotic drive systems. This scheme maintains the consistent +performance of a system in the presence of uncertainty, variations in parameters. +Consequently, asymptotically global synchronization control of the chaotic system +guarantees to converge the error dynamics to the equilibrium point. +In this article, we investigate the three-dimensional chaotic generalized Lotka-Volterra +system, a more general model than the competitive predator-prey examples of +Lotka-Volterra types. We examine the properties of the model, including equilib- +rium analysis, dissipative properties, the maximum Lyapunov exponent, and slow +manifold analysis. We use linear feedback control to stabilize the model at its equi- +librium points. Our goal is to synchronize two identical GLV models with the same +drive variable and different initial conditions using complete replacement synchro- +nization in an ecological context. Prey population is assumed to be in abundance in +such a way that predators from nearby patches also feed upon it. To maintain syn- +chronization indefinitely with only small adjustments within a two-patch system, +we design the active control law(when system parameters are known) and adaptive +control law (when system parameters are unknown) mathematically and validate +the analytical results through numerical simulation. +2. +The Model +The generalized Lotka-Volterra equations are autonomous and deterministic. +The dynamics of the model in a more generalized form are defined as +˙xi(t) = xi(t)(ri + fi(x1, x2, . . . , xn)), +with initial condition +xi(0) = x(i,0), for i ∈ {1, 2, . . ., n}. +where n represents the number of species, xi(t) is the size of population i, ˙xi(t) is +the time derivative of species i, t is the time variable, x(i,0) is the initial population + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +5 +of species i, ri is the self-growth of species i, and fi(x1, x2, x3, . . . , xn) is the nonlin- +ear multi-variable function with intra and inter-species competition terms for each +i. Although the populations are usually measured in integer numbers, xi(t) is real +for each i and can be interpreted as density, biomass or some other measure which +correlates with the number of species. Let Rn denotes the Euclidean space and the +function fi is a continuous, smooth and real-valued function for each i. We make +the following assumptions on fi for i ∈ {1, 2, . . ., n} [15]. +(i) fi, for each i ∈ {1, 2, . . ., n} is bounded on a domain D ⊂ Rn. +(ii) There exist constant Ki > 0 for each i ∈ {1, 2, . . ., n} such that +||fi(X) − fi(Y )|| ≤ Ki ||X − Y || +∀ +X, Y ∈ D ⊂ Rn. +2.1. Model with linear functional response. The generalized Lotka Volterra +(GLV) model and its variant have been studied by many authors [16],[17],[18]. Our +main focus is on three-dimensional GLV chaotic system, which has been devised by +Samardzija and Greller in 1988 [19]. We assume that the vector field for the model +holds the above mentioned properties and takes the following form +˙x1 = x1(1 − x2 + rx1 − px3x1), +˙x2 = x2(−1 + x1), +˙x3 = x3(−q + px2 +1). +(1) +where x1, x2, x3 are the state variables representing prey, middle predator and +top predator populations respectively. Here p, q, r are positive parameters. Au- +thors [18] found the system chaotic in particular parametric range p = 2.9851, q = +3, r = 2 and shown interesting complex dynamical behaviour. +For this set of +parameter values, the orbit of all three states for two different initial conditions +((1.0023, 1.0589, 0.6503) and (1.0023 + 10−3, 1.0589 + 10−3, 0.6503 + 10−3)) has sen- +sitive dependence on initial conditions (SDIC) which is displayed in figure 2.1. Fig- +ure 2.1 characterizes the SDIC in the system where the trajectories with initial +condition (1.0023 + 10−3, 1.0589 + 10−3, 0.6503 + 10−3)) dominate over trajectories +with initial condition ((1.0023, 1.0589, 0.6503) in long run. +Further, we display the dynamics of GLV system for three different set of parame- +ters to characterize its parameter- sensitivity. Different sets of parameters involved + +6 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Time +0.6 +0.8 +1 +1.2 +1.4 +1.6 +Prey +IC: (1.0023, 1.0589, 0.6503) +IC: (1.0033, 1.0599, 0.6513) +(a) +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Time +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +Middle Predator +IC: (1.0023, 1.0589, 0.6503) +IC: (1.0033, 1.0599, 0.6513) +(b) +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +4000 +4500 +5000 +Time +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Top Predator +IC: (1.0023, 1.0589, 0.6503) +IC: (1.0033, 1.0599, 0.6513) +(c) +Figure 1. +(a), (b) and (c): Time series of x1, x2, and x3 for +two nearby initial conditions ((1.0023, 1.0589, 0.6503) and (1.0023+ +10−3, 1.0589 + 10−3, 0.6503 + 10−3)). +in the system are taken as +(p, q, r) ∈ {(2.0451, 2.129, 2), (2.9851, 2.99, 2.1), (2.98098, 2.9799, 2)}. +For simulation, we fix the initial condition at x1(0) = 1.0023, x2(0) = 1.0589, x3(0) = +0.6503 for the GLV system. Figures 2, 3 and 4 show three dimensional attractor +and two dimensional projections of the system on (x1, x2), (x1, x3) and (x2, x3) + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +7 +planes. From these figures, it can be inferred that the sensitivity of the system on +parameters can help in restoring the hidden order out of its chaotic dynamics. +(a) +(b) +(c) +(d) +Figure 2. (a): 3D attractor; (b), (c) and (d): 2D projections of +the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively +for (p, q, r) = (2.0451, 2.129, 2). +Next subsections presents two other variants of GLV model with different func- +tional responses. +2.2. Model with Cyrtoid type (HT II) functional response. To elucidate +the role of functional response, we replace the linear interaction term between prey +and middle predator with Holling type II functional response ( ( +x1 +x1+d)x2). +The +ecological meaning of the non-linear interaction of prey with middle predator is +that the prey’s contribution to the middle predator growth rate is x1x2 +x1+d. Using type +II functional response, the dynamics of new GLV model are proposed as +˙x1 = x1 − x1x2 +x1 + d + rx2 +1 − px1 +2x3, +˙x2 = −x2 + ( x2x1 +x1 + d), +˙x3 = −qx3 + px3x2 +1. +(2) +For simulation, we take the parametric values and initial condition as p = 2.514, q = +2.9089, r = 2.1990507, d = .00198 and x1(0) = 1.78, x2(0) = 0.5020, x3(0) = 1.01 + +(X2,x3) plane2D projection of the attractor on +1.8 +1.6 +1.4.1 +1.2 +1.31.2 +3 +X +1 +0.8 +0.6 +0.4 +0.7 +0.8 +0.9 +1(X,,x,) plane2D projection of the attractor on +1.8 +1.6 +1.41.3 +1.4 +1.5 +1.61.2 +X +1 +0.8 +0.6 +0.4 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2(x,X2) plane2D projection of the attractor on +1.3 +1.21.3 +1.4 +1.5 +1.60.9 +0.8 +0.7 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 AttractorThe Generalised Lotka Volterr1.6 +1.4 +1.2 +1 +X +11.5 +3 +X +0.5 +1.2 +1.1 +1 +0.9 +0.8 +0.8 +0.68 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +(a) +(b) +(c) +(d) +Figure 3. (a): 3D attractor ; (b), (c) and (d): 2D projections of +the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively +for (p, q, r) = (2.9851, 2.99, 2.1). +respectively. Figure 5 displays the three-dimensional phase portrait of GLV system +with HT II functional response which is a ‘stable focus’. Thus, a change in functional +response in GLV system can lead to stable dynamics for a suitable set of parameter +values and initial conditions. +2.3. +Model with Sigmoid type (HT III) functional response. Next, we +change interaction term between prey and top predator population with Holling +type III functional response. The inclusion of Holling type III functional response +increases the search activity of top-predator for prey. With assumption of increasing +prey density, the dynamics of new GLV model with Holling type III functional +response are proposed as +˙x1 = x1 − x1x2 + rx2 +1 − p x2 +1x3 +x2 +1 + d, +˙x2 = −x2 + x1x2, +˙x3 = −qx3 + p x2 +1x3 +x2 +1 + d. +(3) + +(X2,X3) plane2D projection of the attractor on +0.9 +0.81.04 +1.06 +1.08 +1.10.7 +0.6 +0.5 +0.9 +0.92 +0.94 +0.96 +0.98 +1 +1.02 +2(X,x) plane2D projection of the attractor on +0.9 +0.81.1 +1.15 +1.20.6 +0.5 +0.85 +0.9 +0.95 +1.05 +X(X,X2) plane2D projection of the attractor on +1.1 +1.051.1 +1.15 +1.20.95 +0.9 +0.85 +0.9 +0.95 +1.05AttractorThe Generalised Lotka Volterr +0.9 +0.81.2 +1.1 +1 +X +10.6 ~ +0.5 +1.1 +1.05 +1 +0.95 +0.9 +0.9 +0.8SYNCHRONIZATION OF CHAOS IN ECOLOGY +9 +(a) +(b) +(c) +(d) +Figure 4. (a): 3D attractor; (b), (c) and (d): 2D projections of +the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively +for (p, q, r) = (2.98098, 2.9799, 2). +(a) +Figure 5. (a) Three-dimensional attractor of GLV equations with +HT II functional response. +For parameter values p = 7.34, q = 2.0, r = 0.507, d = 3.198, the system (3) has +‘limit cycle’-like attractor which means model (3) has stable dynamics. +For each variation, we have found different parameters values for which models (1), +(2), and (3) show completely different dynamics. We infer that the GLV model’s +unstable dynamics with linear function response can be turned into stable dynamics +when linear functional response is altered by HT II or HT III. However, this may + +tional ResponseThe GL Attractor with HT II Fun2 +1.5 +X +1X +0.5 +0 +0.6 +0.4 +0.2 +0 +0.5(X2,X3) plane2D projection of the attractor on +0.74 +0.72 +0.71 +1.05 +1.10.68 +3 +X +0.66 +0.64 +0.62 +0.6 +0.8 +0.85 +0.9 +0.95(Xj,X3) plane2D projection of the attractor on +0.74 +0.72 +0.7.02 +1.04 +1.060.68 +3 +X +0.66 +0.64 +0.62 +0.6 +0.94 +0.96 +0.98 +1 +1(x,x,) plane2D projection of the attractor on +1.1 +1.05.02 +1.04 +1.06x2 0.95 +0.9 +0.85 +0.8 +0.94 +0.96 +0.98 +1 +1 AttractorThe Generalised Lotka Volterr +0.751.05 +1 +X +10.7 +3 +0.65 +0.6 +1.1 +1 +0.9 +0.8 +0.9510 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +Figure +6. Three-dimensional attractor of +generalised Lotka +Volterra equations with HT III functional response. +not be very effective for arbitrarily given scenario. A case in point- predators usu- +ally do not follow predation rate as HT II or HT III functional response when the +prey population is in abundance. To overcome this problem, we control chaos in the +model (1) through synchronization and achieve complete replacement synchroniza- +tion in two coupled GLV models following linear functional response. Since GLV +system has been shown to have a chaotic attractor for various set of parameters. +Out of these sets, we pick the set of parameters p = 2.9851, q = 3, r = 2 as in [18] +for further study. +3. +Mathematical Properties +In this section, we discuss the dynamical and analytical properties of system +(1) including positive Lyapunov exponent, equation of slow manifold, in-variance, +dissipation, stability of feasible equilibrium points, and control of instability of +unstable equilibrium points. +3.1. +Lyapunov exponents. For three-dimensional system, the local behaviour +of the dynamics varies along three orthogonal directions in state space. In a given +chaotic system , nearby initial conditions may be moving apart along one axis, +and moving together along another. The Lyapunov exponent describes the average +rate of separation between two nearby trajectories with different initial conditions +subject to a flow [3]. Where a positive Lyapunov exponent confirms chaos in the +system. For simulation, we take the parameter values of the system (1) as p = + +The Generalised Lotka Volterra Attractor with Holling Type Ill Functional Response +1.4 +1.22.4 +2.2 +2 +1.8 +1.6 +1.4 +1.20.8~ +3 +0.6 +0.4 +0.2 +0> +1.3 +1.2 +1.1 +1 +0.9 +0.8 +0.7 +1 +0.6 +0.8 +0.6 +X2 +0.5 +0.4SYNCHRONIZATION OF CHAOS IN ECOLOGY +11 +2.0451, q = 2.129, r = 2. The dynamics of Lyapunov exponents are shown in figure +7. The Lyapunov exponents of model (1) are as follows +Figure 7. Dynamics of Lyapunov spectrum of system (1). +L1 = 0.0138667 > 0, L2 = −0.275762 < 0, L3 = −0.293347 < 0. +where L1 is the indicator of chaos in the system (1). +3.2. Equation of slow manifold. The infusion of geometric and topological tech- +niques in chaos theory motivates mathematicians to study the underlying geometric +structures. In this line, expression of slow manifold permits to restore a part of the +deterministic property of the system that was lost because of SDIC. To find an +equation of slow manifold, we consider the system (1) as slow-fast autonomous dy- +namical system (S-FADS). In S-FADS, variables are separated into two groups:, one +is group of fast variable and other is of slow variables where slow variables are used +to determine the behaviour of whole system. To get the equation, we consider that +the slow manifold is locally defined by a plane orthogonal to tangent system’s left +fast eigenvector. Under the set of parameter values p = 2.9851, q = 3, r = 2, the + +onents +-L,=0.0138667 +-L,=-0.275762 +L.=-0.293347 +3Dynamics of Lyapunov Exp +2 +nents800 +1000Lyapunov Expo +0 +-3 +0 +200 +400 +600 +Time12 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +equations of GLV model (1) can be given as +(4) +˙x1 = x1(1 − x2 + 2x1 − 2.9851x3x1), +˙x2 = x2(−1 + x1), +˙x3 = x3(−3 + 2.9851x2 +1). +The Jacobian matrix J at point x = (x1, x2, x3)T is obtained as +J = + + +1 − x2 + 4x1 − 5.9702x1x3 +−x1 +−2.9851x2 +1 +x2 +−1 + x1 +0 +5.9702x1x3 +0 +−3 + 2.9851x2 +1 + + +Let λ1(x1, x2, x3) be a real, negative and dominant Eigen value ( i.e, fast Eigen +value) for Jacobian matrix in a large part of attractor’s phase space domain. +Furthermore, we assume that λ2(x1, x2, x3) and λ3(x1, x2, x3) be two slow Eigen +values. Then Eigen vector ZT +λ1 corresponding to fast Eigen value λ1 of JT (x1, x2, x3) +is given by +(5) +|J − λ1I| Zλ1 = 0. +where I is 3 × 3 identity matrix. Equation (6) gives, +ZT +λ1 = + + +(−1 + x1 − λ1)(−3 + 2.9851x2 +1 − λ1) +x1(−3 + 2.9851x1)2) +2.9851x2 +1(−1 + x1 − λ1) + + . +On the attractive parts of phase space (where J(x) has a fast Eigen value λ1), the +equation of the slow manifold is given by +(6) +˙x(t).ZT +λ1 = 0. +We use the equation (7) to define the equation of slow manifold. With the substi- +tution of ˙x(t) and ZT +λ1 in the equation (7), we write the equation of slow manifold +as +λ2 +1(x1 − x1x2 + 2x2 +1 − 2.9851x2 +1x3) + λ1(−4x1 + 7x2 +1 − 4.9851x3 +1 − 3x1x2 + 2.9851x3 +1x2− +5.97020x4 +1 − 2.985100x2 +1x3 + 2.9851x3 +1x3) + (3x1 + x2 +1 − 8.985100x3 +1 − 2.9851x4 +1 + 5.970200x5 +1 ++ 8.95530x1x3 − 17.910600x2 +1x3 − 0.044478x3 +1x3 + 17.821644x4 +1x3 − 8.91082x3 +1x3) = 0. +(7) + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +13 +where λ1 is fast Eigen value of J(x). Because λ1(x1, x2, x3) is uncertain Eigen value, +it is not easy to use this implicit equation to draw a slow manifold representation +in the three dimensional phase space. +3.3. Invariance property. +Theorem 3.1. Let the system in vector notation is given as +(8) +˙x(t) = H(x(t)) = + + +H1(x1, x2, x3) +H2(x1, x2, x3) +H3(x1, x2, x3) + + +H1(x1, x2, x3) = x1(1 − x2 + rx1 − px3x1), +H2(x1, x2, x3) = x2(−1 + x1), +H3(x1, x2, x3) = x3(−q + px1 +2). +where H1, H2 and H3 are continuously differentiable. Assume that H is locally +Lipschitz and generates a flow φt(x). Let +L : D ⊂ R3 → R3 +be a continuously differentiable function on a domain D ⊂ R3 such that ˙L(x) ≤ 0 +in D, then the largest invariant set Σ ⊂ D is the set; where ∇L.H(x) = 0 ∀x ∈ Σ. +Proof. Consider +L : D ⊂ R3 → R3 +be a continuously differentiable function on a domain D ⊂ R3 and defined as +(9) +L(x1, x2, x3) = x2 +1 + x2 +2 + x2 +3 +2 +. +Equation (9) gives, +(10) +˙L(x1, x2, x3) = x1 ˙x1 + x2 ˙x2 + x3 ˙x3. +The set D ⊂ R3 is said to be an invariant set under the flow φt if for any point +x ∈ D +φt(x) ∈ D ∀ t ∈ R. +Let Σ be a smooth closed surface without boundary in D ⊂ R3 and suppose that +n is a normal vector to the surface Σ at (x1, x2, x3). If we have +(11) +n. < ˙x1, ˙x2, ˙x3 >= 0 ∀ (x1, x2, x3) ∈ Σ. + +14 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +Let us consider Σ be the x1x2 plane i.e. x3 = 0. Note that the vector (0, 0, 1) is +always normal to Σ and at the point (x1, x2, 0) ∈ Σ. So we have, +( ˙x1, ˙x2, ˙x3) = (x1(1 − x2 + rx1 − px3x1), x2(−1 + x1), 0). +Thus, +⟨(0, 0, 1).(x1(1 − x2 + rx1 − px3x1), x2(−1 + x1), 0)⟩ = 0. +Similar arguments can be verified for x1 and x2 planes which directs that each +coordinate plane is an invariant subset. It implies that for any given positive initial +condition, x1(t), x2(t) and x3(t) are positive for all t that is any trajectory starting +in R3 ++ can not cross the co-ordinate planes and it shows that R3 ++ is an invariant set +for the system. +□ +3.4. Dissipation. +Theorem 3.2. Consider the autonomous vector field +˙x(t) = H(x) for x ∈ R3, +and assume that it generates a flow φt(x). +Let D0 is a domain in R3 which is +supposed to have a volume V0, and φt(D0) is its evolution under the flow. If V (t) +is the volume of Dt, then the time rate of change of volume is given as +|dV +dt |t=0 = +� +D0 +∇.Hdx. +The system (1) is dissipative if its time-t map decreases volume for all t > 0. +Proof. Dissipation in any dynamical system manifests itself as contraction of the +phase volume on average. To check this, we express the volume V (t) in the following +form using the definition of the Jacobian of transformation as +(12) +V (t) = +� +D0 +|dφt(x) +dx +|dx. +Expanding φt(x) in the neighbourhood of t = 0. +Since the vector field H(x) is +smooth enough to have a tangent plane in each point on R3 so we can expand φt(x) +by Taylor series expansion. Hence we get, +(13) +φt(x) = x + ˙xt + O(t2) for t → 0 +Since +(14) +˙x(t) = H(x), + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +15 +The equation (15) gives, +(15) +φt(x) = x + H(x)t + O(t2) for t → 0. +It follows that +∂φ +∂x = I + ∂H +∂x t + O(t2), +(16) +|∂φ +∂x| = |I + ∂H +∂x t| + O(t2). +Here I is 3 × 3 identity matrix so det I will be equal to 1. +By expanding the +expression (17) by using expansion of determinant, we get the following +(17) +|∂φ +∂x| = 1 + trace(∂H +∂x )t + O(t2). +Note that +(18) +trace(∂H +∂x ) = ∇.H, +therefore, we have +(19) +V (t) = V0 + +� +D0 +((∇.H)t + O(t2))dx. +It gives +(20) +|dV +dt |t=0 = +� +D0 +∇.Hdx, +i.e. if the volume shrinks then divergence of vector field will be strictly negative [3]. +Now considering the equations of model (1) in vector notation and computing its +Jacobian +(21) +J(x1, x2, x3) = ∂H +∂x . +The Jacobian J(x1, x2, x3) of the model is given by +(22) + + +1 − x2 + 2rx1 − 2px1x3 +−x1 +−px12 +x2 +−1 + x1 +0 +2px1x3 +0 +−q + px2 +1 + + , +we take the parameter values as +(23) +p = 2.9851, +q = 3, +r = 2. + +16 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +The above argument shows that the G.L.V dynamical system will be dissipative if +the generalized divergence should be less than zero, i.e. +(24) +� +i +∂Hi +∂xi +< 0. +where Einstein summation has been used. The divergence of vector field H on R3 +is as follows +(25) +∇.H = ∂H1 +∂x1 + ∂H2 +∂x2 + ∂H3 +∂x3 , +∇.H = −q − x2 + (2r + 1 + px1 − 2px3)x1. +Hence system (1) will be dissipative if the following condition is satisfied, +(26) +(2r + 1 + px1 − 2px3)x1 < q + x2. +□ +3.5. Existence and uniqueness of solution. Since we are dealing with a popula- +tion dynamics model, hence, the existence of at least one solution is must. However, +the uniqueness of existed solution will give more appropriate results. Here we men- +tion two theorems for which solutions of the system (1) uniquely exist for all t > 0 +(complete detail of the proof can be seen in [18]). +Theorem 3.3. If the functions f1, f2 and f3 satisfy assumptions (1) and (2), +mentioned in section , then continuity of functions fi for i ∈ {1, 2, 3} assures that +atleast one solution exists for the dynamics of system (1) in region D × I where +I = (0, T ] and the spatial boundary of region D ⊂ R3 is defined as +D = {x = (x1, x2, x3) : max|xi| ≤ M, for i ∈ {1, 2, 3}} +where M > 0. +Theorem 3.4. Let D be a closed subspace of complete normed linear space R3. +Consider H : D ⊂ R3 → D is Lipschitz continuous so that there exist 0 < K < 1 +such that +||H(χ) − H(ψ)|| < K||χ − ψ|| +with +K = T.max(1 + 2M + 2rM + 4pM 2, 1 + 2M, q + 2pM 2) +For 0 < K < 1, H(t) will be a contraction map. With the help of Banach fixed point +theorem, it can be ensured that 0 < K < 1 is sufficient condition for uniqueness of +solution of the system (1). + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +17 +3.6. +Stability of feasible equilibrium points. The equilibrium points of system +(1) are solutions of following algebraic equations +xx1(1 − x2 + rx1 − px3x1) = 0, +x2(−1 + x1) = 0, +x3(−q + px2 +1) = 0. +(27) +We obtain five equilibrium points by solving the system (27), +(28) +X∗ +0 = + + +0 +0 +0 + + , +X∗ +1 = + + +1 +1 + r +0 + + , X∗ +2 = + + +� +q +p +0 +1+r√ q +p +√pq + + , X∗ +3 = + + +− 1 +r +0 +0 + + , X∗ +4 = + + +− +� +q +p +0 +−1+r√ q +p +√pq + + . +From an ecological point of view, negative population density is not realistic as +the population can not be negative, therefore, we take the vector x = (x1, x2, x3) +as an element of R3 ++. R3 ++ is defined as +(29) +R3 ++ = {X ∈ R3 : xi ≥ 0 for i ∈ {1, 2, 3}}. +Since equilibrium points X∗ +0, X∗ +1 and X∗ +2 are elements of the set Int(R3 ++), therefore, +we study the local stability of ecologically feasible equilibrium points X∗ +0, X∗ +1 and +X∗ +2. +(I) Stability of Trivial Equilibrium Point X∗ +0. +Theorem 3.5. Consider the dynamics of the model (1) in the following form +(30) +˙x = H(x) = Ax + f(x) for x ∈ R3. +If following three conditions are satisfied +(i) Constant matrix A3×3 has 3 Eigen-values with non-zero real part, +(ii) f(x) is smooth and +(iii) lim||x||→0 +||f(x)|| +||x|| += 0, +then in a neighbourhood of the critical point X∗ +0 = (0, 0, 0), there exists stable +and unstable manifolds Ws and Wu with the same dimensions ns and nu as +the stable and unstable manifolds Es and Eu of the system +˙Z(t) = AZ. +In x = 0, Es and Eu are tangent to Ws and Wu[20]. + +18 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +Proof. For GLV system, X∗ +0 = (0, 0, 0) is trivial equilibrium point. Here we +check all three mentioned conditions of theorem I. +(i) Note that for model (1), the constant matrix A3×3 is +A = + + + + +1 +0 +0 +0 +−1 +0 +0 +0 +−q + + + + +The determinant of the matrix A is non zero if q ̸= 0. Since, we have +taken q = 3, therefore, all Eigen values of A have non zero real part. +(ii) Since functions f1(x1, x2, x3), f2(x1, x2, x3) and f3(x1, x2, x3) for model +(1) are considered as +x1(−x2 + rx1 − px3x1) += f1(x1, x2, x3), +x2x1 += f2(x1, x2, x3), +x3(px2 +1) += f3(x1, x2, x3). +All three functions are continuous and have continuous partial derivative +for all x ∈ R3 which implies that f(x) is smooth on R3. Hence, the +second condition also holds. +(iii) For any x ∈ R3, converting the Cartesian coordinates into spherical +coordinates by making the following transformation + + +x1 = r sin θ cos φ +x2 = r sin θ sin φ +x3 = r cos θ + + , +where r ≥ 0, 0 ≤ θ ≤ π and 0 ≤ φ ≤ π. +Using this transformation in model (1), we have +lim +||x||→0 +||f(x)|| +||x|| += 0. +Hence, the critical point (0, 0, 0) of system (1) is of the same type of +critical point of the system +˙Z(t) = AZ. + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +19 +The Eigen values of A are 1, −1 and −q which implies that (0, 0, 0) is +saddle node for the system ˙Z(t) = AZ. Therefore, the trivial steady +state X∗ +0 = (0, 0, 0) of the model (1) is a saddle point. +□ +(II) Stability of Axial Equilibrium Point X∗ +1. +The Jacobian matrix of model (1) for parameter values p = 2.9851, q = +3, r = 2 is given as +(31) +J = + + +1 − x2 + 4x1 − 5.9702x1x3 +−x1 +−2.9851x2 +1 +x2 +−1 + x1 +0 +5.9702x1x3 +0 +−3 + 2.9851x2 +1 + + . +Jacobian matrix (31) of the model (1) about X∗ +1 = (1, 3, 0) yields the following +Jacobian matrix +(32) +JX∗ +1 = + + +2 +−1 +−2.9851 +3 +0 +0 +0 +0 +−.0149 + + . +The characteristic equation |JX∗ +1 − λI| = 0 of matrix (32) is given as +(33) +λ3 − (1.9851)λ2 + (2.9702)λ + 0.0447 = 0. +The characteristic equation (33) has the following Eigen values +(34) +λ1 = −0.014900, λ2 = 1 + +√ +2ι, λ3 = 1 − +√ +2ι. +Since λ2 and λ3 have positive real parts, it implies that X∗ +1 = (1, 3, 0) is +unstable equilibrium point. +(III) Stability of Planer Equilibrium Point X∗ +2. +The Jacobian matrix of model (1) for parameter values p = 2.9851, q = +3, r = 2 is given as +(35) +J = + + +1 − x2 + 4x1 − 5.9702x1x3 +−x1 +−2.9851x2 +1 +x2 +−1 + x1 +0 +5.9702x1x3 +0 +−3 + 2.9851x2 +1 + + . + +20 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +Jacobian matrix (35) of the model (1) about X∗ +2 = (1.002493, 0, 1.4159) yields +the following Jacobian matrix +(36) +JX∗ +2 = + + +3.004986 +−1.002493 +3.000002 +0 +.002493 +0 +6.00887 +0 +.000002 + + . +The characteristic equation |JX∗ +2 − λI| = 0 of matrix (36) is given as +(37) +λ3 + (0.997507)λ2 + (18.027423)λ − 0.044949 = 0. +The characteristic equation (37) has the following Eigen values +(38) +λ1 = −0.002493, λ2 = −0.5 + 4.216ι, λ3 = −0.5 − 4.216ι. +Since all three Eigen-values of matrix (36) have negative real parts, it shows +that X∗ +2 = (1002493, 0, 1.4159) is locally stable equilibrium point. +It is clear that planer equilibrium point is stable whereas trivial and axial equilib- +rium points are unstable equilibrium points. Since trivial equilibrium point refers +the zero density of all three species, therefore, we neglect the instability of trivial +equilibrium point. From an ecological point of view, we mainly focus on non-trivial +unstable equilibrium point X∗ +1 and try to stabilize it by adding some external control +inputs. +3.7. +Control of instability of axial equilibrium point. In order to suppress +instability to X∗ +1 = (1, 3, 0), we consider the controlled GLV system in the following +form +(39) +˙x1 += x1(1 − x2 + rx1 − px3x1) + u1, +˙x2 += x2(−1 + x1) + u2, +˙x3 += x3(−q + px2 +1) + u3. +We introduce the external control law +(40) +u1 += −µ1(x1 − 1), +u2 += −µ2(x2 − 3), +u3 += −µ3(x3 − 0). +with x1, x2, x3 as the feedback variable and µ1, µ2, µ3 as the positive feedback +gains. We substitute control law (40) into (39) and hence, the controlled system + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +21 +(39) takes the following form +(41) +˙x1 = x1(1 − x2 + rx1 − px3x1)) − µ1(x1 − 1), +˙x2 = x2(−1 + x1) − µ2(x2 − 3), +˙x3 = x3(−q + px2 +1) − µ3(x3 − 0). +Theorem 3.6. The equilibrium point X∗ +1 = (1, 3, 0) of the model (1) will be asymp- +totically stable if positive gains µ1, µ2 and µ3 satisfy the following inequalities[21] +(42) +µ1 +> 2, +µ1µ2 +> 1 + 2µ2, +µ1µ2(µ3 + 0.0149) +> µ2(2µ3 + 2.2528) + µ3 + 0.0149. +Proof. The Jacobian matrix J of the system (41) is given by +(43) + + +1 − x2 + 2x1(r − px3) − µ1 +−x1 +−px2 +1 +x2 +−1 + x1 − µ2 +0 +2px1x3 +0 +−q + px2 +1 − µ3 + + +Let us consider that +(44) +e1 += (x1 − 1), +e2 += (x2 − 3), +e3 += (x3 − 0). +From (44), we get the error system as +(45) +˙e1 = (2 − µ1)e1 − e2 − 2.9851e3, +˙e2 = 3e1 − µ2e2, +˙e3 = −(0.014900 + µ3)e3. +The system (1) with constant and known parameters, will be stabilized to steady +state X∗ +1 = (1, 3, 0), if error system (45) stabilized to (0, 0, 0). +To study the stability of equilibrium point (0, 0, 0) of error system, we consider the +Lyapunov function L(e1, e2, e3) as: +(46) +L = 1 +2(e2 +1 + e2 +2 + e2 +3). +The time derivative of L in the neighbourhood of (0, 0, 0) is given as +(47) +˙L = (2 − µ1)e2 +1 + 2e1e2 − µ2e2 +2 − 2.9851e1e3 − (0.0149 + µ2)e2 +3. + +22 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +The time derivative of Lyapunov function can be re-written in the following form +(48) +˙L = eT Me. +where e = ((x1 − 1), (x2 − 3), (x3 − 0)) is the error vector in R3, eT is the transpose +of error vector e and the matrix M is 3 × 3 is given as +M = + + +2 − µ1 +1 +−1.49250 +1 +−µ2 +0 +−1.49250 +0 +−(0.0149 + µ3) + + +According to Lyapunov stability theory, the equilibrium point (0, 0, 0) of system +(45) will be asymptotically stable if ˙L < 0. And ˙L < 0 if matrix M will be negative +definite. +Considering this, we find that mentioned condition will be fulfilled if +positive feedback gains µ1, and µ2 satisfy the following inequalities, +(49) +µ1 +> 2, +µ1µ2 +> 1 + 2µ2, +µ1µ2(µ3 + 0.0149) +> µ2(2µ3 + 2.2528) + µ3 + 0.0149. +□ +4. Complete Replacement Synchronization +To investigate complete replacement synchronization techniques, we consider two +identical chaotic GLV systems having the same parameter but different initial con- +ditions. Since the coupling between models is needed to maintain the synchronous +state, we couple the states of both models with two controllers and drive the re- +sponse system with prey species x1. For this, we remove prey from response system, +and drive its counterpart. Here, we can think of prey species x1 as a driving vari- +able for response system with an assumption that it is superfluous in the system of +two coupled GLV models. This construction gives us a new five-dimensional drive- +response system having drive and response variables as (x1d, x2d, x3d) and (x2r, x3r) + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +23 +respectively. The coupled chaotic system with x1d drive configuration is as follows: +(50) + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(Drive System) +˙x1d += x1d(1 − x2d + rx1d − px3dx1d), +˙x2d += x2d(−1 + x1d), +˙x3d += x3d(−q + px2 +1d), +( Response System) +˙x2r += x2d(−1 + x1d) + u1, +˙x3r += x3r(−q + px2 +1d) + u2. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +x2d(0) ̸= x2r(0) and x3d(0) ̸= x3r(0). +4.1. Active control law for stability of synchronization manifold. +Theorem 4.1. The identical synchronization manifold Ω = [x2d = x2r, x3d = x3r] +is globally asymptotically stable for the coupling between drive and response system +in equation (24) for positive µ1 and µ2, where µ1 and µ2 are large enough such that +µ1 + 1 > x1d, +µ2 + q > px2 +1d. +Proof. We consider drive-response system given by equations (50) and add the uni- +directional controllers to the response system through the linear positive constants +µ1 and µ2. We choose two controllers for response system as +(51) +u1 += −µ1(x2r(t) − x2d(t)), +u2 += −µ2(x3r(t) − x3d(t)). +Existence of all forms of identical synchronization in any dynamical system +(chaotic or not), are really manifestations of dynamical behaviour restricted to +a flat hyper-plane in the phase space i.e. to say motion is continually confined to a +hyper-plane which can be referred as synchronization manifold [22]. Therefore, we +consider the identical synchronization manifold of the systems equation (50) as +Ω = [x2d = x2r, x3d = x3r]. +Further, we consider the errors between states of drive and response systems of +system (50) as +(52) +e2(t) += x2r(t) − x2d(t), +e3(t) += x3r(t) − x3d(t). + +24 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +The dynamics of error system is as follows +(53) +˙e2 += (−1 − µ1 + x1d)e2, +˙e3 += (−q − µ2 + px2 +1d)e3. +where, we are now interested in the stability of origin. The Jacobian of right side +of (53) is given by +(54) +J(e2, e3) = +� +−1 + µ1 + x1d +0 +0 +−q + µ2 + px2 +1d +� +We treat the response system (x2d(t), x3r(t)) as a separate system driven by x1d, +then the solutions of equation (54) convey us about convergence and divergence of +two initially nearby trajectories of {x2r(t), x2d(t)} and {x3r(t), x3d(t)}. +Next, We analyse the possibility of synchronization using the Lyapunov function +construction method. We consider the Lyapunov function as +(55) +L(e2, e3) = 1 +2(e2 +2 + e2 +3). +(56) +dL +dt = e2 ˙e2 + e3 ˙e3. +Plugging dynamics of errors (53) into (55), we get +(57) +dL +dt = −[(µ1 + 1 − x1d)e2 +2 + (µ2 + q − px2 +1d)e2 +3], +which will be strictly negative for following conditions on µ1 and µ2 +(58) +µ1 + 1 > x1d, +µ2 + q > px2 +1d. +for all t > 0. Condition (58) ensures that we consider the bounded density of prey +species, then we can bound the positive feedback gains. Thus, if µ1 and µ2 satisfy +(58), then it can be assured that dL +dt < 0 for all t > 0 or in other words, the complete +replacement synchronization follows as e2 and e3 → 0 as t → ∞. +□ +4.1.1. Numerical Simulation. Since, a suitable coupling can influence both fre- +quency as well as chaotic amplitude, therefore, the states coincide (or nearby co- +incide) and regime of synchronization sets in. Thus, it is pre-arranged that the +chosen coupling should assist the coupled states in coincidence without perturbing +their chaotic rhythm. We numerically integrate the System (50) and display results + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +25 +in figure 4.1.1, where drive and response systems are shown to synchronize when +considered positive feed-back gain are chosen as µ1 = 0.000024, µ2 = 1.345. +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Time +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +Population +x1d +x2d + x3d +y2r +y3r +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Time +-0.14 +-0.12 +-0.1 +-0.08 +-0.06 +-0.04 +-0.02 +0 +Errors +(e2) +(e3) +(b) +x2d +y2r +x2d vs y2r +(c) +(d) +Figure 8. (a): solutions of drive and response systems plotted +over time, (b): errors between drive and response systems over +time,(c): synchronization plot of {x2d, x2r}, (d): synchronization +plot of {x3d, x3r} +4.1.2. Lyapunov Spectrum For Drive-Response System. Since the necessary condi- +tion for the stability of the synchronization manifold is the negative largest trans- +verse Lyapunov exponent. In the case of complete replacement synchronization, +the transverse Lyapunov exponents are also known as conditional Lyapunov expo- +nents. It is because Lyapunov exponents for the new system depend on the coupling +from the drive[23]. The Lyapunov spectrum is a global indicator of the system’s + +X +VS +3d3 +X +3d26 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +state, aggregating over the behaviour of the entire system trajectory in phase space. +The typical approach for observing these transitions is to see a change in sign of +the Lyapunov exponents of the system, as obtained from ensemble averages of the +eigenvalues of the Jacobian matrix of system (50) [24]. We solve equations of system +(50) and get the Jacobian matrix as +(59) +J(x1d, x2d, x3d, x2d, x3r) = A5×5, where A = [aij]. +The entries of the matrix A are given as: +a11 = 1 − x2d + 2(r − px3d)x1d, +a12 = −x1d, +a13 = −px2 +1d, +a21 = x2d, +a22 = −1 + x1d, +a31 = 2px1dx3d, +a33 = −q + px2 +1d, +a41 = x2d, +a42 = µ1, +a44 = −1 + x1d − µ1x2d, +a51 = 2px3rx1d, +a53 = µ2, +a55 = −q + px2 +1d − µ2, +a14 = +a15 = +a23 = +a24 = +a25 = +a32 = 0, +a34 = +a35 = +a43 = +a45 = +a52 = +a54 = 0. +Averaging the eigenvalues of Jacobian J over all phase space configurations set-up +by the chaotic trajectory, we get the five Lyapunov exponents of the system of two +coupled GLV models. +For set of parameter values {p, q, r} = {2.9851, 3, 2}, Lyapunov exponents of the +drive-response system are obtained as +(60) +L1 = −0.011320, +L2 = −0.174464, +L3 = −0.22221, +L4 = −5.011, +L5 = −5.0059. +Since, all Lyapunov exponents are negative which confirms the stable synchroniza- +tion manifold, therefore, it can be concluded that the states of coupled GLV systems +are synchronized. + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +27 +0 +20 +40 +60 +80 +100 +Time +-6 +-5 +-4 +-3 +-2 +-1 +0 +Lyapunov exponents +Dynamics of Lyapunov exponents +L1=-0.01132 +L2=-0.17446 +L3=-0.22221 +L4=-5.011 +L5=-5.0059 +Figure 9. Lyapunov Exponents of two GLV models coupled with +positive feedback gains µ1 = 0.000024 and µ2 = 1.345. +4.2. Adaptive control law for stability of synchronization manifold. Us- +ing the method [25], we design non-linear adaptive controller for global complete- +replacement synchronization of two chaotic GLV systems with unknown parameters. +We consider the drive system as +(61) +˙x1d += x1d(1 − x2d + rx1d − px3dx1d), +˙x2d += x2d(−1 + x1d), +˙x3d += x3d(−q + px2 +1d). +The response system is given by controlled chaotic system +(62) +˙x2r += x2r(−1 + x1d) + u1, +˙x3r += x3r(−q + px2 +1d) + u2. +The synchronization error between drive and response systems is defined as +(63) +e2(t) += x2r(t) − x2d(t) +e3(t) += x3r(t) − x3d(t). +The error dynamics between drive and response systems is calculated as : +(64) +˙e2 += −e2 + x1de2 + u1, +˙e3 += −qe3 + px2 +1de3 + u2. +In (64), unknown parameters p and q are to be determined by using parameter +estimates P(t) and Q(t) respectively. For this purpose, we consider adaptive control + +28 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +laws u1 and u2 with positive feedback gains µ1 and µ2 as +(65) +u1 += e2 − x1de2 − µ1e2, +u2 += Q(t)e3 − P(t)x2 +1de3 − µ2e3. +Using control law (65) into the error dynamics (64), we get +(66) +˙e2 += −µ1e2, +˙e3 += −(q − Q(t))e3 + (p − P(t))x2 +1de3 − µ2e3. +We define parameter estimation error as +(67) +ep(t) = (p − P(t)), +eq(t) = (q − Q(t)). +Using (67), we can simplify the error dynamics (66) as +(68) +˙e2 = −µ1e2, +˙e3 = −eq(t)e3 + ep(t)x2 +1de3 − µ2e3. +Differentiating (67) with respect to t, we get +(69) +˙ep += − ˙P(t), +˙eq += − ˙Q(t) +Theorem 4.2. The identical synchronization manifold Ω = [x2d = x2r, x3d = x3r] +is globally asymptotically stable for the coupling between derive and response system +in equation (50) for positive µ1 and µ2. +Proof. The identical synchronization manifold for the systems equation (24) can be +written as +Ω = [x2d = x2r, x3d = x3r]. +Using change of coordinates +(70) +e = +� +e2 +e3 +� += +� +x2d − x2d +x3r − x3d +� +, +such that Ω can be written +Ω = (0, 0). +Next, we use Lyapunov stability theory for finding an update law for the parameter +estimates. we consider the quadratic Lyapunov function as +(71) +L = 1 +2(e2 +2 + e2 +3 + e2 +p + e2 +q) + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +29 +Note that Lyapunov function L is positive definite on R4. Differentiating L along +the trajectories of (66) and (69). We get, +(72) +dL +dt += e2 ˙e2 + e3 ˙e3 + ep ˙ep + eq ˙eq, +dL +dt += −µ1e2 +2 − µ2e2 +3 + ep(t)(− ˙P(t) + x2 +1de3) + eq(t)(− ˙Q(t) − e2 +3). +We want error system to be asymptotically stable i.e. +(73) +dL +dt < 0. +In view of (72), we take the parameter update law as +(74) +˙P(t) = x2 +1de3, +˙Q(t) = −e2 +3. +By substituting the parameter update law (74) into Lyapunov function, we obtain +time derivative of L as +(75) +dL +dt = −µ1e2 +2 − µ2e2 +3, +From (75), it is clear that ˙L is negative semi-definite function on R4. Thus, we can +conclude that the synchronization error vector e(t) and the parameter estimation +error are globally bounded, i.e. +(76) +[e2, e3, ep, eq]T ∈ L∞. +We define µ = min{µ1, µ2}, then it follows from (75) that +(77) +dL +dt ≤ −µ||e||2. +Integrating the inequality (77) with respect to τ from 0 to t. We get, +(78) +� t +0 +µ||e(τ)||2dτ ≤ L(0) − L(t). +From (78) it follows that e ∈ L2 and hence, ˙e(t) ∈ L∞. +With the help of Barbalat’s lemma [26],[27], we conclude that e(t) → 0 exponentially +as t → ∞ for all initial conditions e(0) ∈ R2. +□ + +30 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +4.2.1. Numerical Simulation. For numerical simulations, we use the classical fourth- +order Runge-Kutta method to solve the GLV system. The initial value of parameter +estimates are taken as P(0) = 3.9, Q(0) = 4. The initial values of states of drive and +response systems are taken as (x1d(0), x2d(0), x3d(0)) = (4, 1.4, 1.41) and (x2r(0), +x3r(0)) = (1, 1.414) respectively. The effectiveness of control law is verified through +simulation results which are shown in figure 4.2.1. Figure shows the solutions of +system (50). +It is clear that although the initial values are different, the error +dynamics approach to zero as time goes to ∞. Therefore, our numerical results +confirm that the amplitude and frequency of state variables of response system +become same with the drive system under the designed control law. +5. Conclusion +This work examines predator-prey systems in ecosystems to understand how +they contribute to sustainable environment. Our focus is on the use of Generalized +Lotka-Volterra (GLV) equations to model the competition and trophic relationships +between various species. We consider three different forms of three-dimensional GLV +models, each with different functional responses (linear, Holling type II, and Holling +type III). We find that the model with the linear functional response exhibits un- +stable dynamics, where alteration in functional response can stabilize the system +dynamics for a particular scenario. To stabilize the dynamics in patchy ecosystem, +we focus on the GLV model with the linear functional response for the remainder of +the study. We investigate its fundamental properties and also examine the stability +of equilibrium points and the suppression of instability at equilibrium. Through +computation of Lyapunov exponent, we find that the model is chaotic due to one +positive Lyapunov exponent and has two unstable equilibrium points for the con- +stant parameters p = 2.9851, q = 3, r = 2. +Further, we investigate the synchronization of two chaotic GLV models using two +control schemes: the Active Control Technique and the Adaptive Control Technique. +We consider a configuration in which the prey population in the drive system acts +as a driving variable for the response system, allowing the other two predator popu- +lations to depend only on the prey population. Using the Active Control Technique, +we apply two simple linear controllers to synchronize the states of the GLV systems. + +SYNCHRONIZATION OF CHAOS IN ECOLOGY +31 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Time +-1 +0 +1 +2 +3 +4 +5 +Population +x1d +x2d + x3d +y2r +y3r +(a) +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +Time +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Errors +(e2) +(e3) +(b) +x2d +0 +y2r +x2d vs y2r +(c) +0 +x3d +0 +y3r +x3d vs y3r +(d) +Figure 10. (a): solutions of drive and response systems over time, +(b): +errors between drive and response systems over time, (c): +synchronization plot for {x2d, x2r}, (d): synchronization plot for +{x3d, x3r} for coupling strengths µ1 = 0.0038 and µ2 = 2. +These controllers are easy to implement and more straightforward than previous re- +sults. The stability of synchronization manifold is ensured through the transition +of positive conditional Lyapunov exponent to negative one. We also examine the +synchronization of two chaotic GLV systems with unknown parameters using the +Adaptive Control Technique. We design two adaptive laws of parameters using the +Lyapunov stability theory to ensure global and exponential synchronization of the +systems. Our results show that both the Active and Adaptive Control Techniques +are effective for achieving global synchronization in chaotic systems. + +32 +SHUBHANGI, NITU KUMARI, AND R.K.UPADHYAY +Funding +The second author’s research was funded by the Science and Engineering Re- +search Board (SERB), under two separate grants with grant numbers MTR/2018/000727 +and EMR/2017/005203. +Disclosure statement +The authors declare that they have no conflict of interest. +References +[1] Lorenz EN. Deterministic nonperiodic flow. Journal of the atmospheric sciences. 1963; +20(2):130–141. +[2] Hunt BR, Ott E. Defining chaos. 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IEEE Transactions on +Automatic Control. 2009;54(9):2222–2225. + diff --git a/BtE1T4oBgHgl3EQf9gZu/content/tmp_files/load_file.txt b/BtE1T4oBgHgl3EQf9gZu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..29a4dc2e5d47155eb7b77ddd9154190541ba2e49 --- /dev/null +++ b/BtE1T4oBgHgl3EQf9gZu/content/tmp_files/load_file.txt @@ -0,0 +1,1008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf,len=1007 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='03557v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='DS] 9 Jan 2023 USING PREY ABUNDANCE TO SYNCHRONIZE TWO CHAOTIC GLV MODELS SHUBHANGI DWIVEDI E-MAIL: SHUBHANGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='DWIVEDI176@GMAIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='COM , NITU KUMARI E-MAIL: NITU@IITMANDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='IN SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES, KAMAND, INDIAN INSTITUTE OF TECHNOLOGY, MANDI-HIMACHAL PRADESH, 175005, INDIA , AND RANJIT KUMAR UPADHYAY DEPARTMENT OF APPLIED MATHEMATICS, INDIAN INSTITUTE OF TECHNOLOGY (ISM), DHANBAD-JHARKHAND, 826004 , INDIA E-MAIL: RANJIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='CHAOS@GMAIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='COM ( Accepted: 02 November 2021 ) Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The concept of superfluous prey, or an excess of prey in certain areas within a patchy ecosystem, has significant implications for the synchronization of the predator population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' These areas, known as ”hotspots,” have a higher density of prey compared to other areas and attract a higher concentration of predators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' As a result, the preda- tor population becomes more stable and predictable, as they are less likely to migrate to other areas in search of food.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' This phenomenon can have important consequences for both the predators and their prey, as well as the overall functioning of the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' This work investigates the synchronization between two chaotic food webs using the general- ized Lotka-Volterra (GLV) model consisting of one prey and two predator populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We, first, examine the impact of three functional responses (linear, Holling type II, and Holling type III) on system dynamics For the study, we consider the model with a linear functional response consisting of chaotic oscillations and apply controllers to stabilize its unstable fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' This research contributes to the understanding of how to apply chaotic ecological models to predict the population of competing species in one habitat using information about similar populations in another system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To do this, we configure a drive-response system where prey acts as the driving variable and both predators depend only on the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We use active and adaptive control methods to synchronize two coupled GLV models and verify the analytical results through numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' AMS Classification: — Keywords: Lyapunov exponents, slow manifold equation, chaos control, complete replacement synchronization, active and adaptive control techniques JOURNAL OF DYNAMICAL SYSTEMS & GEOMETRIC THEORIES ©TARU PUBLICATIONS 1 2 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Introduction In ancient philosophy and mythology, the word chaos meant the disordered state of unformed matter supposed to have existed before the ordered universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In the course of its journey to truth, science has met with startling phenomena called chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since the 1960s, with the discovery of chaotic systems, chaos has set a nonlinear dynamics research boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Chaos theory is attributed to the work of Edward Lorenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' His 1963 paper, “Deterministic Non-periodic Flow” [1], is credited for laying the foundation for chaos theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Hunt and Ott [2] reviewed the problem and proposed a computationally feasible entropy-based good definition of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' They define chaos as “ the existence of positive Expansion Entropy (EE) (equal to topological entropy for infinitely differentiable maps) on a given restraining region (bounded positive volume subset),” which confirms both the notions (COS as well as OS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since EE enjoys the properties of simplicity, computability, and generality, so they call it a “ good ” definition of chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Chaotic systems are sensitive to initial conditions, topologically mixing and with dense periodic orbits [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Because of slightest dif- ference, chaotic dynamical systems can lead to entirely different trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The main characteristic of chaos is that the system does not repeat its past behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Mathematically, chaotic dynamical systems are classified as non-linear dynamical systems having at least one positive Lyapunov exponent[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Chaos theory is not just about chaos - it has two sides to it: chaos control and chaos synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The study of chaos control and understanding chaotic model be- haviour has gained significant interest, with applications in various fields such as secure communications, biology, neural networks, finance, and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Whereas chaos synchronization refers to the alignment in time of different chaotic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' At first glance, chaotic systems may seem to defy synchronization, but it has been observed and studied in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In 1665, Dutch physicist Huygens observed the adjustment of rhythms via a cou- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' He noticed that pendulum clocks suspended from the same beam would slowly adjust their phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In 1984, Kuramoto set theory for the onset of sync, and Pecora and Carroll [6] reviewed the area of synchronization in chaotic systems and presented a more geometric view using synchronization manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In 1999, Blasius explained the theoretical analysis of seasonally synchronized chaotic population cy- cles [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' All these contributions help the researcher think that synchronization is SYNCHRONIZATION OF CHAOS IN ECOLOGY 3 an essential phenomenon in physical and biological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In literature, this phe- nomenon has been nominated by various types, such as phase locking, frequency pulling, generalized synchrony, and complete locking, depending on the degree or type of synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Synchronization of chaotic systems can be achieved by configuring drive and re- sponse systems, with the goal of using the output of the drive system to control the response system so that the output tracks the drive system asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' However, creating identical chaotic synchronized ecological systems is questionable as it has a potential threat to biodiversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' As discovered in Pecora and Carroll’s pioneering work, another way of achieving complete synchronization between two systems is what is now called the technique of complete replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The complete replacement synchronization is helpful in a network of patchy ecosystems as it can help in identifying the underlying mechanism that derive the group co-ordination in the present of severely chaotic oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In population biology, the chaotic dynamics may synchronize if populations are coupled through environmental or bi- ological interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' From the ecological aspect, it is crucial to figure out the ecologically feasible cou- pling scheme that guarantee the permanence and global attractiveness of all species in a multi-patch ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Upadhyay and Rai [8] have demonstrated that the two non-identical chaotic ecological systems having different kinds of top-predators can be synchronized using an algorithm proposed by Lu and Cao[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The idea of this approach is that it takes care of the uncertainties involved in the parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' There are many methods in control theory for synchronizing chaotic systems, including the Adaptive Control Method, Back-stepping Method, Active Control Method, Time-Delay feedback approach, and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Among above-listed methods, the linear active control technique for chaos syn- chronization is popular and effective for synchronizing identical and non-identical chaotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In this work, we will use the active and adaptive control meth- ods for achieving synchronization of chaos within either identical or non-identical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Active control method was first used for chaos synchronization by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Bai and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Lonngren [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In this method, non-linear controllers are designed based on the Lyapunov stability theory to achieve synchronization in cou- pled systems using the known parameters of the drive and response systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 4 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY Since we will be dealing with an ecological model, the system’s parameters cannot be precisely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The adaptive control is one of the popular technique for controlling and synchronizing non-linear systems with uncertain parameters [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The method allows the model to adapt data assimilation along the way that may be useful for predicting the real system’s future behaviour [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In this method, control law and parameter update law are designed in such a way that the chaotic response system to behave like chaotic drive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' This scheme maintains the consistent performance of a system in the presence of uncertainty, variations in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Consequently, asymptotically global synchronization control of the chaotic system guarantees to converge the error dynamics to the equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In this article, we investigate the three-dimensional chaotic generalized Lotka-Volterra system, a more general model than the competitive predator-prey examples of Lotka-Volterra types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We examine the properties of the model, including equilib- rium analysis, dissipative properties, the maximum Lyapunov exponent, and slow manifold analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We use linear feedback control to stabilize the model at its equi- librium points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Our goal is to synchronize two identical GLV models with the same drive variable and different initial conditions using complete replacement synchro- nization in an ecological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Prey population is assumed to be in abundance in such a way that predators from nearby patches also feed upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To maintain syn- chronization indefinitely with only small adjustments within a two-patch system, we design the active control law(when system parameters are known) and adaptive control law (when system parameters are unknown) mathematically and validate the analytical results through numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Model The generalized Lotka-Volterra equations are autonomous and deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The dynamics of the model in a more generalized form are defined as ˙xi(t) = xi(t)(ri + fi(x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' , xn)), with initial condition xi(0) = x(i,0), for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where n represents the number of species, xi(t) is the size of population i, ˙xi(t) is the time derivative of species i, t is the time variable, x(i,0) is the initial population SYNCHRONIZATION OF CHAOS IN ECOLOGY 5 of species i, ri is the self-growth of species i, and fi(x1, x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' , xn) is the nonlin- ear multi-variable function with intra and inter-species competition terms for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Although the populations are usually measured in integer numbers, xi(t) is real for each i and can be interpreted as density, biomass or some other measure which correlates with the number of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Let Rn denotes the Euclidean space and the function fi is a continuous, smooth and real-valued function for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We make the following assumptions on fi for i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=', n} [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (i) fi, for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=', n} is bounded on a domain D ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (ii) There exist constant Ki > 0 for each i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=', n} such that ||fi(X) − fi(Y )|| ≤ Ki ||X − Y || ∀ X, Y ∈ D ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Model with linear functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The generalized Lotka Volterra (GLV) model and its variant have been studied by many authors [16],[17],[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Our main focus is on three-dimensional GLV chaotic system, which has been devised by Samardzija and Greller in 1988 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We assume that the vector field for the model holds the above mentioned properties and takes the following form ˙x1 = x1(1 − x2 + rx1 − px3x1), ˙x2 = x2(−1 + x1), ˙x3 = x3(−q + px2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (1) where x1, x2, x3 are the state variables representing prey, middle predator and top predator populations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Here p, q, r are positive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Au- thors [18] found the system chaotic in particular parametric range p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2 and shown interesting complex dynamical behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For this set of parameter values, the orbit of all three states for two different initial conditions ((1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023 + 10−3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589 + 10−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503 + 10−3)) has sen- sitive dependence on initial conditions (SDIC) which is displayed in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Fig- ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 characterizes the SDIC in the system where the trajectories with initial condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023 + 10−3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589 + 10−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503 + 10−3)) dominate over trajectories with initial condition ((1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503) in long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Further, we display the dynamics of GLV system for three different set of parame- ters to characterize its parameter- sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Different sets of parameters involved 6 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 Prey IC: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503) IC: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0033, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0599, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6513) (a) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 Middle Predator IC: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503) IC: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0033, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0599, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6513) (b) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 Top Predator IC: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503) IC: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0033, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0599, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6513) (c) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a), (b) and (c): Time series of x1, x2, and x3 for two nearby initial conditions ((1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023+ 10−3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589 + 10−3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503 + 10−3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' in the system are taken as (p, q, r) ∈ {(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0451, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='129, 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='99, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='98098, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9799, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For simulation, we fix the initial condition at x1(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0023, x2(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0589, x3(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6503 for the GLV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Figures 2, 3 and 4 show three dimensional attractor and two dimensional projections of the system on (x1, x2), (x1, x3) and (x2, x3) SYNCHRONIZATION OF CHAOS IN ECOLOGY 7 planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' From these figures, it can be inferred that the sensitivity of the system on parameters can help in restoring the hidden order out of its chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a) (b) (c) (d) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a): 3D attractor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (b), (c) and (d): 2D projections of the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively for (p, q, r) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0451, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='129, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Next subsections presents two other variants of GLV model with different func- tional responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Model with Cyrtoid type (HT II) functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To elucidate the role of functional response, we replace the linear interaction term between prey and middle predator with Holling type II functional response ( ( x1 x1+d)x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The ecological meaning of the non-linear interaction of prey with middle predator is that the prey’s contribution to the middle predator growth rate is x1x2 x1+d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Using type II functional response, the dynamics of new GLV model are proposed as ˙x1 = x1 − x1x2 x1 + d + rx2 1 − px1 2x3, ˙x2 = −x2 + ( x2x1 x1 + d), ˙x3 = −qx3 + px3x2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (2) For simulation, we take the parametric values and initial condition as p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='514, q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9089, r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1990507, d = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='00198 and x1(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='78, x2(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5020, x3(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='01 (X2,x3) plane2D projection of the attractor on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 3 X 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 1(X,,x,) plane2D projection of the attractor on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 X 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2(x,X2) plane2D projection of the attractor on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 AttractorThe Generalised Lotka Volterr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1 X 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='68 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY (a) (b) (c) (d) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a): 3D attractor ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (b), (c) and (d): 2D projections of the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively for (p, q, r) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='99, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Figure 5 displays the three-dimensional phase portrait of GLV system with HT II functional response which is a ‘stable focus’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Thus, a change in functional response in GLV system can lead to stable dynamics for a suitable set of parameter values and initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Model with Sigmoid type (HT III) functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Next, we change interaction term between prey and top predator population with Holling type III functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The inclusion of Holling type III functional response increases the search activity of top-predator for prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' With assumption of increasing prey density, the dynamics of new GLV model with Holling type III functional response are proposed as ˙x1 = x1 − x1x2 + rx2 1 − p x2 1x3 x2 1 + d, ˙x2 = −x2 + x1x2, ˙x3 = −qx3 + p x2 1x3 x2 1 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (3) (X2,X3) plane2D projection of the attractor on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='98 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='02 2(X,x) plane2D projection of the attractor on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='05 X(X,X2) plane2D projection of the attractor on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='05AttractorThe Generalised Lotka Volterr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1 X 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='05 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8SYNCHRONIZATION OF CHAOS IN ECOLOGY 9 (a) (b) (c) (d) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a): 3D attractor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (b), (c) and (d): 2D projections of the attractor on (x1, x2), (x1, x3) and (x2, x3) planes respectively for (p, q, r) = (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='98098, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9799, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a) Three-dimensional attractor of GLV equations with HT II functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For parameter values p = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='34, q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0, r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='507, d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='198, the system (3) has ‘limit cycle’-like attractor which means model (3) has stable dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For each variation, we have found different parameters values for which models (1), (2), and (3) show completely different dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We infer that the GLV model’s unstable dynamics with linear function response can be turned into stable dynamics when linear functional response is altered by HT II or HT III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' However, this may tional ResponseThe GL Attractor with HT II Fun2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 X 1X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5(X2,X3) plane2D projection of the attractor on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='71 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='68 3 X 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='98 1 1(x,x,) plane2D projection of the attractor on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='06x2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='98 1 1 AttractorThe Generalised Lotka Volterr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='05 1 X 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9510 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Three-dimensional attractor of generalised Lotka Volterra equations with HT III functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' not be very effective for arbitrarily given scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' A case in point- predators usu- ally do not follow predation rate as HT II or HT III functional response when the prey population is in abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To overcome this problem, we control chaos in the model (1) through synchronization and achieve complete replacement synchroniza- tion in two coupled GLV models following linear functional response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since GLV system has been shown to have a chaotic attractor for various set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Out of these sets, we pick the set of parameters p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2 as in [18] for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Mathematical Properties In this section, we discuss the dynamical and analytical properties of system (1) including positive Lyapunov exponent, equation of slow manifold, in-variance, dissipation, stability of feasible equilibrium points, and control of instability of unstable equilibrium points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Lyapunov exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For three-dimensional system, the local behaviour of the dynamics varies along three orthogonal directions in state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In a given chaotic system , nearby initial conditions may be moving apart along one axis, and moving together along another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Lyapunov exponent describes the average rate of separation between two nearby trajectories with different initial conditions subject to a flow [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Where a positive Lyapunov exponent confirms chaos in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For simulation, we take the parameter values of the system (1) as p = The Generalised Lotka Volterra Attractor with Holling Type Ill Functional Response 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8~ 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 0> 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 X2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4SYNCHRONIZATION OF CHAOS IN ECOLOGY 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0451, q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='129, r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The dynamics of Lyapunov exponents are shown in figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Lyapunov exponents of model (1) are as follows Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Dynamics of Lyapunov spectrum of system (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' L1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0138667 > 0, L2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='275762 < 0, L3 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='293347 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where L1 is the indicator of chaos in the system (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Equation of slow manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The infusion of geometric and topological tech- niques in chaos theory motivates mathematicians to study the underlying geometric structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In this line, expression of slow manifold permits to restore a part of the deterministic property of the system that was lost because of SDIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To find an equation of slow manifold, we consider the system (1) as slow-fast autonomous dy- namical system (S-FADS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In S-FADS, variables are separated into two groups:, one is group of fast variable and other is of slow variables where slow variables are used to determine the behaviour of whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To get the equation, we consider that the slow manifold is locally defined by a plane orthogonal to tangent system’s left fast eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Under the set of parameter values p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2, the onents L,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0138667 L,=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='275762 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='293347 3Dynamics of Lyapunov Exp 2 nents800 1000Lyapunov Expo 0 3 0 200 400 600 Time12 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY equations of GLV model (1) can be given as (4) ˙x1 = x1(1 − x2 + 2x1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x3x1), ˙x2 = x2(−1 + x1), ˙x3 = x3(−3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Jacobian matrix J at point x = (x1, x2, x3)T is obtained as J = \uf8ee \uf8ef\uf8ef\uf8f0 1 − x2 + 4x1 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702x1x3 −x1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 x2 −1 + x1 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702x1x3 0 −3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 \uf8f9 \uf8fa\uf8fa\uf8fb Let λ1(x1, x2, x3) be a real, negative and dominant Eigen value ( i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e, fast Eigen value) for Jacobian matrix in a large part of attractor’s phase space domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Furthermore, we assume that λ2(x1, x2, x3) and λ3(x1, x2, x3) be two slow Eigen values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Then Eigen vector ZT λ1 corresponding to fast Eigen value λ1 of JT (x1, x2, x3) is given by (5) |J − λ1I| Zλ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where I is 3 × 3 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Equation (6) gives, ZT λ1 = \uf8ee \uf8ef\uf8ef\uf8f0 (−1 + x1 − λ1)(−3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 − λ1) x1(−3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x1)2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1(−1 + x1 − λ1) \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' On the attractive parts of phase space (where J(x) has a fast Eigen value λ1), the equation of the slow manifold is given by (6) ˙x(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='ZT λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We use the equation (7) to define the equation of slow manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' With the substi- tution of ˙x(t) and ZT λ1 in the equation (7), we write the equation of slow manifold as λ2 1(x1 − x1x2 + 2x2 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1x3) + λ1(−4x1 + 7x2 1 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x3 1 − 3x1x2 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x3 1x2− 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='97020x4 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='985100x2 1x3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x3 1x3) + (3x1 + x2 1 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='985100x3 1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x4 1 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='970200x5 1 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='95530x1x3 − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='910600x2 1x3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='044478x3 1x3 + 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='821644x4 1x3 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='91082x3 1x3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (7) SYNCHRONIZATION OF CHAOS IN ECOLOGY 13 where λ1 is fast Eigen value of J(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Because λ1(x1, x2, x3) is uncertain Eigen value, it is not easy to use this implicit equation to draw a slow manifold representation in the three dimensional phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Invariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Let the system in vector notation is given as (8) ˙x(t) = H(x(t)) = \uf8ee \uf8ef\uf8ef\uf8f0 H1(x1, x2, x3) H2(x1, x2, x3) H3(x1, x2, x3) \uf8f9 \uf8fa\uf8fa\uf8fb H1(x1, x2, x3) = x1(1 − x2 + rx1 − px3x1), H2(x1, x2, x3) = x2(−1 + x1), H3(x1, x2, x3) = x3(−q + px1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where H1, H2 and H3 are continuously differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Assume that H is locally Lipschitz and generates a flow φt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Let L : D ⊂ R3 → R3 be a continuously differentiable function on a domain D ⊂ R3 such that ˙L(x) ≤ 0 in D, then the largest invariant set Σ ⊂ D is the set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where ∇L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='H(x) = 0 ∀x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Consider L : D ⊂ R3 → R3 be a continuously differentiable function on a domain D ⊂ R3 and defined as (9) L(x1, x2, x3) = x2 1 + x2 2 + x2 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Equation (9) gives, (10) ˙L(x1, x2, x3) = x1 ˙x1 + x2 ˙x2 + x3 ˙x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The set D ⊂ R3 is said to be an invariant set under the flow φt if for any point x ∈ D φt(x) ∈ D ∀ t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Let Σ be a smooth closed surface without boundary in D ⊂ R3 and suppose that n is a normal vector to the surface Σ at (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' If we have (11) n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' < ˙x1, ˙x2, ˙x3 >= 0 ∀ (x1, x2, x3) ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 14 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY Let us consider Σ be the x1x2 plane i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Note that the vector (0, 0, 1) is always normal to Σ and at the point (x1, x2, 0) ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' So we have, ( ˙x1, ˙x2, ˙x3) = (x1(1 − x2 + rx1 − px3x1), x2(−1 + x1), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Thus, ⟨(0, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (x1(1 − x2 + rx1 − px3x1), x2(−1 + x1), 0)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Similar arguments can be verified for x1 and x2 planes which directs that each coordinate plane is an invariant subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' It implies that for any given positive initial condition, x1(t), x2(t) and x3(t) are positive for all t that is any trajectory starting in R3 + can not cross the co-ordinate planes and it shows that R3 + is an invariant set for the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Consider the autonomous vector field ˙x(t) = H(x) for x ∈ R3, and assume that it generates a flow φt(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Let D0 is a domain in R3 which is supposed to have a volume V0, and φt(D0) is its evolution under the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' If V (t) is the volume of Dt, then the time rate of change of volume is given as |dV dt |t=0 = � D0 ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='Hdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The system (1) is dissipative if its time-t map decreases volume for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Dissipation in any dynamical system manifests itself as contraction of the phase volume on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To check this, we express the volume V (t) in the following form using the definition of the Jacobian of transformation as (12) V (t) = � D0 |dφt(x) dx |dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Expanding φt(x) in the neighbourhood of t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since the vector field H(x) is smooth enough to have a tangent plane in each point on R3 so we can expand φt(x) by Taylor series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Hence we get, (13) φt(x) = x + ˙xt + O(t2) for t → 0 Since (14) ˙x(t) = H(x), SYNCHRONIZATION OF CHAOS IN ECOLOGY 15 The equation (15) gives, (15) φt(x) = x + H(x)t + O(t2) for t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' It follows that ∂φ ∂x = I + ∂H ∂x t + O(t2), (16) |∂φ ∂x| = |I + ∂H ∂x t| + O(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Here I is 3 × 3 identity matrix so det I will be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' By expanding the expression (17) by using expansion of determinant, we get the following (17) |∂φ ∂x| = 1 + trace(∂H ∂x )t + O(t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Note that (18) trace(∂H ∂x ) = ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='H, therefore, we have (19) V (t) = V0 + � D0 ((∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='H)t + O(t2))dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' It gives (20) |dV dt |t=0 = � D0 ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='Hdx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' if the volume shrinks then divergence of vector field will be strictly negative [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Now considering the equations of model (1) in vector notation and computing its Jacobian (21) J(x1, x2, x3) = ∂H ∂x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Jacobian J(x1, x2, x3) of the model is given by (22) \uf8ee \uf8ef\uf8ef\uf8f0 1 − x2 + 2rx1 − 2px1x3 −x1 −px12 x2 −1 + x1 0 2px1x3 0 −q + px2 1 \uf8f9 \uf8fa\uf8fa\uf8fb , we take the parameter values as (23) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 16 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY The above argument shows that the G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='V dynamical system will be dissipative if the generalized divergence should be less than zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (24) � i ∂Hi ∂xi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where Einstein summation has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The divergence of vector field H on R3 is as follows (25) ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='H = ∂H1 ∂x1 + ∂H2 ∂x2 + ∂H3 ∂x3 , ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='H = −q − x2 + (2r + 1 + px1 − 2px3)x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Hence system (1) will be dissipative if the following condition is satisfied, (26) (2r + 1 + px1 − 2px3)x1 < q + x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Existence and uniqueness of solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since we are dealing with a popula- tion dynamics model, hence, the existence of at least one solution is must.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' However, the uniqueness of existed solution will give more appropriate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Here we men- tion two theorems for which solutions of the system (1) uniquely exist for all t > 0 (complete detail of the proof can be seen in [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' If the functions f1, f2 and f3 satisfy assumptions (1) and (2), mentioned in section , then continuity of functions fi for i ∈ {1, 2, 3} assures that atleast one solution exists for the dynamics of system (1) in region D × I where I = (0, T ] and the spatial boundary of region D ⊂ R3 is defined as D = {x = (x1, x2, x3) : max|xi| ≤ M, for i ∈ {1, 2, 3}} where M > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Let D be a closed subspace of complete normed linear space R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Consider H : D ⊂ R3 → D is Lipschitz continuous so that there exist 0 < K < 1 such that ||H(χ) − H(ψ)|| < K||χ − ψ|| with K = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='max(1 + 2M + 2rM + 4pM 2, 1 + 2M, q + 2pM 2) For 0 < K < 1, H(t) will be a contraction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' With the help of Banach fixed point theorem, it can be ensured that 0 < K < 1 is sufficient condition for uniqueness of solution of the system (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' SYNCHRONIZATION OF CHAOS IN ECOLOGY 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Stability of feasible equilibrium points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The equilibrium points of system (1) are solutions of following algebraic equations xx1(1 − x2 + rx1 − px3x1) = 0, x2(−1 + x1) = 0, x3(−q + px2 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (27) We obtain five equilibrium points by solving the system (27), (28) X∗ 0 = \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , X∗ 1 = \uf8ee \uf8ef\uf8ef\uf8f0 1 1 + r 0 \uf8f9 \uf8fa\uf8fa\uf8fb , X∗ 2 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 � q p 0 1+r√ q p √pq \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb , X∗ 3 = \uf8ee \uf8ef\uf8ef\uf8f0 − 1 r 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , X∗ 4 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 − � q p 0 −1+r√ q p √pq \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' From an ecological point of view, negative population density is not realistic as the population can not be negative, therefore, we take the vector x = (x1, x2, x3) as an element of R3 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' R3 + is defined as (29) R3 + = {X ∈ R3 : xi ≥ 0 for i ∈ {1, 2, 3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since equilibrium points X∗ 0, X∗ 1 and X∗ 2 are elements of the set Int(R3 +), therefore, we study the local stability of ecologically feasible equilibrium points X∗ 0, X∗ 1 and X∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (I) Stability of Trivial Equilibrium Point X∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Consider the dynamics of the model (1) in the following form (30) ˙x = H(x) = Ax + f(x) for x ∈ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' If following three conditions are satisfied (i) Constant matrix A3×3 has 3 Eigen-values with non-zero real part, (ii) f(x) is smooth and (iii) lim||x||→0 ||f(x)|| ||x|| = 0, then in a neighbourhood of the critical point X∗ 0 = (0, 0, 0), there exists stable and unstable manifolds Ws and Wu with the same dimensions ns and nu as the stable and unstable manifolds Es and Eu of the system ˙Z(t) = AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In x = 0, Es and Eu are tangent to Ws and Wu[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 18 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For GLV system, X∗ 0 = (0, 0, 0) is trivial equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Here we check all three mentioned conditions of theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (i) Note that for model (1), the constant matrix A3×3 is A = \uf8eb \uf8ec \uf8ec \uf8ed 1 0 0 0 −1 0 0 0 −q \uf8f6 \uf8f7 \uf8f7 \uf8f8 The determinant of the matrix A is non zero if q ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since, we have taken q = 3, therefore, all Eigen values of A have non zero real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (ii) Since functions f1(x1, x2, x3), f2(x1, x2, x3) and f3(x1, x2, x3) for model (1) are considered as x1(−x2 + rx1 − px3x1) = f1(x1, x2, x3), x2x1 = f2(x1, x2, x3), x3(px2 1) = f3(x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' All three functions are continuous and have continuous partial derivative for all x ∈ R3 which implies that f(x) is smooth on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Hence, the second condition also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (iii) For any x ∈ R3, converting the Cartesian coordinates into spherical coordinates by making the following transformation \uf8ee \uf8ef\uf8ef\uf8f0 x1 = r sin θ cos φ x2 = r sin θ sin φ x3 = r cos θ \uf8f9 \uf8fa\uf8fa\uf8fb , where r ≥ 0, 0 ≤ θ ≤ π and 0 ≤ φ ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Using this transformation in model (1), we have lim ||x||→0 ||f(x)|| ||x|| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Hence, the critical point (0, 0, 0) of system (1) is of the same type of critical point of the system ˙Z(t) = AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' SYNCHRONIZATION OF CHAOS IN ECOLOGY 19 The Eigen values of A are 1, −1 and −q which implies that (0, 0, 0) is saddle node for the system ˙Z(t) = AZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Therefore, the trivial steady state X∗ 0 = (0, 0, 0) of the model (1) is a saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' □ (II) Stability of Axial Equilibrium Point X∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Jacobian matrix of model (1) for parameter values p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2 is given as (31) J = \uf8ee \uf8ef\uf8ef\uf8f0 1 − x2 + 4x1 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702x1x3 −x1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 x2 −1 + x1 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702x1x3 0 −3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Jacobian matrix (31) of the model (1) about X∗ 1 = (1, 3, 0) yields the following Jacobian matrix (32) JX∗ 1 = \uf8ee \uf8ef\uf8ef\uf8f0 2 −1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851 3 0 0 0 0 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The characteristic equation |JX∗ 1 − λI| = 0 of matrix (32) is given as (33) λ3 − (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851)λ2 + (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702)λ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0447 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The characteristic equation (33) has the following Eigen values (34) λ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='014900, λ2 = 1 + √ 2ι, λ3 = 1 − √ 2ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since λ2 and λ3 have positive real parts, it implies that X∗ 1 = (1, 3, 0) is unstable equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (III) Stability of Planer Equilibrium Point X∗ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Jacobian matrix of model (1) for parameter values p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2 is given as (35) J = \uf8ee \uf8ef\uf8ef\uf8f0 1 − x2 + 4x1 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702x1x3 −x1 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 x2 −1 + x1 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9702x1x3 0 −3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851x2 1 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 20 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY Jacobian matrix (35) of the model (1) about X∗ 2 = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='002493, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4159) yields the following Jacobian matrix (36) JX∗ 2 = \uf8ee \uf8ef\uf8ef\uf8f0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='004986 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='002493 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='000002 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='002493 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='00887 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='000002 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The characteristic equation |JX∗ 2 − λI| = 0 of matrix (36) is given as (37) λ3 + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='997507)λ2 + (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='027423)λ − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='044949 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The characteristic equation (37) has the following Eigen values (38) λ1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='002493, λ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='216ι, λ3 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='5 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='216ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since all three Eigen-values of matrix (36) have negative real parts, it shows that X∗ 2 = (1002493, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4159) is locally stable equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' It is clear that planer equilibrium point is stable whereas trivial and axial equilib- rium points are unstable equilibrium points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since trivial equilibrium point refers the zero density of all three species, therefore, we neglect the instability of trivial equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' From an ecological point of view, we mainly focus on non-trivial unstable equilibrium point X∗ 1 and try to stabilize it by adding some external control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Control of instability of axial equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In order to suppress instability to X∗ 1 = (1, 3, 0), we consider the controlled GLV system in the following form (39) ˙x1 = x1(1 − x2 + rx1 − px3x1) + u1, ˙x2 = x2(−1 + x1) + u2, ˙x3 = x3(−q + px2 1) + u3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We introduce the external control law (40) u1 = −µ1(x1 − 1), u2 = −µ2(x2 − 3), u3 = −µ3(x3 − 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' with x1, x2, x3 as the feedback variable and µ1, µ2, µ3 as the positive feedback gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We substitute control law (40) into (39) and hence, the controlled system SYNCHRONIZATION OF CHAOS IN ECOLOGY 21 (39) takes the following form (41) ˙x1 = x1(1 − x2 + rx1 − px3x1)) − µ1(x1 − 1), ˙x2 = x2(−1 + x1) − µ2(x2 − 3), ˙x3 = x3(−q + px2 1) − µ3(x3 − 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The equilibrium point X∗ 1 = (1, 3, 0) of the model (1) will be asymp- totically stable if positive gains µ1, µ2 and µ3 satisfy the following inequalities[21] (42) µ1 > 2, µ1µ2 > 1 + 2µ2, µ1µ2(µ3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149) > µ2(2µ3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2528) + µ3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Jacobian matrix J of the system (41) is given by (43) \uf8ee \uf8ef\uf8ef\uf8f0 1 − x2 + 2x1(r − px3) − µ1 −x1 −px2 1 x2 −1 + x1 − µ2 0 2px1x3 0 −q + px2 1 − µ3 \uf8f9 \uf8fa\uf8fa\uf8fb Let us consider that (44) e1 = (x1 − 1), e2 = (x2 − 3), e3 = (x3 − 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' From (44), we get the error system as (45) ˙e1 = (2 − µ1)e1 − e2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851e3, ˙e2 = 3e1 − µ2e2, ˙e3 = −(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='014900 + µ3)e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The system (1) with constant and known parameters, will be stabilized to steady state X∗ 1 = (1, 3, 0), if error system (45) stabilized to (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To study the stability of equilibrium point (0, 0, 0) of error system, we consider the Lyapunov function L(e1, e2, e3) as: (46) L = 1 2(e2 1 + e2 2 + e2 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The time derivative of L in the neighbourhood of (0, 0, 0) is given as (47) ˙L = (2 − µ1)e2 1 + 2e1e2 − µ2e2 2 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851e1e3 − (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149 + µ2)e2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 22 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY The time derivative of Lyapunov function can be re-written in the following form (48) ˙L = eT Me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where e = ((x1 − 1), (x2 − 3), (x3 − 0)) is the error vector in R3, eT is the transpose of error vector e and the matrix M is 3 × 3 is given as M = \uf8ee \uf8ef\uf8ef\uf8f0 2 − µ1 1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='49250 1 −µ2 0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='49250 0 −(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149 + µ3) \uf8f9 \uf8fa\uf8fa\uf8fb According to Lyapunov stability theory, the equilibrium point (0, 0, 0) of system (45) will be asymptotically stable if ˙L < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' And ˙L < 0 if matrix M will be negative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Considering this, we find that mentioned condition will be fulfilled if positive feedback gains µ1, and µ2 satisfy the following inequalities, (49) µ1 > 2, µ1µ2 > 1 + 2µ2, µ1µ2(µ3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149) > µ2(2µ3 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2528) + µ3 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Complete Replacement Synchronization To investigate complete replacement synchronization techniques, we consider two identical chaotic GLV systems having the same parameter but different initial con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since the coupling between models is needed to maintain the synchronous state, we couple the states of both models with two controllers and drive the re- sponse system with prey species x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For this, we remove prey from response system, and drive its counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Here, we can think of prey species x1 as a driving vari- able for response system with an assumption that it is superfluous in the system of two coupled GLV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' This construction gives us a new five-dimensional drive- response system having drive and response variables as (x1d, x2d, x3d) and (x2r, x3r) SYNCHRONIZATION OF CHAOS IN ECOLOGY 23 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The coupled chaotic system with x1d drive configuration is as follows: (50) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 (Drive System) ˙x1d = x1d(1 − x2d + rx1d − px3dx1d), ˙x2d = x2d(−1 + x1d), ˙x3d = x3d(−q + px2 1d), ( Response System) ˙x2r = x2d(−1 + x1d) + u1, ˙x3r = x3r(−q + px2 1d) + u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8fe x2d(0) ̸= x2r(0) and x3d(0) ̸= x3r(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Active control law for stability of synchronization manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The identical synchronization manifold Ω = [x2d = x2r, x3d = x3r] is globally asymptotically stable for the coupling between drive and response system in equation (24) for positive µ1 and µ2, where µ1 and µ2 are large enough such that µ1 + 1 > x1d, µ2 + q > px2 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We consider drive-response system given by equations (50) and add the uni- directional controllers to the response system through the linear positive constants µ1 and µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We choose two controllers for response system as (51) u1 = −µ1(x2r(t) − x2d(t)), u2 = −µ2(x3r(t) − x3d(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Existence of all forms of identical synchronization in any dynamical system (chaotic or not), are really manifestations of dynamical behaviour restricted to a flat hyper-plane in the phase space i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' to say motion is continually confined to a hyper-plane which can be referred as synchronization manifold [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Therefore, we consider the identical synchronization manifold of the systems equation (50) as Ω = [x2d = x2r, x3d = x3r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Further, we consider the errors between states of drive and response systems of system (50) as (52) e2(t) = x2r(t) − x2d(t), e3(t) = x3r(t) − x3d(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 24 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY The dynamics of error system is as follows (53) ˙e2 = (−1 − µ1 + x1d)e2, ˙e3 = (−q − µ2 + px2 1d)e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' where, we are now interested in the stability of origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Jacobian of right side of (53) is given by (54) J(e2, e3) = � −1 + µ1 + x1d 0 0 −q + µ2 + px2 1d � We treat the response system (x2d(t), x3r(t)) as a separate system driven by x1d, then the solutions of equation (54) convey us about convergence and divergence of two initially nearby trajectories of {x2r(t), x2d(t)} and {x3r(t), x3d(t)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Next, We analyse the possibility of synchronization using the Lyapunov function construction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We consider the Lyapunov function as (55) L(e2, e3) = 1 2(e2 2 + e2 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (56) dL dt = e2 ˙e2 + e3 ˙e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Plugging dynamics of errors (53) into (55), we get (57) dL dt = −[(µ1 + 1 − x1d)e2 2 + (µ2 + q − px2 1d)e2 3], which will be strictly negative for following conditions on µ1 and µ2 (58) µ1 + 1 > x1d, µ2 + q > px2 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Condition (58) ensures that we consider the bounded density of prey species, then we can bound the positive feedback gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Thus, if µ1 and µ2 satisfy (58), then it can be assured that dL dt < 0 for all t > 0 or in other words, the complete replacement synchronization follows as e2 and e3 → 0 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Numerical Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since, a suitable coupling can influence both fre- quency as well as chaotic amplitude, therefore, the states coincide (or nearby co- incide) and regime of synchronization sets in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Thus, it is pre-arranged that the chosen coupling should assist the coupled states in coincidence without perturbing their chaotic rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We numerically integrate the System (50) and display results SYNCHRONIZATION OF CHAOS IN ECOLOGY 25 in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1, where drive and response systems are shown to synchronize when considered positive feed-back gain are chosen as µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='000024, µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='6 Population x1d x2d x3d y2r y3r (a) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='02 0 Errors (e2) (e3) (b) x2d y2r x2d vs y2r (c) (d) Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a): solutions of drive and response systems plotted over time, (b): errors between drive and response systems over time,(c): synchronization plot of {x2d, x2r}, (d): synchronization plot of {x3d, x3r} 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Lyapunov Spectrum For Drive-Response System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since the necessary condi- tion for the stability of the synchronization manifold is the negative largest trans- verse Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In the case of complete replacement synchronization, the transverse Lyapunov exponents are also known as conditional Lyapunov expo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' It is because Lyapunov exponents for the new system depend on the coupling from the drive[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The Lyapunov spectrum is a global indicator of the system’s X VS 3d3 X 3d26 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY state, aggregating over the behaviour of the entire system trajectory in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The typical approach for observing these transitions is to see a change in sign of the Lyapunov exponents of the system, as obtained from ensemble averages of the eigenvalues of the Jacobian matrix of system (50) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We solve equations of system (50) and get the Jacobian matrix as (59) J(x1d, x2d, x3d, x2d, x3r) = A5×5, where A = [aij].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The entries of the matrix A are given as: a11 = 1 − x2d + 2(r − px3d)x1d, a12 = −x1d, a13 = −px2 1d, a21 = x2d, a22 = −1 + x1d, a31 = 2px1dx3d, a33 = −q + px2 1d, a41 = x2d, a42 = µ1, a44 = −1 + x1d − µ1x2d, a51 = 2px3rx1d, a53 = µ2, a55 = −q + px2 1d − µ2, a14 = a15 = a23 = a24 = a25 = a32 = 0, a34 = a35 = a43 = a45 = a52 = a54 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Averaging the eigenvalues of Jacobian J over all phase space configurations set-up by the chaotic trajectory, we get the five Lyapunov exponents of the system of two coupled GLV models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For set of parameter values {p, q, r} = {2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, 3, 2}, Lyapunov exponents of the drive-response system are obtained as (60) L1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='011320, L2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='174464, L3 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='22221, L4 = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='011, L5 = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Since, all Lyapunov exponents are negative which confirms the stable synchroniza- tion manifold, therefore, it can be concluded that the states of coupled GLV systems are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' SYNCHRONIZATION OF CHAOS IN ECOLOGY 27 0 20 40 60 80 100 Time 6 5 4 3 2 1 0 Lyapunov exponents Dynamics of Lyapunov exponents L1=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='01132 L2=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='17446 L3=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='22221 L4=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='011 L5=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0059 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Lyapunov Exponents of two GLV models coupled with positive feedback gains µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='000024 and µ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Adaptive control law for stability of synchronization manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Us- ing the method [25], we design non-linear adaptive controller for global complete- replacement synchronization of two chaotic GLV systems with unknown parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We consider the drive system as (61) ˙x1d = x1d(1 − x2d + rx1d − px3dx1d), ˙x2d = x2d(−1 + x1d), ˙x3d = x3d(−q + px2 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The response system is given by controlled chaotic system (62) ˙x2r = x2r(−1 + x1d) + u1, ˙x3r = x3r(−q + px2 1d) + u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The synchronization error between drive and response systems is defined as (63) e2(t) = x2r(t) − x2d(t) e3(t) = x3r(t) − x3d(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The error dynamics between drive and response systems is calculated as : (64) ˙e2 = −e2 + x1de2 + u1, ˙e3 = −qe3 + px2 1de3 + u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In (64), unknown parameters p and q are to be determined by using parameter estimates P(t) and Q(t) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For this purpose, we consider adaptive control 28 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY laws u1 and u2 with positive feedback gains µ1 and µ2 as (65) u1 = e2 − x1de2 − µ1e2, u2 = Q(t)e3 − P(t)x2 1de3 − µ2e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Using control law (65) into the error dynamics (64), we get (66) ˙e2 = −µ1e2, ˙e3 = −(q − Q(t))e3 + (p − P(t))x2 1de3 − µ2e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We define parameter estimation error as (67) ep(t) = (p − P(t)), eq(t) = (q − Q(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Using (67), we can simplify the error dynamics (66) as (68) ˙e2 = −µ1e2, ˙e3 = −eq(t)e3 + ep(t)x2 1de3 − µ2e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Differentiating (67) with respect to t, we get (69) ˙ep = − ˙P(t), ˙eq = − ˙Q(t) Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The identical synchronization manifold Ω = [x2d = x2r, x3d = x3r] is globally asymptotically stable for the coupling between derive and response system in equation (50) for positive µ1 and µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The identical synchronization manifold for the systems equation (24) can be written as Ω = [x2d = x2r, x3d = x3r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Using change of coordinates (70) e = � e2 e3 � = � x2d − x2d x3r − x3d � , such that Ω can be written Ω = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Next, we use Lyapunov stability theory for finding an update law for the parameter estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' we consider the quadratic Lyapunov function as (71) L = 1 2(e2 2 + e2 3 + e2 p + e2 q) SYNCHRONIZATION OF CHAOS IN ECOLOGY 29 Note that Lyapunov function L is positive definite on R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Differentiating L along the trajectories of (66) and (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We get, (72) dL dt = e2 ˙e2 + e3 ˙e3 + ep ˙ep + eq ˙eq, dL dt = −µ1e2 2 − µ2e2 3 + ep(t)(− ˙P(t) + x2 1de3) + eq(t)(− ˙Q(t) − e2 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We want error system to be asymptotically stable i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (73) dL dt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' In view of (72), we take the parameter update law as (74) ˙P(t) = x2 1de3, ˙Q(t) = −e2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' By substituting the parameter update law (74) into Lyapunov function, we obtain time derivative of L as (75) dL dt = −µ1e2 2 − µ2e2 3, From (75), it is clear that ˙L is negative semi-definite function on R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Thus, we can conclude that the synchronization error vector e(t) and the parameter estimation error are globally bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (76) [e2, e3, ep, eq]T ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We define µ = min{µ1, µ2}, then it follows from (75) that (77) dL dt ≤ −µ||e||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Integrating the inequality (77) with respect to τ from 0 to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We get, (78) � t 0 µ||e(τ)||2dτ ≤ L(0) − L(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' From (78) it follows that e ∈ L2 and hence, ˙e(t) ∈ L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' With the help of Barbalat’s lemma [26],[27], we conclude that e(t) → 0 exponentially as t → ∞ for all initial conditions e(0) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' □ 30 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Numerical Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' For numerical simulations, we use the classical fourth- order Runge-Kutta method to solve the GLV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The initial value of parameter estimates are taken as P(0) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9, Q(0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The initial values of states of drive and response systems are taken as (x1d(0), x2d(0), x3d(0)) = (4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='41) and (x2r(0), x3r(0)) = (1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='414) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The effectiveness of control law is verified through simulation results which are shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Figure shows the solutions of system (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' It is clear that although the initial values are different, the error dynamics approach to zero as time goes to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Therefore, our numerical results confirm that the amplitude and frequency of state variables of response system become same with the drive system under the designed control law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Conclusion This work examines predator-prey systems in ecosystems to understand how they contribute to sustainable environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Our focus is on the use of Generalized Lotka-Volterra (GLV) equations to model the competition and trophic relationships between various species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We consider three different forms of three-dimensional GLV models, each with different functional responses (linear, Holling type II, and Holling type III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We find that the model with the linear functional response exhibits un- stable dynamics, where alteration in functional response can stabilize the system dynamics for a particular scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' To stabilize the dynamics in patchy ecosystem, we focus on the GLV model with the linear functional response for the remainder of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We investigate its fundamental properties and also examine the stability of equilibrium points and the suppression of instability at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Through computation of Lyapunov exponent, we find that the model is chaotic due to one positive Lyapunov exponent and has two unstable equilibrium points for the con- stant parameters p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='9851, q = 3, r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Further, we investigate the synchronization of two chaotic GLV models using two control schemes: the Active Control Technique and the Adaptive Control Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We consider a configuration in which the prey population in the drive system acts as a driving variable for the response system, allowing the other two predator popu- lations to depend only on the prey population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Using the Active Control Technique, we apply two simple linear controllers to synchronize the states of the GLV systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' SYNCHRONIZATION OF CHAOS IN ECOLOGY 31 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time 1 0 1 2 3 4 5 Population x1d x2d x3d y2r y3r (a) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='4 Errors (e2) (e3) (b) x2d 0 y2r x2d vs y2r (c) 0 x3d 0 y3r x3d vs y3r (d) Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' (a): solutions of drive and response systems over time, (b): errors between drive and response systems over time, (c): synchronization plot for {x2d, x2r}, (d): synchronization plot for {x3d, x3r} for coupling strengths µ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='0038 and µ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' These controllers are easy to implement and more straightforward than previous re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' The stability of synchronization manifold is ensured through the transition of positive conditional Lyapunov exponent to negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We also examine the synchronization of two chaotic GLV systems with unknown parameters using the Adaptive Control Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' We design two adaptive laws of parameters using the Lyapunov stability theory to ensure global and exponential synchronization of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Our results show that both the Active and Adaptive Control Techniques are effective for achieving global synchronization in chaotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 32 SHUBHANGI, NITU KUMARI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='UPADHYAY Funding The second author’s research was funded by the Science and Engineering Re- search Board (SERB), under two separate grants with grant numbers MTR/2018/000727 and EMR/2017/005203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Disclosure statement The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' References [1] Lorenz EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Deterministic nonperiodic flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' Journal of the atmospheric sciences.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' IEEE Transactions on Automatic Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} +page_content='54(9):2222–2225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE1T4oBgHgl3EQf9gZu/content/2301.03557v1.pdf'} diff --git a/CNAyT4oBgHgl3EQfePiT/content/tmp_files/2301.00318v1.pdf.txt b/CNAyT4oBgHgl3EQfePiT/content/tmp_files/2301.00318v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a0c8e66d9a98c9960a58c47a3aacd7456dcd700d --- /dev/null +++ b/CNAyT4oBgHgl3EQfePiT/content/tmp_files/2301.00318v1.pdf.txt @@ -0,0 +1,456 @@ +1 +Effect of Edge Roughness on resistance and +switching voltage of Magnetic Tunnel Junctions +Rachit R. Pandey1, Sutapa Dutta1, Heston A. Mendonca1, Ashwin A. Tulapurkar1 +1Solid State devices group, Department of Electrical Engineering, Indian Institute of Technology Bombay, +Mumbai 400076,India +Abstract—We investigate the impact of edge roughness on the +electrical transport properties of magnetic tunnel junctions using +non-equilibrium Green’s function formalism. We have modeled +edge roughness as a stochastic variation in the cross-sectional +profile of magnetic tunnel junction characterized by the stretched +exponential decay of the correlation function. The stochastic +variation in the shape and size changes the transverse energy +mode profile and gives rise to the variations in the resistance +and switching voltage of the magnetic tunnel junction. We find +that the variations are larger as the magnetic tunnel junction +size is scaled down due to the quantum confinement effect. A +model is proposed for the efficient calculation of edge roughness +effects by approximating the cross-sectional geometry to a circle +with the same cross-sectional area. Further improvement can be +obtained by approximating the cross-sectional area to an ellipse +with an aspect ratio determined by the first transverse eigenvalue +corresponding to the 2D cross section. These results would be +useful for reliable design of the spin transfer torque- magnetic +random access memory (STT-MRAM) with ultra-small magnetic +tunnel junctions. +Index Terms—Magnetic Tunnel Junction, spin transfer torque, +circular edge roughness, non-equilibrium Green’s function +I. INTRODUCTION +Magnetic tunnel junction (MTJ) comprises two ferromag- +netic layers (free layer and pinned layer) separated by a +tunneling barrier. Binary information can be stored in MTJs +corresponding to parallel (P) and anti-parallel (AP) configu- +rations of the magnetizations. The information can be read +by measuring the resistance which is low for P and high +for AP configurations respectively. Spin transfer torque (STT) +produced by application of large positive and negative voltages +to free layer with respect to the fixed layer, stabilizes P and +AP configurations respectively, and thus can be used for writ- +ing the memory. As ferromagnetic layers with perpendicular +magnetic anisotropy (PMA) have lower threshold switching +voltage (Vc) with enhanced thermal stability, they are preferred +over in-plane magnetized layers [1]. Reliability analysis of +STT-MRAM (Magnetic Random Access Memory) in terms of +write error, tunnel oxide breakdown, temperature variations, +etc. has been carried out before [2], [3], [4], [5], [6], [7]. +In this paper, we have investigated the effect of lithographic +imperfections on the performance of MTJ, which becomes +more evident with the technology scaling. The circular edge +roughness (CER) is defined as the straying of a pattern from +its expected circular shape and is used to characterize the +unwanted sidewall roughness emerging during fabrication pro- +cesses [8], [9], [10]. The threshold voltages and resistances of +the MTJ have been calculated using non-equilibrium Green’s +function (NEGF) method, for 250 realizations of the sidewalls +for fixed CER parameters. CER affects the area as well as +the shape of MTJ, which in turn changes the transverse mode +energies of the electrons tunneling across the barrier and thus +gives rise to variations in the resistance and threshold voltage. +We have calculated these variations for a range of CER +parameters. The NEGF calculation needs transverse energy +eigenvalues which were obtained by solving the Schrodinger +equation for a 2d potential well with a random boundary +corresponding to each realization. +II. SIMULATION METHODOLOGY +In the first step, charge current and spin current for a given +applied voltage across MTJ is calculated using 1d NEGF +formalism, as a function of transverse mode energy at 300 K +temperature. The device Hamiltonian matrix is modeled using +an effective mass tight binding approach. The transport of +electrons across the device is assumed to be coherent. The +effect of the contacts is taken into account as self-energy +contributions to the Hamiltonian. Charge and spin currents +are calculated from the energy-resolved electron correlation +matrix [11], [12]. We used CoFeB as the ferromagnet for both +the fixed and free layers with the Fermi energy EF =2.25 eV +and the exchange splitting, ∆ = 2.15 eV. The barrier height +from the Fermi level is taken as UB “ 0.76 eV. The effective +mass of MgO (tunnelling barrier) and FM are taken as 0.16 +me and 0.38 me, respectively, where me is the free electron +mass. The thickness of the oxide layer is set to 0.9 nm. The +charge and parallel spin current (spin current along the fixed +layer direction) as a function of the transverse mode energy +are tabulated for a range of voltage values ranging from -0.6 +to 0.6 V for both P and AP configurations. In the second step, +the transverse energy modes are found from the solution of +the Schrodinger equation for 2d infinite well with a boundary +given by the cross-section of the MTJ. If the cross-section is +a perfect circle, the eigenvalues of Hamiltonian are known +analytically. For an arbitrary cross-section, the eigenvalues +can be found numerically using finite difference method by +discretizing the area into a square grid. In the third step, +the charge current and spin current for each transverse mode +are summed up to get the net charge and spin current for a +range of voltage values ranging from -0.6 to 0.6 V for both +P and AP configurations. The resistance-area (RA) product +arXiv:2301.00318v1 [cond-mat.mes-hall] 1 Jan 2023 + +2 +calculated at 0.01 V, for MTJ with elliptical cross-section for +different aspect ratios as a function of corresponding areas is +shown in Fig. 1b. From this figure, we can see that as the area +reduces, the RA product shows dependence on area as well as +shape [13]. The critical spin current can be calculated from the +Gilbert damping (αG) and the energy barrier between P and +AP states (∆E), as Isc “ p4qαG{¯hq∆E. Further, the energy +barrier is given by, ∆E +“ p1{2qµ0MsAtF MHK, where +Ms, A, tF M, HK denote the saturation magnetization, cross- +sectional area, free layer thickness and effective perpendicular +anisotropy respectively. The critical voltage can be found by +interpolating spin current vs voltage data. If the radius of MTJ +is 10 nm, assuming ∆E “ 40kBT (T=300 K), αG=0.08, +tF M=2 nm and Ms “ 1.2 ˆ 106A{m, the HK comes out +to be 3.5 ˆ 105A{m. The critical voltage for P to AP and AP +to P switching as a function of area assuming circular cross- +section and the same HK is shown by the magenta curve +in Fig. 1c. Similar calculations for 8 nm and 6 nm radii are +shown by green and blue curves respectively. Fig. 1d shows +the critical voltage (assuming HK of 6 nm radius MTJ) for +elliptical cross-section of different aspect ratios as a function +of the area. We can see that as the area reduces, the threshold +voltage shows dependence on area as well as shape. +Fig. 1. (a) Schematic of MTJ without edge roughness. (b) RA +product vs area for ellipse with different aspect ratios (AR=1 +corresponds to a circle).(c) Vc vs area for circle with energy +barrier of 40kBT for radii=6 nm (blue), 8nm (green) and 10 +nm (magenta). (d) Vc vs area for ellipse with different aspect +ratios for energy barrier of 40kBT for radius=6 nm. +For incorporation of circular edge roughness into a circular +cross-section of radius R0, we make a random line segment +of length 2πR0 with auto-correlation function (R) given by +the equation, Rpxq “ σ2e´pd{ξq2α, where the chord length +d is given by d “ 2R0|sinpx{2R0q|. ξ, α, and σ denote the +correlation length, roughness parameter and standard deviation +respectively [14], [15] . A realization of random line segment +is obtained as follows [16]: We numerically generate white +noise series with unit power spectral density (PSD) and take +its Fourier transform. This is then multiplied by the PSD of +the correlation function. The inverse FT of the product gives +us a random line segment. The random shape is constructed by +Fig. 2. Coefficient of Variation plots for: (a) Resistance (P) (b) +Resistance (AP) (c) Vc (P) (d) Vc (AP). +Fig. 3. (a) Schematic of MTJ with edge roughness. (b) +comparison of Vc (AP) for 20 trials obtained from detailed +calculation(blue), circle approximation (red), ellipse approxi- +mation (green). +taking R0 `x as the radii distribution for angles from 0 to 2π. +The coefficient of variation (CV=standard deviation/mean) for +the quantities to be analyzed is obtained from 250 samples. +III. RESULTS AND DISCUSSIONS +Variation in the area and shape of MTJ cross-section due +to the CER produces variation in the transverse energy mode +profile. This in turn produces variation in the charge current +and spin current flowing across the MTJ for a given applied +voltage. The coefficient of variation of resistance and switch- +ing voltage as a function of σ and ξ for α “ 0.5 and average +radius 6 nm obtained from detailed calculation is shown as +a 2D plot in Fig. 2. We can see that the variations become +larger as σ and ξ increase. CV for different parameters at the +centre of 2D plot (σ “ 0.67nm, ξ “ 15nm) are shown in the +table I for different average radii of the cross-section under +“detailed calculation” column heading. We can see that the +variations increase as the MTJ size is scaled down. To find out +the influence of area variation, for each of the 250 samples, we +mapped the random shape to a perfect circle of the same area +and found out the resistance and switching voltage (See Fig. +1c). The CV obtained from this procedure is shown in table +I under “circle approximation” coumn heading and it matches +well with values obtained from detailed calculation. + +(a) +(b) +.AR=1 +(AP) +RA product (2 um +-"AR=0.6 (AP) +AR=0.4 +(AP) +5.9 +CoFeB :0 +AR=1 (P) +(free) +AR=0.6 (P) +1.2 +MgO +AR=0.4 (P) +1.1 +CoFeB +(pinned) +100 +300 +Area(nm2) +(c) +(d) +..AR=1 (AP) +0.16 +R=6 nm (AP) +--AR=0.6 (AP) +.AR=0.4 (AP) +0.1 +M +0.14 +M +-0.22 +AR=1 (P) +R=8 nm +(P) +-0.3 +R=10 nm +AR=0.6 (P) +-R=6 nm +(P) +-0.25 +AR=0.4 (P) +100 +300 +100 +300 +Area(nm?) +Area(nm?)25 +25 +(a) +(b) +0.5 +0.3 +15 +15 +0.3 +cS +cS +0.1 +0.1 +5 +5 +0.4 +1 +0.4 +1 +25 +25 +(c) +(d) +0.05 +0.05 +0.03 +15 +15 +0.03 +cS +cS +0.01 +0.01 +5 +5 +0.4 +1 +0.4 +0 +1(a) +(b) +detailed calculation +ellipse approximation +circle approximation +0.159 +(v) +CoFeB +P +(free) +0.155 +U +> +MgO +0.151 +CoFeB +(pinned) +5 +10 +15 +203 +TABLE I: % CV for α “ 0.5 σ “ 0.67nm ξ “ 15nm +% CV of +R0pnmq +Detailed +calculation +Estimated +from eq. 1 +Circle +approx. +6 +21.73 +20.5 +21.62 +RP +8 +12.94 +13.86 +12.95 +10 +10.12 +10.20 +9.99 +6 +25.63 +23.32 +25.33 +RAP +8 +14.26 +15.04 +14.15 +10 +10.89 +10.93 +10.69 +6 +2.18 +2.07 +1.99 +V cP +8 +0.93 +0.92 +0.94 +10 +0.57 +0.57 +0.56 +6 +2.02 +1.83 +1.85 +V cAP +8 +0.90 +0.90 +0.89 +10 +0.55 +0.53 +0.54 +The circle approximation is expected to work well when +the ratio, (σ{R0) is small. Further, for the approximation +to work well, the minimum normalized correlation function +e´p2R0{ξq2α should be close to 1 i.e.p2R0{ξq2α should be +small. If area variation due to CER plays a dominant role, +we can estimate the variance in a quantity Q as, +varpQq « pdQ{dAq2r2 +ż L +0 +pL ´ xqRpxqdxs +(1) +where L “ 2πR0 is the average perimeter. The term in the +square bracket in the above equation is the area variance. The +CV of various parameters estimated with above equation is +given under “estimated” column heading in table I. We can +see that values estimated from area variation are fairly close to +the numerically calculated values. These equations imply that +the area variance is proportional to σ2 and it is an increasing +function of ξ, which is consistent with trends seen in the 2d +plots in Fig. 2. (area variance saturates at large values of ξ{L). +To see if the circle approximation can be further improved, +we mapped a given random shape to an ellipse. This is done as +follows: We first note down the area. We calculate numerically +the ground state energy of the 2d infinite well with boundary +given by the random edge. We then compare ground state +energy with the tabulated ground state energies of ellipses with +the same area and different aspect ratios. An aspect ratio is +assigned to the random figure by interpolation. Using tabulated +data of Vc and resistance as a function of area for different +aspect ratios (see Fig. 1), we can calculate the switching +voltage and resistance of the random cross-section MTJ by +interpolation. Fig. 3 b shows the Vc for AP to P state for 20 +different realizations (out of 250). The blue bar corresponds +to Vc calculated by numerically “exact” way i.e. getting all +the transverse energy modes to form the numerical solution of +2d Schrodinger equation and summing up transverse currents +for each mode. The green bar corresponds to the calculation +by mapping the shape to an ellipse which needs only the +ground state energy calculation and is hence faster. However, +for large values of σ{R0 and R0{ξ, the contribution from the +non-elliptical shape variation should be taken into account. It +should be also noted that the area variation arising from CER +gives rise to variation in the thermal stability as the energy +barrier ∆E, depends on the area. +IV. CONCLUSION +We have demonstrated that edge roughness gives rise to +variance in the area and shape of a magnetic tunnel junction. +This in turn produces variance in the resistance and switching +voltage. The variance becomes larger as the MTJ size is scaled +down. These results would be useful for designing reliable +MRAM cells. +REFERENCES +[1] Ikeda, S., Miura, H., Mizunuma, K., Gan, H. 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ISSN. +Elsevier Science, 2000. + diff --git a/CNAyT4oBgHgl3EQfePiT/content/tmp_files/load_file.txt b/CNAyT4oBgHgl3EQfePiT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a9d26292ebab92cdad85de814221a28b7463d8e --- /dev/null +++ b/CNAyT4oBgHgl3EQfePiT/content/tmp_files/load_file.txt @@ -0,0 +1,381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf,len=380 +page_content='1 Effect of Edge Roughness on resistance and switching voltage of Magnetic Tunnel Junctions Rachit R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Pandey1, Sutapa Dutta1, Heston A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Mendonca1, Ashwin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Tulapurkar1 1Solid State devices group, Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai 400076,India Abstract—We investigate the impact of edge roughness on the electrical transport properties of magnetic tunnel junctions using non-equilibrium Green’s function formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We have modeled edge roughness as a stochastic variation in the cross-sectional profile of magnetic tunnel junction characterized by the stretched exponential decay of the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The stochastic variation in the shape and size changes the transverse energy mode profile and gives rise to the variations in the resistance and switching voltage of the magnetic tunnel junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We find that the variations are larger as the magnetic tunnel junction size is scaled down due to the quantum confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' A model is proposed for the efficient calculation of edge roughness effects by approximating the cross-sectional geometry to a circle with the same cross-sectional area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Further improvement can be obtained by approximating the cross-sectional area to an ellipse with an aspect ratio determined by the first transverse eigenvalue corresponding to the 2D cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' These results would be useful for reliable design of the spin transfer torque- magnetic random access memory (STT-MRAM) with ultra-small magnetic tunnel junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Index Terms—Magnetic Tunnel Junction, spin transfer torque, circular edge roughness, non-equilibrium Green’s function I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' INTRODUCTION Magnetic tunnel junction (MTJ) comprises two ferromag- netic layers (free layer and pinned layer) separated by a tunneling barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Binary information can be stored in MTJs corresponding to parallel (P) and anti-parallel (AP) configu- rations of the magnetizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The information can be read by measuring the resistance which is low for P and high for AP configurations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Spin transfer torque (STT) produced by application of large positive and negative voltages to free layer with respect to the fixed layer, stabilizes P and AP configurations respectively, and thus can be used for writ- ing the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' As ferromagnetic layers with perpendicular magnetic anisotropy (PMA) have lower threshold switching voltage (Vc) with enhanced thermal stability, they are preferred over in-plane magnetized layers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Reliability analysis of STT-MRAM (Magnetic Random Access Memory) in terms of write error, tunnel oxide breakdown, temperature variations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' has been carried out before [2], [3], [4], [5], [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' In this paper, we have investigated the effect of lithographic imperfections on the performance of MTJ, which becomes more evident with the technology scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The circular edge roughness (CER) is defined as the straying of a pattern from its expected circular shape and is used to characterize the unwanted sidewall roughness emerging during fabrication pro- cesses [8], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The threshold voltages and resistances of the MTJ have been calculated using non-equilibrium Green’s function (NEGF) method, for 250 realizations of the sidewalls for fixed CER parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' CER affects the area as well as the shape of MTJ, which in turn changes the transverse mode energies of the electrons tunneling across the barrier and thus gives rise to variations in the resistance and threshold voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We have calculated these variations for a range of CER parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The NEGF calculation needs transverse energy eigenvalues which were obtained by solving the Schrodinger equation for a 2d potential well with a random boundary corresponding to each realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' SIMULATION METHODOLOGY In the first step, charge current and spin current for a given applied voltage across MTJ is calculated using 1d NEGF formalism, as a function of transverse mode energy at 300 K temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The device Hamiltonian matrix is modeled using an effective mass tight binding approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The transport of electrons across the device is assumed to be coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The effect of the contacts is taken into account as self-energy contributions to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Charge and spin currents are calculated from the energy-resolved electron correlation matrix [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We used CoFeB as the ferromagnet for both the fixed and free layers with the Fermi energy EF =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='25 eV and the exchange splitting, ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='15 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The barrier height from the Fermi level is taken as UB “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='76 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The effective mass of MgO (tunnelling barrier) and FM are taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='16 me and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='38 me, respectively, where me is the free electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The thickness of the oxide layer is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The charge and parallel spin current (spin current along the fixed layer direction) as a function of the transverse mode energy are tabulated for a range of voltage values ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 V for both P and AP configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' In the second step, the transverse energy modes are found from the solution of the Schrodinger equation for 2d infinite well with a boundary given by the cross-section of the MTJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' If the cross-section is a perfect circle, the eigenvalues of Hamiltonian are known analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' For an arbitrary cross-section, the eigenvalues can be found numerically using finite difference method by discretizing the area into a square grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' In the third step, the charge current and spin current for each transverse mode are summed up to get the net charge and spin current for a range of voltage values ranging from -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 V for both P and AP configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The resistance-area (RA) product arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='00318v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='mes-hall] 1 Jan 2023 2 calculated at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='01 V, for MTJ with elliptical cross-section for different aspect ratios as a function of corresponding areas is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' From this figure, we can see that as the area reduces, the RA product shows dependence on area as well as shape [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The critical spin current can be calculated from the Gilbert damping (αG) and the energy barrier between P and AP states (∆E), as Isc “ p4qαG{¯hq∆E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Further, the energy barrier is given by, ∆E “ p1{2qµ0MsAtF MHK, where Ms, A, tF M, HK denote the saturation magnetization, cross- sectional area, free layer thickness and effective perpendicular anisotropy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The critical voltage can be found by interpolating spin current vs voltage data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' If the radius of MTJ is 10 nm, assuming ∆E “ 40kBT (T=300 K), αG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='08, tF M=2 nm and Ms “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='2 ˆ 106A{m, the HK comes out to be 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='5 ˆ 105A{m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The critical voltage for P to AP and AP to P switching as a function of area assuming circular cross- section and the same HK is shown by the magenta curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Similar calculations for 8 nm and 6 nm radii are shown by green and blue curves respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1d shows the critical voltage (assuming HK of 6 nm radius MTJ) for elliptical cross-section of different aspect ratios as a function of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We can see that as the area reduces, the threshold voltage shows dependence on area as well as shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (a) Schematic of MTJ without edge roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (b) RA product vs area for ellipse with different aspect ratios (AR=1 corresponds to a circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (c) Vc vs area for circle with energy barrier of 40kBT for radii=6 nm (blue), 8nm (green) and 10 nm (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (d) Vc vs area for ellipse with different aspect ratios for energy barrier of 40kBT for radius=6 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' For incorporation of circular edge roughness into a circular cross-section of radius R0, we make a random line segment of length 2πR0 with auto-correlation function (R) given by the equation, Rpxq “ σ2e´pd{ξq2α, where the chord length d is given by d “ 2R0|sinpx{2R0q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' ξ, α, and σ denote the correlation length, roughness parameter and standard deviation respectively [14], [15] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' A realization of random line segment is obtained as follows [16]: We numerically generate white noise series with unit power spectral density (PSD) and take its Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' This is then multiplied by the PSD of the correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The inverse FT of the product gives us a random line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The random shape is constructed by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Coefficient of Variation plots for: (a) Resistance (P) (b) Resistance (AP) (c) Vc (P) (d) Vc (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (a) Schematic of MTJ with edge roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (b) comparison of Vc (AP) for 20 trials obtained from detailed calculation(blue), circle approximation (red), ellipse approxi- mation (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' taking R0 `x as the radii distribution for angles from 0 to 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The coefficient of variation (CV=standard deviation/mean) for the quantities to be analyzed is obtained from 250 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS Variation in the area and shape of MTJ cross-section due to the CER produces variation in the transverse energy mode profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' This in turn produces variation in the charge current and spin current flowing across the MTJ for a given applied voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The coefficient of variation of resistance and switch- ing voltage as a function of σ and ξ for α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='5 and average radius 6 nm obtained from detailed calculation is shown as a 2D plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We can see that the variations become larger as σ and ξ increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' CV for different parameters at the centre of 2D plot (σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='67nm, ξ “ 15nm) are shown in the table I for different average radii of the cross-section under “detailed calculation” column heading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We can see that the variations increase as the MTJ size is scaled down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' To find out the influence of area variation, for each of the 250 samples, we mapped the random shape to a perfect circle of the same area and found out the resistance and switching voltage (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The CV obtained from this procedure is shown in table I under “circle approximation” coumn heading and it matches well with values obtained from detailed calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (a) (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='AR=1 (AP) RA product (2 um "AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 (AP) AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 (AP) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='9 CoFeB :0 AR=1 (P) (free) AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 (P) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='2 MgO AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 (P) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='1 CoFeB (pinned) 100 300 Area(nm2) (c) (d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='.AR=1 (AP) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='16 R=6 nm (AP) --AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 (AP) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 (AP) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='1 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='14 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='22 AR=1 (P) R=8 nm (P) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='3 R=10 nm AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='6 (P) R=6 nm (P) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='25 AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 (P) 100 300 100 300 Area(nm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=') Area(nm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' )25 25 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='3 15 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='3 cS cS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='1 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 1 25 25 (c) (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='03 15 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='03 cS cS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='01 5 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='4 0 1(a) (b) detailed calculation ellipse approximation circle approximation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='159 (v) CoFeB P (free) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='155 U > MgO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='151 CoFeB (pinned) 5 10 15 203 TABLE I: % CV for α “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='5 σ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='67nm ξ “ 15nm % CV of R0pnmq Detailed calculation Estimated from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1 Circle approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 6 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='73 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='62 RP 8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='94 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='86 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='95 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='12 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='99 6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='63 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='32 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='33 RAP 8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='26 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='04 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='15 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='89 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='93 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='69 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='99 V cP 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='94 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='56 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='85 V cAP 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='89 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='54 The circle approximation is expected to work well when the ratio, (σ{R0) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Further, for the approximation to work well, the minimum normalized correlation function e´p2R0{ξq2α should be close to 1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='p2R0{ξq2α should be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' If area variation due to CER plays a dominant role, we can estimate the variance in a quantity Q as, varpQq « pdQ{dAq2r2 ż L 0 pL ´ xqRpxqdxs (1) where L “ 2πR0 is the average perimeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The term in the square bracket in the above equation is the area variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The CV of various parameters estimated with above equation is given under “estimated” column heading in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We can see that values estimated from area variation are fairly close to the numerically calculated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' These equations imply that the area variance is proportional to σ2 and it is an increasing function of ξ, which is consistent with trends seen in the 2d plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' (area variance saturates at large values of ξ{L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' To see if the circle approximation can be further improved, we mapped a given random shape to an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' This is done as follows: We first note down the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We calculate numerically the ground state energy of the 2d infinite well with boundary given by the random edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' We then compare ground state energy with the tabulated ground state energies of ellipses with the same area and different aspect ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' An aspect ratio is assigned to the random figure by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Using tabulated data of Vc and resistance as a function of area for different aspect ratios (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 1), we can calculate the switching voltage and resistance of the random cross-section MTJ by interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' 3 b shows the Vc for AP to P state for 20 different realizations (out of 250).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The blue bar corresponds to Vc calculated by numerically “exact” way i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' getting all the transverse energy modes to form the numerical solution of 2d Schrodinger equation and summing up transverse currents for each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The green bar corresponds to the calculation by mapping the shape to an ellipse which needs only the ground state energy calculation and is hence faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' However, for large values of σ{R0 and R0{ξ, the contribution from the non-elliptical shape variation should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' It should be also noted that the area variation arising from CER gives rise to variation in the thermal stability as the energy barrier ∆E, depends on the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' CONCLUSION We have demonstrated that edge roughness gives rise to variance in the area and shape of a magnetic tunnel junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' This in turn produces variance in the resistance and switching voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' The variance becomes larger as the MTJ size is scaled down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' These results would be useful for designing reliable MRAM cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' REFERENCES [1] Ikeda, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=', Miura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=', Mizunuma, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=', Gan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' D.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} +page_content=' Elsevier Science, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CNAyT4oBgHgl3EQfePiT/content/2301.00318v1.pdf'} diff --git a/EdFKT4oBgHgl3EQfZy6F/content/tmp_files/2301.11805v1.pdf.txt b/EdFKT4oBgHgl3EQfZy6F/content/tmp_files/2301.11805v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d13247786d9e5bb7fe153664a5bcd5651edb69c1 --- /dev/null +++ b/EdFKT4oBgHgl3EQfZy6F/content/tmp_files/2301.11805v1.pdf.txt @@ -0,0 +1,587 @@ +arXiv:2301.11805v1 [math.LO] 27 Jan 2023 +A GAME FOR BAIRE’S GRAND THEOREM +LORENZO NOTARO +Abstract. Generalizing a result of Kiss, we provide a game that char- +acterizes Baire class 1 functions between arbitrary separable metrizable +spaces. We show that the determinacy of our game is equivalent to a +generalization of Baire’s grand theorem, and that both these statements +hold under AD and in Solovay’s model. +1. Introduction +A Polish space is a separable, completely metrizable topological space. Given +two topological spaces X, Y , a function f : X → Y is said to be Baire class 1 if, +for every open subset V of Y , the pre-image f −1(V ) is an Fσ subset of X, i.e. a +countable union of closed subsets. If X is metrizable, then the open subsets of X +are also Fσ subsets. All continuous functions with metrizable domain are Baire +class 1. +A classical result concerning this class of functions is the following theorem of +Baire — known as Baire’s grand theorem — which provides a characterization of +Baire class 1 functions from a Polish space to a separable metrizable space (e.g. see +[5, Theorem 24.15]). +Theorem (Baire). Let X be a Polish space, Y a separable metrizable space and +f : X → Y . Then the following are equivalent: +1) f is Baire class 1 +2) f↾K has a point of continuity for every compact K ⊆ X +Actually, the separability hypothesis on X can be avoided [4, Corollary 1; 12, Ch. +II, §31, X], but in this article we are interested in the separable case. +We note that Baire class 1 functions have been, and still are, sometimes defined +as pointwise limits of continuous functions — e.g. [16–18], Baire himself originally +stated his grand theorem for pointwise limits of continuous real functions [14]. +This definition and ours are equivalent only under certain hypotheses — e.g. see +[5, Theorem 24.10; 20]. +In this article, we study the generalization of Baire’s grand theorem in which +the domain’s hypothesis is weakened from Polish to separable metrizable, and its +relationship with the determinacy of a two-player game. The use of infinite two- +player, perfect information games to characterize certain classes of functions has a +2020 Mathematics Subject Classification. Primary 03E15, Secondary 26A21, 91A44. +Key words and phrases. Baire class 1, Baire characterization theorem, topological game, re- +duction game . +The author would like to thank Rapha¨el Carroy for his careful readings and useful suggestions, +which helped shaping this article. The author would like to acknowledge INDAM for the financial +support. This research was also partially supported by the project PRIN 2017 “Mathematical +Logic: models, sets, computability”, prot. 2017NWTM8R. +1 + +2 +LORENZO NOTARO +long and established history — e.g. [1, 2, 6, 7, 13, 19, 21], see Motto Ros [11] for a +detailed introduction on this subject. +In Section 2 we define our game G(f), where f is a function between separable +metrizable spaces. We prove that Player II has a winning strategy in G(f) if and +only if f is Baire class 1 (Theorem 2.2). Then we show that Player I has a winning +strategy in G(f) if and only if there is a compact K ⊆ X such that f↾K has no +point of continuity (Theorem 2.3). +In Section 3 we discuss the determinacy of our game. We start by observing +that the determinacy of our game for every function is equivalent to GBT, the gen- +eralization of Baire’s grand theorem in which the domain’s hypothesis is weakened +from Polish to separable metrizable (Corollary 3.1). We note that AC and GBT are +mutually inconsistent (Proposition 3.2). Then we show that GBT is equivalent to a +separation property coming from Descriptive Set Theory (Theorem 3.4), and that +both these statements hold under AD, the axiom of determinacy, and in Solovay’s +model. +2. The game +Given X a topological space and (Un)n∈N a sequence of open subsets of X, we +say that (Un)n∈N is convergent if it is decreasing with respect to ⊆, and if it is a +local basis of some x ∈ X. In that case we write limn→∞ Un = x. +Definition 2.1. Let X, Y be separable metrizable spaces and let f : X → Y . In +our game G(f), at the nth round, Player I plays a nonempty open subset Un of X, +and then Player II plays yn ∈ ran(f), +I +U0 +U1 +U2 +... +II +y0 +y1 +y2 +... +with the rule: Un+1 ⊆ Un for each n ∈ N. +At the end of a game run, Player +I and II have produced a sequence (Un)n∈N of nonempty open subsets of X and +a sequence (yn)n∈N in ran(f), respectively. Player II wins the run if either the +sequence (Un)n∈N is not convergent or it converges to an x ∈ X and (yn)n∈N +converges to f(x). +This game is an elaboration of Kiss’ game [1] and a further generalization of +Duparc’s eraser game [7]. +Since we use trees and operations over finite sequences throughout, we briefly +recall their classical definition and notation. Let X be a nonempty set. We denote +by X 0; otherwise, we let it be the empty sequence. Given an x ∈ X, we write +u⌢x to denote the finite sequence u⌢⟨x⟩, where ⟨x⟩ is the sequence of length 1 +containing only x. Sometimes we use the notation ⃗x to denote an element of X σ⋆ +N(⃗U) such that diam(fn[Uk]) ≤ 2−n: fix an n satisfying +the condition and an m such that d(qm, fn[Uk]) ≤ 2−n; set σ⋆(⃗U ⌢Uk) = +(qm, n). +2) otherwise: we set σ⋆(⃗U ⌢Uk) = σ⋆(⃗U). +We now show that σ⋆ +Y is a winning strategy for Player II in G(f). Fix an infinite +play (Uk)k∈N of Player I in G(f). If (Uk)k∈N is not convergent, then Player II wins. +Assume that (Uk)k∈N converges to an x ∈ X, and set yk = σ⋆ +Y (U0, . . . , Uk), nk = +σ⋆ +N(U0, . . . , Uk) for each k ∈ N. We now need to show limk→∞ yk = f(x). +Claim 2.2.1. The sequence (nk)k∈N is nondecreasing and unbounded in N. +Proof. The fact that (nk)k∈N is nondecreasing is a direct consequence of the defini- +tion of σ⋆. Next note that, for all n ∈ N, the diameters of the sets in the sequence +(fn[Uk])k∈N converge to 0, as fn is continuous and (Uk)k∈N is a local basis of x, +decreasing with respect to ⊆. +Fix a k ∈ N and an n > nk. By the previous observation, there exist a k′ > k such +that diam(fn[Uk′]) ≤ 2−n. Fix one such k′, there are two cases: either nk′−1 > nk +or nk′−1 = nk. In the latter case, the first condition in the inductive definition of +σ⋆ happens at the k′-th round, hence nk′ > nk′−1 = nk. In either case, nk′ > nk. +We just proved that for every k there is a k′ > k such that nk′ > nk, therefore +(nk)k∈N is unbounded. +□ +Let ¯k be the least k such that nk > 0. +Claim 2.2.2. For all k ≥ ¯k, d(yk, fnk(x)) ≤ 21−nk. +Proof. Fix a k ≥ ¯k and pick the smallest l ≤ k such that nl = nk. Note that +yk = yl, as from the l-th round to the k-th σ⋆ does not change its response. From + +4 +LORENZO NOTARO +the minimality of l it follows that the first condition of the inductive definition of σ⋆ +happens at the l-th round, therefore d(yl, fnl[Ul]) ≤ 2−nl and diam(fnl[Ul]) ≤ 2−nl. +Since we assumed x = limn→∞ Un, x belongs to Ul and fnl(x) belongs to fnl[Ul], +hence d(yl, fnl(x)) ≤ 21−nl. As nl = nk and yl = yk we are done. +□ +Then, for each k ≥ ¯k, +d(yk, f(x)) ≤ d(fnk(x), f(x)) + d(yk, fnk(x)) ≤ d(fnk(x), f(x)) + 21−nk. +Since (nk)k∈N is unbounded and the fn’s pointwise converge to f, these inequalities +imply that (yk)k∈N converges to f(x) and therefore σ⋆ +Y wins the run. As (Uk)k∈N +was an arbitrary play of Player I, we have showed that σ⋆ +Y is a winning strategy for +Player II in G(f). +(=⇒): Suppose that Player II has a winning strategy in G(f), we show that the +function f is Baire class 1. +Fix a winning strategy σ for Player II in G(f) and fix a compatible metric d +on X. As X is separable, there exists a scheme (Us)s∈N 0 the set {x ∈ X | +oscf(x) ≥ ǫ} is closed. +Lemma 2.4. Let X, Y be separable metric spaces, ǫ > 0 and f : X → Y such that +oscf(x) ≥ ǫ for all x ∈ X. Then there is a countable Q ⊆ X such that oscf↾Q(x) ≥ ǫ +for all x ∈ X. +Proof. Let dY be the metric on Y and fix a sequence (yn)n∈N dense in Y . For each +n, m ∈ N, let Qn,m be a countable and dense subset of f −1(B(yn, 2−m)). We claim +that the countable set Q = � +n,m Qn,m satisfies the wanted property. +Fix x ∈ X, m ∈ N and an open neighborhood U of x. By assumption, there +are x0, x1 ∈ U such that dY (f(x0), f(x1)) ≥ ǫ − 2−m. Let n0, n1 be such that +f(xi) ∈ B(yni, 2−m) for i = 0, 1. In particular U ∩ f −1(B(yni, 2−m)) ̸= ∅, and +therefore U ∩ Qni,m ̸= ∅ for i = 0, 1. Pick q0, q1 in U ∩ Qn0,m and U ∩ Qn1,m, +respectively. Then, +dY (f(q0), f(q1)) ≥ dY (f(x0), f(x1)) − dY (f(x0), f(q0)) − dY (f(x1), f(q1)) +≥ (ǫ − 2−m) − 21−m − 21−m = ǫ − 5 · 2−m. +Indeed, for i = 0, 1, f(xi) and f(qi) both belong to B(yni, 2−m), and therefore their +distance is less or equal to 21−m. +We have showed that for each x ∈ X, for every open neighborhood U of x and for +all m there are q0, q1 ∈ U ∩ Q such that dY (f(q0), f(q1)) is greater than ǫ − 5 · 2−m. +In particular diam(f(U ∩ Q)) ≥ ǫ. Hence, for all x ∈ X, oscf↾Q(x) ≥ ǫ. +□ +proof of Theorem 2.3. (⇐=) : Fix a compact set K ⊆ X such that f↾K has no +continuity point. The winning strategy for Player I that we define is essentially the +one defined by Kiss1 in [1, §2], the only difference being that we deal with a bit +more care the amount of choice used in the construction (see Remark 2.5). +1Kiss’ strategy, in turn, is based on the one defined by Carroy in [2, Theorem 4.1] + +6 +LORENZO NOTARO +Fix a compatible metric dX on X and dY on Y . Since f↾K has no point of +continuity, it follows that oscf↾K(x) > 0 for every x ∈ K. In particular, K = +� +n Kn, where +Kn = +� +x ∈ K | oscf↾K(x) ≥ 1 +n +� +. +By Baire’s category theorem, there are a nonempty open U ⊆ X and an n +such that Kn ∩ U = K ∩ U. Let C be the closure of Kn ∩ U and ǫ = 1/n, then +oscf↾C(x) ≥ ǫ for every x ∈ C. +By Lemma 2.4, we know that there is a countable Q ⊆ C such that oscf↾Q(x) ≥ ǫ +for every x ∈ Q. Let (qn)n∈N be an enumeration of Q. We now define a winning +strategy τ for Player I by induction on the lengths of Player II’s partial plays. In +particular, the map τ ranges among the open balls of X centered in Q, i.e. open +sets of the form B(x, ρ) for some x ∈ Q and radius ρ > 0: first set τ(∅) = B(q0, 1) +— we are setting the first move of Player I; fix k ∈ N, suppose that we have defined +τ for all partial plays of Player II of lengths up to k and consider the partial play +⃗y ⌢yk of length k + 1 with B(qnk, ρk) = τ(⃗y), then +1) if dY (yk, f(qnk)) ≤ ǫ/8: +let nk+1 be the least n such that qn ∈ B(qnk, ρk) and dY (f(qn), f(qnk)) ≥ +ǫ/3; let ρ be the greatest ρ ≤ ρk such that B(qnk+1, ρ) ⊆ B(qnk, ρk) and set +τ(⃗y ⌢yk) = B(qnk+1, ρ/2). +2) otherwise: +τ(⃗y ⌢yk) = B(qnk, ρk/2). +We now prove that τ is a winning strategy for Player I. Fix an infinite play (yk)k∈N +of Player II and set Bk = B(xk, ρk) = τ(y0, . . . , yk) for every k. First we show that +the sequence (Bk)k∈N converges to an x ∈ K. Indeed, it follows directly from τ’s +inductive definition that � +k Bk = � +k Bk; the compactness of K guarantees that +K ∩ � +k Bk ̸= ∅; finally, the radii of (Bk)k∈N converge to 0, hence K ∩ � +k Bk is a +singleton {x} and (Bk)k∈N converges to x. +So we are left to prove that the sequence (yk)k∈N does not converge to f(x). +Suppose first that condition 1) of τ’s inductive definition happens only finitely +many times during this game run. This means that there exists an n such that for +all k ≥ n, xk = x, and therefore dY (yk, f(x)) > ǫ/8 for all k ≥ n. In this case +(yk)k∈N certainly does not converge to f(x). +Now suppose otherwise, and let the increasing sequence (kn)n∈N be such that +condition 1) happens at the kn+1-th round for each n. More precisely, (kn)n∈N +is the increasing sequence such that dY (yk, f(xk)) ≤ ǫ/8 if and only if k = kn for +some (unique) n. For every n, +dY (ykn, ykn+1) ≥ dY (f(xkn), f(xkn+1)) − dY (f(xkn), ykn) − dY (f(xkn+1), ykn+1) += dY (f(xkn), f(xkn+1)) − dY (f(xkn), ykn) − dY (f(xkn+1), ykn+1) +≥ ǫ/3 − ǫ/8 − ǫ/8 = ǫ/12 +where the equality follows from xkn+1 = xkn+1, which holds because in the rounds +between kn + 1 and kn+1 the strategy τ does not change the center of its balls; the +last inequality follows directly from the definition of τ. Therefore, as (kn)n∈N is +unbounded, the sequence (yk)k∈N does not converge. + +A GAME FOR BAIRE’S GRAND THEOREM +7 +In either case (yk)k∈N does not converge to f(x), therefore τ wins the run. As +(yk)k∈N was an arbitrary play of Player II, we have showed that τ is a winning +strategy for Player I in G(f). +(=⇒) : Suppose that Player I has a winning strategy in G(f), we want to prove +that there exists a compact set K ⊆ X such that f↾K has no point of continuity. +We show instead that there exists a compact K ⊆ X such that Player I has a +winning strategy in G(f↾K). Indeed, if we do so, it would mean that the function +f↾K is not Baire class 1, as otherwise Player II would have a winning strategy in +G(f↾K) by Theorem 2.2. Then, by Baire’s grand theorem — which can be applied +as K, being a compact separable metrizable space, is a Polish space — there would +be a compact K′ ⊆ K such that f↾K′ has no point of continuity. +Fix a winning strategy τ for Player I and fix also an enumeration (qn)n∈N of +a countable dense subset of ran(f). +Denote by S the tree {s ∈ N 0. Rdir +t+1 is positive if the positive +prediction has a higher score than the negative (dt,3 > dt,1) and the current inventory is positive; or if dt,3 < dt,1 and +4 + +A PREPRINT - JANUARY 23, 2023 +Xt < 0. Further, if the signal [−1, 0, 1] · dt has an opposite sign than inventory Xt, Rdir +t+1 is negative. This can be +summarised as follows: +Mark-to-Market Value +Mt = Ct + Xtpm +t +∆Mt = ∆Ct + Xt−1∆pm +t + ∆xtpm +t +PnL Reward +Rpnl +t+1 = ln(Mt) − ln(Mt−1) +Directional Reward +Rdir +t+1 = κ[−1, 0, 1] · dtXt +Total Reward +rt+1 = wdirRdir +t+1 + (1 − wdir)Rpnl +t+1 +(3) +The weight on the directional reward wdir ∈ [0, 1) is reduced every learning step by a factor ψ ∈ (0, 1), +wdir ← ψwdir +so that initially the agent quickly learns to trade in the signal direction. Over the course of the learning process, Rpnl +t +becomes dominant and the agent maximises its mark-to-market profits. +5 +Experimental Results +We train all RL policies using the problem setup discussed in section 4.2 on 5.5 months of Apple (AAPL) limit order +book data (2012-01-01 to 2012-05-16) and evaluate performance on 1.5 months of out-of-sample data (2012-05-17 to +2012-06-31). We only use the first hour of every trading day (09:30 to 10:30) as the opening hour exhibits higher than +average trading volume and price moves. Each hour of the data corresponds to a single RL episode. +Our neural network architecture consists of 3 feed-forward layers, followed by an LSTM layer, for both the value- and +advantage stream of the duelling architecture. The LSTM layer allows the agent to efficiently learn a memory-based +policy with observations including 100 LOB states. +We compare the resulting learned RL policies to a baseline trading algorithm, which receives the same artificially +perturbed high-frequency signal of future mid-prices. The baseline policy trades aggressively by crossing the spread +whenever the signal indicates a directional price move up or down until the inventory constraint is reached. The signal +direction in the baseline algorithm is determined as the prediction class with the highest score (down, neutral, or up). +When the signal changes from up or down to neutral, indicating no immediate expected price move, the baseline strategy +places a passive order to slowly reduce position size until the inventory is cleared. This heuristic utilises the same action +space as the RL agent and yielded better performance than trading using only passive orders (placed at the near touch), +or only aggressive orders (at the far touch). +Figure 1 plots a 17 second simulation window from the test period, comparing the simulated baseline strategy with +the RL strategy. It can be seen that prices in the LOB are affected by the trading activity as both strategies inject new +order flow into the market, in addition to the historical orders, thereby consuming or adding liquidity at the best bid and +ask. During the plotted period, the baseline strategy incurs small losses due to the signal switching between predicting +decreasing and increasing future prices. This causes the baseline strategy to trade aggressively, paying the spread with +every trade. The RL strategy, on the other hand, navigates this difficult period better by trading more passively out of its +long position, and again when building up a new position. Especially in the second half of the depicted time period, the +RL strategy adds a large number of passive buy orders (green circles in the second panel of figure 1). This is shown +by the green straight lines, which connect the orders to their execution or cancellation, some of which occur after the +depicted period. +5.1 +Oracle Signal +The RL agent receives a noisy oracle signal of the mean return h = 10 seconds into the future (see equation 4.1). It +chooses an action every 0.1s, allowing a sufficiently quick build-up of long or short positions using repeated limit +orders of single stocks. The algorithm is constrained to keep the stock inventory within bounds of [posmin, posmax] = +[−10, 10]. To change the amount of noise in the signal, we vary the aH parameter of the Dirichlet distribution, keeping +aL = 1 constant in all cases. To keep the notation simple, we hence drop the H superscript and refer to the variable +Dirichlet parameter aH simply as a. We consider three different noise levels, parametrising the Dirichlet distribution +with a = 1.6 (low noise), a = 1.3 (mid noise), and a = 1.1 (high noise). A fixed return classification threshold +k = 4 · 10−5 was chosen to achieve good performance of the baseline algorithm, placing around 85% of observations +in the up or down category. The signal process persistence parameter is set to φ = 0.9. +5 + +A PREPRINT - JANUARY 23, 2023 +Figure 1: A short snapshot of simulation results (AAPL on 2012-06-14), comparing the RL policy (second panel) with +the baseline (first panel). The first two panels plot the best bid, ask, and mid-price, overlaying trading events of buy +orders (green) and sell orders (red). Circles mark new unmarketable limit orders entering the book. Crosses mark order +executions (trades) and triangles order cancellations. Open orders are connected by lines to either cancellations or +trades. Since we are simulating the entire LOB, trading activity can be seen to affect bid and ask prices. The third panel +plots the evolution of the inventory position of both strategies, and the last panel the trading profits over the period in +USD. +Out-of-sample trading performance is visualised by the account curves in figure 2. The curves show the evolution of +the portfolio value for a chronological evaluation of all test episodes. Every account curve shows the mean episodic +log-return µ and corresponding Sharpe ratio S next to it. We show that all RL-derived policies are able to outperform +their respective baseline strategies for the three noise levels investigated. Over the 31 test episodes, the cumulative RL +algorithm out-performance over the baseline strategy ranges between 14.8 (a = 1.3) and 32.2 (a = 1.1) percentage +points (and 20.7 for a = 1.6). In the case of the signal with the lowest signal-to-noise ratio (a=1.1), for which the +baseline strategy incurs a loss for the test period, the RL agent has learned a trading strategy with an approximately +zero mean return. Temporarily, the strategy even produces positive gains. Overall, it produces a sufficiently strong +performance to not lose money while still trading actively and incurring transaction costs. Compared to a buy-and-hold +strategy over the same time period, the noisy RL strategy similarly produces temporary out-performance, with both +account curves ending up flat with a return around zero. Inspecting Sharpe ratios, we find that using RL to optimise the +trading strategy is able to increase Sharpe ratios significantly. The increase in returns of the RL strategies is hence not +simply explained by taking on more market risk. +Figure 3a compares the mean return between the buy & hold, baseline, and RL policies for all out-of-sample episodes +across the three noise levels. A single dashed grey line connects the return for a single test episode across the three +trading strategies: buy & hold, baseline, and the RL policy. The solid blue lines representing the mean return across all +episodes. Error bars represent the 95% bootstrapped confidence intervals for the means. Testing for the significance of +the differences between RL and baseline returns across all episodes (t-test) is statistically significant (p ≪ 0.1) for all +noise levels. Differences in Sharpe ratios are similarly significant. We can thus conclude that the high frequency trading +strategies learned by RL outperform our baseline strategy for all levels of noise we have considered. +6 + +bid +trade +order +ask +- mid +order +V +canc. +X +canc. +trade +O +O +V +baseline +570.8 +Pric +570.4 +RL +570.8 +Pric +570.4 +Position +10 +0 +baseline +-10 +RL +Profit +0 +-10 +09:54:35 +09:54:40 +09:54:45 +09:54:50A PREPRINT - JANUARY 23, 2023 +Figure 2: Account curves, trading the noisy oracle signal in the test set, comparing the learned RL policies (solid lines) +with the baseline trading strategy (dashed). The black line shows the performance of the buy & hold strategy over the +same period. Different colours correspond to different signal noise levels. The RL policy is able to improve the trading +performance across all signal noise levels. +(a) Episodic mean strategy return of buy & hold, baseline, and +RL strategies for high (a = 1.1), mid (a = 1.3), and low noise +(a = 1.6) in 31 evaluation episodes. The grey dashed lines con- +nect mean log-returns across strategies for all individual episodes. +The blue line connects the mean of all episodes with 95% boot- +strapped confidence intervals. +(b) Turnover per episode: comparison between baseline and RL +strategy. Lower noise results in a more persistent signal, decreas- +ing baseline turnover, but a higher quality signal, resulting in the +RL policy to increase trading activity and turnover. +Figure 3: Mean return and turnover of the baseline and RL trading strategies. +It is also informative to compare the amount of trading activity between the baseline and RL strategies (see figure +3b). The baseline turnover decreases with an increasing signal-to-noise ratio (higher a), as the signal remains more +stable over time, resulting in fewer trades. In contrast, the turnover of the RL trading agent increases with a higher +signal-to-noise ratio, suggesting that the agent learns to trust the signal more and reflecting that higher transaction +costs, resulting from the higher trading activity, can be sustained, given a higher quality signal. In the high noise +case (a = 1.1), the RL agent learns to reduce trading activity relative to the other RL strategies, thereby essentially +filtering the signal. The turnover is high in all cases due to the high frequency of the signal and the fact that we are only +trading a small inventory. Nonetheless, performance is calculated net of spread-based transaction costs as our simulator +adequately accounts for the execution of individual orders. +Table 1 lists action statistics for all RL policies, including how often actions are skipped, and the price levels at which +limit orders are placed, grouped by buy and sell orders. With the least informative signal, the strategy almost exclusively +uses marketable limit orders, with buy orders being placed at the bid and sell orders at the ask price. With better signals +being available (a = 1.3 and a = 1.6), buy orders are more often placed at the mid-quote price, thereby trading less +aggressively and saving on transaction costs. Overall, the strategies trained on different signals all place the majority of +sell orders at the best bid price, with the amount of skipped actions varying considerably across the signals. +7 + +(μ=0.25, S=19.30) +1.2 +RL (a=1.1) +baseline (a=l.1) +(μ=0.21, S=14.69) +1.0 +RL (a=1.3) +baseline (a=1.3) +0.8 +RL (a=1.6) +(μ=0.14, S=11.28) +baseline (a=1.6) +Return +0.6 +-(μ=0.11, S=7.34) +Buy & Hold +0.4 +0.2 +S=-%724) +0.0 +=%.%0. +-0.2 + (μ=-0.07, S=-5.68) +T +T +T +T +0 +5 +10 +15 +20 +25 +30 +35 +Hours Tradinga=1.1 +a=1.3 +a=1.6 +0.4- +Return +0.2 +0.0 +-0.2 +buy & hold +buy & hold +buy & hold +baselin +baseline +baselinea=1.1 +a=1.3 +a=1.6 +800 +Turnover +600 +400 +200 +T +baseline +RL +baseline +RL +baseline +RLA PREPRINT - JANUARY 23, 2023 +a=1.1 +a=1.3 +a=1.6 +action skipped [%] +24.5 +43.8 +7.8 +sell levels (bid, mid, ask) [%] +(95.4, 3.1, 1.5) +(94.6, 2.8, 2.65) +(97.2, 1.9, 0.9) +buy levels (bid, mid, ask) [%] +(1.1, 1.3, 97.5) +(1.6, 52.9, 45.5) +(1.7, 13.0, 85.3) +Table 1: Actions taken by RL policy for the three different noise levels: the first row shows how often the policy +chooses the “skip” action. Not choosing this action does however not necessarily result in an order being placed, e.g. if +inventory constraints are binding. The last two rows show the relative proportion of limit order placement levels for sell +orders, and buy orders, respectively. +6 +Conclusions +Using Deep Double Duelling Q-learning with asynchronous experience replay, a state-of-the-art off-policy reinforcement +learning algorithm, we train a limit order trading strategy in an environment using historic market-by-order (MBO) +exchange message data. For this purpose we develop an RL environment based on the ABIDES [7] market simulator, +which reconstructs order book states dynamically from MBO data. Observing an artificial high-frequency signal of +the mean return over the following 10 seconds, the RL policy successfully transforms a directional signal into a limit +order trading strategy. The policies acquired by RL outperform our baseline trading algorithm, which places marketable +limit orders to trade into positions and passive limit orders to exit positions, both in terms of mean return and Sharpe +ratio. We investigate the effect of different levels of noise in the alpha signal on the RL performance. Unsurprisingly, +more accurate signals lead to higher trading returns but we also find that RL provides a similar added benefit to trading +performance across all noise levels investigated. +The task of converting high-frequency forecasts into tradeable and profitable strategies is difficult to solve as transaction +costs, due to high portfolio turnover, can have a prohibitively large impact on the bottom line profits. We suggest that +RL can be a useful tool to perform this translational role and learn optimal strategies for a specific signal and market +combination. We have shown that tailoring strategies in this way can significantly improve performance, and eliminates +the need for manually fine-tuning execution strategies for different markets and signals. For practical applications, +multiple different signals could even be combined into a single observation space. That way the problem of integrating +different forecasts into a single coherent trading strategy could be directly integrated into the RL problem. +While we here show an interesting use-case of RL in limit order book markets, we also want to motivate the need for +further research in this area. There are many years of high-frequency market data available, which ought to be utilised +to make further progress in LOB-based tasks and improve RL in noisy environments. This, together with the newest +type of neural network architectures, such as attention-based transformers [29, 30], enables learning tasks in LOB +environments directly from raw data with even better performance. For the task we have considered in this paper, future +research could enlarge the action space, allowing for placement of limit orders deeper into the book and larger orders +sizes. Allowing for larger sizes however would require a realistic model of market impact, considering the reaction of +other market participants. +8 + +A PREPRINT - JANUARY 23, 2023 +References +[1] Zihao Zhang, Stefan Zohren, and Stephen Roberts. DeepLOB: Deep convolutional neural networks for limit order +books. IEEE Transactions on Signal Processing, 67(11):3001–3012, 2019. +[2] Zihao Zhang and Stefan Zohren. Multi-horizon forecasting for limit order books: Novel deep learning approaches +and hardware acceleration using intelligent processing units. arXiv preprint arXiv:2105.10430, 2021. +[3] Petter N Kolm, Jeremy Turiel, and Nicholas Westray. Deep order flow imbalance: Extracting alpha at multiple +horizons from the limit order book. 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Attention is all you need. Advances in Neural Information Processing Systems, 30, 2017. +[30] Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. +arXiv preprint arXiv:1904.10509, 2019. +[31] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox, Ken Goldberg, Joseph Gonzalez, Michael +Jordan, and Ion Stoica. Rllib: Abstractions for distributed reinforcement learning. In International Conference on +Machine Learning, pages 3053–3062. PMLR, 2018. +7 +Appendix +We use the RLlib library [31] for a reference implementation of the APEX algorithm. Table 2 shows a selection of +relevant parameters we used for RL training. +Paramter +Value +timesteps_total +300e6 +framework +torch +num_gpus +1 +num_workers +42 +batch_mode +truncate_episode +gamma +.99 +lr_schedule +[[0,2e-5], [1e6, 5e-6]] +buffer_size +2e6 +learning_starts +5000 +train_batch_size +50 +rollout_fragment_length +50 +target_network_update_freq +5000 +n_step +3 +prioritized_replay +False +Table 2: Selected RL parameters for APEX algorithm using RLlib [31] library for training. +Figure 4 shows confusion matrices interpreting the oracle signal scores as probabilities over the three classes: down, +stationary, and up. The predicted class is thus the one with the highest score. +Figure 4: Confusion matrices of the artificial oracle signal for three noise levels, from low to high noise. +10 + +a=1.1 +a=1.3 +a=1.6 +0.5 +0.23 +0.27 +0.69 0.11 +0.2 +0.77 +0.06 0.17 +down +True label +stationary +0.350.31 +0.34 +0.34 0.340.32 +0.35 0.360.29 +up 0.29 +0.24 +0.48 +0.21 +0.11 +0.68 +0.2 +0.0580.74 +down +stationary +down +stationary +down +stationary +dn +Tabe \ No newline at end of file diff --git a/M9FAT4oBgHgl3EQfxh78/content/tmp_files/load_file.txt b/M9FAT4oBgHgl3EQfxh78/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3204e65950b7f7711611adf11cc69ec9184d9935 --- /dev/null +++ b/M9FAT4oBgHgl3EQfxh78/content/tmp_files/load_file.txt @@ -0,0 +1,448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf,len=447 +page_content='ASYNCHRONOUS DEEP DOUBLE DUELLING Q-LEARNING FOR TRADING-SIGNAL EXECUTION IN LIMIT ORDER BOOK MARKETS Peer Nagy Oxford-Man Institute of Quantitative Finance University of Oxford peer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='nagy@eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='uk Jan-Peter Calliess Oxford-Man Institute of Quantitative Finance University of Oxford Stefan Zohren Oxford-Man Institute of Quantitative Finance University of Oxford January 23, 2023 ABSTRACT We employ deep reinforcement learning (RL) to train an agent to successfully translate a high- frequency trading signal into a trading strategy that places individual limit orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Based on the ABIDES limit order book simulator, we build a reinforcement learning OpenAI gym environment and utilise it to simulate a realistic trading environment for NASDAQ equities based on historic order book messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To train a trading agent that learns to maximise its trading return in this environment, we use Deep Duelling Double Q-learning with the APEX (asynchronous prioritised experience replay) architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The agent observes the current limit order book state, its recent history, and a short-term directional forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To investigate the performance of RL for adaptive trading independently from a concrete forecasting algorithm, we study the performance of our approach utilising synthetic alpha signals obtained by perturbing forward-looking returns with varying levels of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Here, we find that the RL agent learns an effective trading strategy for inventory management and order placing that outperforms a heuristic benchmark trading strategy having access to the same signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Keywords Limit Order Books · Quantitative Finance · Reinforcement Learning · LOBSTER · 1 Introduction Successful quantitative trading strategies often work by generating trading signals, which exhibit a statistically significant correlation with future prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' These signals are then turned into actions, aiming to assume positions in order to gain from future price changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The higher the signal frequency and strategy turnover, the more critical is the execution component of the strategy, which translates the signal into concrete orders that can be submitted to a market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Such markets are oftentimes organised as an order ledger represented by a limit order book (LOB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Limit order book prices have been shown to be predictable over short time periods, predicting a few successive ticks into the future with some accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This has been done by either utilising the recent history of order book states [1, 2], order-flow data [3], or market-by-order (MBO) data directly as features [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' However, given the short time horizons over which these predictions are performed, and correspondingly small price movements, predictability does not directly translate into trading profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Transaction costs, strategy implementation details, and time delays add up to the challenging problem of translating high-frequency forecasts into a trading strategy, which determines when and which orders to send to the exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Moreover, different predictive signals have to be traded differently to achieve optimal results, depending on the forecast horizon, signal stability, and predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='08688v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='TR] 20 Jan 2023 A PREPRINT - JANUARY 23, 2023 In this paper, we use asynchronous off-policy reinforcement learning (RL), specifically Deep Duelling Double Q- learning with the APEX architecture [5], to learn an optimal trading strategy, given a noisy directional signal of short-term forward mid-quote returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For this purpose, we developed an OpenAI gym [6] limit order book environment based on the ABIDES [7] market simulator, similar to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We use this simulator to replay NASDAQ price-time priority limit order book markets using message data from the LOBSTER data set [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We study the case of an artificial or synthetic signal, taking the future price as known and adding varying levels of noise, allowing us to investigate learning performance and to quantify the benefit of an RL-derived trading policy compared to a baseline strategy using the same noisy signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This is not an unrealistic setup when choosing the correct level of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Practitioners often have dedicated teams researching and deriving alpha signals, often over many years, while other teams might work on translating those signals into profitable strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Our aim is to focus on the latter problem which becomes increasingly more difficult as signals become faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' It is thus interesting to see how an RL framework can be used to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In particular, we show that the RL agent learns superior policies to the baselines, both in terms of strategy return and Sharpe ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Machine learning methods, such as RL, have become increasingly important to automate trade execution in the financial industry in recent years [10], underlining the practical use of research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We make a number of contributions to the existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' By defining a novel action and state space in a LOB trading environment, we allow for the placement of limit orders at different prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This allows the agent to learn a concrete high-frequency trading strategy for a given signal, trading either aggressively by crossing the spread, or conservatively, implicitly trading off execution probability and cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In addition to the timing and level placement of limit orders, our RL agent also learns to use limit orders of single units of stock to manage its inventory as it holds variably sized long or short positions over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' More broadly, we demonstrate the practical use case of RL to translate predictive signals into limit order trading strategies, which is still usually a hand-crafted component of a trading system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We thus show that simulating limit order book markets and using RL to further automate the investment process is a promising direction for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To the best of our knowledge, this is also the first study applying the APEX [5] algorithm to limit order book environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The remaining paper is structured as follows: Section 2 surveys related literature, section 3 explains the mechanics of limit order book markets and the APEX algorithm, section 4 details the construction of the artificial price signal, section 5 showcases our empirical results, and section 6 concludes our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 2 Related Work Reinforcement learning has been applied to learn different tasks in limit order book market environments, such as optimal trade execution [11, 12, 13, 14, 15], market making [16, 17], portfolio optimisation [18] or trading [19, 20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The objective of optimal trade execution is to minimise the cost of trading a predetermined amount of shares over a given time frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Trading direction and the number of shares is already pre-defined in the execution problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Market makers, on the other hand, place limit orders on both sides of the order book and set out to maximise profits from capturing the spread, while minimising the risk of inventory accumulation and adverse selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We summarise using the term “RL for trading” such tasks which maximise profit from taking directional bets in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This is a hard problem for RL to solve as the space of potential trading strategies is large, leading to potentially many local optima in the loss landscape, and actionable directional market forecasts are notoriously difficult due to arbitrage in the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The work of [19] is an early study of RL for market microstructure tasks, including trade execution and predicting price movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' While the authors achieve some predictive power of directional price moves, forecasts are determined to be too erroneous for profitable trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The most similar work to ours is [21] that provides the first end-to-end DRL framework for high-frequency trading, using PPO [22] to trade Intel stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To model price impact, [21] use an approximation, moving prices proportionately to the square-root of traded volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The action space is essentially limited to market orders, so there is no decision made on limit prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The trained policy is able to produce a profitable trading strategy on the evaluated 20 test days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' However, this is not compared to baseline strategies and the resulting performance is not statistically tested for significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In contrast, we consider a larger action space, allowing for the placement of limit orders at different prices, thereby potentially lowering transaction costs of the learned HFT strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For a broader survey of deep RL (DRL) for trading, including portfolio optimisation, model-based and hierarchical RL approaches the reader is referred to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 2 A PREPRINT - JANUARY 23, 2023 3 Background 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 Limit Order Book Data Limit order books (LOBs) are one of the most popular financial market mechanisms used by exchanges around the world [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Market participants submit limit buy or sell orders, specifying a maximum (minimum) price at which they are willing to buy (sell), and the size of the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The exchange’s limit order book then keeps track of unfilled limit orders on the buy side (bids) and the sell side (asks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' If an incoming order is marketable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' there are open orders on the opposing side of the order book at acceptable prices, the order is matched immediately, thereby removing liquidity from the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The most popular matching prioritisation scheme is price-time priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Here, limit orders are matched first based on price, starting with the most favourable price for the incoming order, and then based on arrival time, starting with the oldest resting limit order in the book, at each price level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For a more complete review of limit order book dynamics and pertaining models, we refer the reader to [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In this paper, we consider equity limit order book data from the NASDAQ exchange [9], which also uses a price-time priority prioritisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Our market simulator keeps track of the state of the LOB by replaying historical message data, consisting of new incoming limit orders, order cancellations or modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The RL agent can then inject new messages into the order flow and thereby, change the LOB state from its observed historical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Our simulator reconstructs LOB dynamics from message data, so every marketable order takes liquidity from the book and thus has a direct price impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Beyond that, we make no further assumptions on permanent market impact or reactions of other agents in the market, which we leave to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 Double DQN with Distributed Experience Replay We model the trader’s problem as a Markov Decision Process (MDP) [24, 25], described by the tuple ⟨S, A, T , r, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' S denotes a state space, A an action space, T a stochastic transition function, r a reward function and γ a discount factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Observing the current environment state st ∈ S at time t, the trader takes action at ∈ A, which causes the environment to transition state according to the stochastic transition function T (st+1|st, at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' After transitioning from st to st+1, the agent receives a reward rt+1 = r(st, at, st+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We use Deep Double Q-learning [26] with a duelling network architecture [27] to approximate the optimal Q-function Q∗(s, a) = E[rt+1+γ maxa′ Q∗(st+1, a′)|at = a, st = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To speed up the learning process we employ the APEX training architecture [5], which combines asynchronous experience sampling using parallel environments with off-policy learning from experience replay buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Every episode i results in an experience trajectory τi = {st, at}T t=1, many of which are sampled from parallel environment instances and are then stored in the replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The environment sampling is done asynchronously using parallel processes running on CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Experience data from the buffer is then sampled randomly and batched to perform a policy improvement step of the Q-network on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Prioritised sampling from the experience buffer has proven to degrade performance in our noisy problem setting, hence we are sampling uniformly from the buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 After a sufficient number of training steps, the new policy is then copied to every CPU worker to update the behavioural policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Double Q-learning [28, 26] stabilises the learning process by keeping separate Q-network weights for action selection (main network) and action validation (target network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The target network weights are then updated gradually in the direction of the main network’s weight every few iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The duelling network architecture [27] on the other hand uses two separate network branches (for both main and target Q-networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' One branch estimates the value function V (s) = maxa Q(s, a), while the other estimates the advantage function A(s, a) = Q(s, a) − V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The benefit of this architecture choice lies therein that the advantage of individual actions in some states might be irrelevant, and the state value, which can be learnt more easily, suffices for an action-value approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 4 Framework 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 Artificial Price Signal The artificial directional price signal dt ∈ ∆2 = {x ∈ R3 : x1 + x2 + x3 = 1, xi ≥ 0 for i = 1, 2, 3} the agent receives is modelled as a discrete probability distribution over 3 classes, corresponding to the averaged mid-quote price decreasing, remaining stable, or increasing over a fixed future time horizon of h ∈ N+ seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To achieve realistic levels of temporal stability of the signal process, dt is an exponentially weighted average, with persistence coefficient 1In many application domains prioritised sampling, whereby we resample instances more frequently where the model initially performs poorly tends to aide learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' However, in our low signal-to-noise application domain, we noted poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Investigating the matter, we found that prioritised sampling caused more frequent resampling of highly noisy instances where learning was particularly difficult, hence degrading performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 3 A PREPRINT - JANUARY 23, 2023 φ ∈ (0, 1), of Dirichlet random variables ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The Dirichlet parameters α depend on the realised smoothed future return rt+h, specifically on whether the return lies within a neighbourhood of size k around zero, or above or below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Thus we have: dt = φdt−1 + (1 − φ)ϵt ϵt = Dirichlet (α(rt+h)) rt+h = pt+h − pt pt where pt+h = 1 h h � i=1 pt+i (1) and prices pt refer to the mid-quote price at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The Dirichlet distribution is parametrised, so that, in expectation, the signal dt updates in the direction of future returns, where aH and aL determine the variance of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The Dirichlet parameter vector is thus: α(rt+h) = � � � (aH, aL, aL) if rt+h < −k (aL, aH, aL) if − k ≤ rt+h < k (aL, aL, aH) if k ≤ rt+h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' (2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 RL Problem Specification At each time step t, the agent receives a new state observation st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' st consists of the time left in the current episode T − t given the episode’s duration of T, the agent’s cash balance Ct, stock inventory Xt, the directional signal dt ∈ ∆2, encoded as probabilities of prices decreasing, remaining approximately constant, or increasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' and price and volume quantities for the best bid and ask (level 1), including the agent’s own volume posted at bid and ask: ob,t and oa,t respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In addition to the most current observable variables at time t, the agent also observes a history of the previous l values, which are updated whenever there is an observed change in the LOB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Putting all this together, we obtain the following state observation: st = � � � � � T − u Cu Xu (d1 u, d2 u, d3 u)′ (pa,u, va,u, oa,u, pb,u, vb,u, ob,u)′ � � � � � u={t−l,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=',t} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' After receiving the state observation, the agent then chooses an action at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' It can place a buy or sell limit order of a single share at bid, mid-quote, or ask price;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' or do nothing and advance to the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Actions, which would immediately result in positions outside the allowed inventory constraints [posmin, posmax] are disallowed and do not trigger an order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Whenever the execution of a resting limit order takes the inventory outside the allowed constraints, a market order in the opposing direction is triggered to reduce the position back to posmin for short positions or posmax for long positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Hence, we define at ∈ A = ({−1, 1} × {−1, 0, 1}) ∪ {skip} so that in total there are 7 discrete actions available, three levels for both buy and sell orders, and a skip action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For the six actions besides the “skip” action, the first dimension encodes the trading direction (sell or buy) and the second dimension the price level (bid, mid-price, or ask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For example, a = (1, 0) describes the action to place a buy order at the mid price, and a = (−1, 1) a sell order at best ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Rewards Rt+1 consist of a convex combination of a profit-and-loss-based reward Rpnl t+1 and a directional reward Rdir t+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Rpnl t+1 is the log return of the agent’s mark-to-market portfolio value Mt, encompassing cash and the current inventory value, marked at the mid-price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The benefit of log-returns is that they are additive over time, rather than multiplicative like gross returns, so that, without discounting (γ = 1) the total profit-and-loss return �T s=t+1 Rpnl s = MT − Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The directional reward term Rdir t+1 incentivizes the agent to hold inventory in the direction of the signal and penalises the agent for inventory positions opposing the signal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The size of the directional reward can be scaled by the parameter κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Rdir t+1 is positive if the positive prediction has a higher score than the negative (dt,3 > dt,1) and the current inventory is positive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' or if dt,3 < dt,1 and 4 A PREPRINT - JANUARY 23, 2023 Xt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Further, if the signal [−1, 0, 1] · dt has an opposite sign than inventory Xt, Rdir t+1 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This can be summarised as follows: Mark-to-Market Value Mt = Ct + Xtpm t ∆Mt = ∆Ct + Xt−1∆pm t + ∆xtpm t PnL Reward Rpnl t+1 = ln(Mt) − ln(Mt−1) Directional Reward Rdir t+1 = κ[−1, 0, 1] · dtXt Total Reward rt+1 = wdirRdir t+1 + (1 − wdir)Rpnl t+1 (3) The weight on the directional reward wdir ∈ [0, 1) is reduced every learning step by a factor ψ ∈ (0, 1), wdir ← ψwdir so that initially the agent quickly learns to trade in the signal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Over the course of the learning process, Rpnl t becomes dominant and the agent maximises its mark-to-market profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 5 Experimental Results We train all RL policies using the problem setup discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 on 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='5 months of Apple (AAPL) limit order book data (2012-01-01 to 2012-05-16) and evaluate performance on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='5 months of out-of-sample data (2012-05-17 to 2012-06-31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We only use the first hour of every trading day (09:30 to 10:30) as the opening hour exhibits higher than average trading volume and price moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Each hour of the data corresponds to a single RL episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Our neural network architecture consists of 3 feed-forward layers, followed by an LSTM layer, for both the value- and advantage stream of the duelling architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The LSTM layer allows the agent to efficiently learn a memory-based policy with observations including 100 LOB states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We compare the resulting learned RL policies to a baseline trading algorithm, which receives the same artificially perturbed high-frequency signal of future mid-prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The baseline policy trades aggressively by crossing the spread whenever the signal indicates a directional price move up or down until the inventory constraint is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The signal direction in the baseline algorithm is determined as the prediction class with the highest score (down, neutral, or up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' When the signal changes from up or down to neutral, indicating no immediate expected price move, the baseline strategy places a passive order to slowly reduce position size until the inventory is cleared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This heuristic utilises the same action space as the RL agent and yielded better performance than trading using only passive orders (placed at the near touch), or only aggressive orders (at the far touch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Figure 1 plots a 17 second simulation window from the test period, comparing the simulated baseline strategy with the RL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' It can be seen that prices in the LOB are affected by the trading activity as both strategies inject new order flow into the market, in addition to the historical orders, thereby consuming or adding liquidity at the best bid and ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' During the plotted period, the baseline strategy incurs small losses due to the signal switching between predicting decreasing and increasing future prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This causes the baseline strategy to trade aggressively, paying the spread with every trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The RL strategy, on the other hand, navigates this difficult period better by trading more passively out of its long position, and again when building up a new position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Especially in the second half of the depicted time period, the RL strategy adds a large number of passive buy orders (green circles in the second panel of figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This is shown by the green straight lines, which connect the orders to their execution or cancellation, some of which occur after the depicted period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 Oracle Signal The RL agent receives a noisy oracle signal of the mean return h = 10 seconds into the future (see equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' It chooses an action every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1s, allowing a sufficiently quick build-up of long or short positions using repeated limit orders of single stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The algorithm is constrained to keep the stock inventory within bounds of [posmin, posmax] = [−10, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To change the amount of noise in the signal, we vary the aH parameter of the Dirichlet distribution, keeping aL = 1 constant in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' To keep the notation simple, we hence drop the H superscript and refer to the variable Dirichlet parameter aH simply as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We consider three different noise levels, parametrising the Dirichlet distribution with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6 (low noise), a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3 (mid noise), and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 (high noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' A fixed return classification threshold k = 4 · 10−5 was chosen to achieve good performance of the baseline algorithm, placing around 85% of observations in the up or down category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The signal process persistence parameter is set to φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 5 A PREPRINT - JANUARY 23, 2023 Figure 1: A short snapshot of simulation results (AAPL on 2012-06-14), comparing the RL policy (second panel) with the baseline (first panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The first two panels plot the best bid, ask, and mid-price, overlaying trading events of buy orders (green) and sell orders (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Circles mark new unmarketable limit orders entering the book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Crosses mark order executions (trades) and triangles order cancellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Open orders are connected by lines to either cancellations or trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Since we are simulating the entire LOB, trading activity can be seen to affect bid and ask prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The third panel plots the evolution of the inventory position of both strategies, and the last panel the trading profits over the period in USD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Out-of-sample trading performance is visualised by the account curves in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The curves show the evolution of the portfolio value for a chronological evaluation of all test episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Every account curve shows the mean episodic log-return µ and corresponding Sharpe ratio S next to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We show that all RL-derived policies are able to outperform their respective baseline strategies for the three noise levels investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Over the 31 test episodes, the cumulative RL algorithm out-performance over the baseline strategy ranges between 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8 (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3) and 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1) percentage points (and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='7 for a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In the case of the signal with the lowest signal-to-noise ratio (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1), for which the baseline strategy incurs a loss for the test period, the RL agent has learned a trading strategy with an approximately zero mean return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Temporarily, the strategy even produces positive gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Overall, it produces a sufficiently strong performance to not lose money while still trading actively and incurring transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Compared to a buy-and-hold strategy over the same time period, the noisy RL strategy similarly produces temporary out-performance, with both account curves ending up flat with a return around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Inspecting Sharpe ratios, we find that using RL to optimise the trading strategy is able to increase Sharpe ratios significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The increase in returns of the RL strategies is hence not simply explained by taking on more market risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Figure 3a compares the mean return between the buy & hold, baseline, and RL policies for all out-of-sample episodes across the three noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' A single dashed grey line connects the return for a single test episode across the three trading strategies: buy & hold, baseline, and the RL policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The solid blue lines representing the mean return across all episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Error bars represent the 95% bootstrapped confidence intervals for the means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Testing for the significance of the differences between RL and baseline returns across all episodes (t-test) is statistically significant (p ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1) for all noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Differences in Sharpe ratios are similarly significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We can thus conclude that the high frequency trading strategies learned by RL outperform our baseline strategy for all levels of noise we have considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 6 bid trade order ask mid order V canc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' X canc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' trade O O V baseline 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8 Pric 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='4 RL 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8 Pric 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='4 Position 10 0 baseline 10 RL Profit 0 10 09:54:35 09:54:40 09:54:45 09:54:50A PREPRINT - JANUARY 23, 2023 Figure 2: Account curves, trading the noisy oracle signal in the test set, comparing the learned RL policies (solid lines) with the baseline trading strategy (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The black line shows the performance of the buy & hold strategy over the same period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Different colours correspond to different signal noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The RL policy is able to improve the trading performance across all signal noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' (a) Episodic mean strategy return of buy & hold, baseline, and RL strategies for high (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1), mid (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3), and low noise (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6) in 31 evaluation episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The grey dashed lines con- nect mean log-returns across strategies for all individual episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The blue line connects the mean of all episodes with 95% boot- strapped confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' (b) Turnover per episode: comparison between baseline and RL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Lower noise results in a more persistent signal, decreas- ing baseline turnover, but a higher quality signal, resulting in the RL policy to increase trading activity and turnover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Figure 3: Mean return and turnover of the baseline and RL trading strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' It is also informative to compare the amount of trading activity between the baseline and RL strategies (see figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The baseline turnover decreases with an increasing signal-to-noise ratio (higher a), as the signal remains more stable over time, resulting in fewer trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In contrast, the turnover of the RL trading agent increases with a higher signal-to-noise ratio, suggesting that the agent learns to trust the signal more and reflecting that higher transaction costs, resulting from the higher trading activity, can be sustained, given a higher quality signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In the high noise case (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1), the RL agent learns to reduce trading activity relative to the other RL strategies, thereby essentially filtering the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The turnover is high in all cases due to the high frequency of the signal and the fact that we are only trading a small inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Nonetheless, performance is calculated net of spread-based transaction costs as our simulator adequately accounts for the execution of individual orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Table 1 lists action statistics for all RL policies, including how often actions are skipped, and the price levels at which limit orders are placed, grouped by buy and sell orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' With the least informative signal, the strategy almost exclusively uses marketable limit orders, with buy orders being placed at the bid and sell orders at the ask price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' With better signals being available (a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3 and a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6), buy orders are more often placed at the mid-quote price, thereby trading less aggressively and saving on transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Overall, the strategies trained on different signals all place the majority of sell orders at the best bid price, with the amount of skipped actions varying considerably across the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 7 (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='25, S=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='30) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 RL (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1) baseline (a=l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1) (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='21, S=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='69) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='0 RL (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3) baseline (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8 RL (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6) (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='14, S=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='28) baseline (a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6) Return 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6 (μ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='11, S=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='34) Buy & Hold 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 S=-%724) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='0 =%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='%0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 (μ=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='07, S=-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='68) T T T T 0 5 10 15 20 25 30 35 Hours Tradinga=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='4- Return 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2 buy & hold buy & hold buy & hold baselin baseline baselinea=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6 800 Turnover 600 400 200 T baseline RL baseline RL baseline RLA PREPRINT - JANUARY 23, 2023 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6 action skipped [%] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8 sell levels (bid, mid, ask) [%] (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='5) (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='65) (97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='9) buy levels (bid, mid, ask) [%] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='5) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='9, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='5) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='7, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='0, 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3) Table 1: Actions taken by RL policy for the three different noise levels: the first row shows how often the policy chooses the “skip” action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Not choosing this action does however not necessarily result in an order being placed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' if inventory constraints are binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The last two rows show the relative proportion of limit order placement levels for sell orders, and buy orders, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 6 Conclusions Using Deep Double Duelling Q-learning with asynchronous experience replay, a state-of-the-art off-policy reinforcement learning algorithm, we train a limit order trading strategy in an environment using historic market-by-order (MBO) exchange message data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For this purpose we develop an RL environment based on the ABIDES [7] market simulator, which reconstructs order book states dynamically from MBO data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Observing an artificial high-frequency signal of the mean return over the following 10 seconds, the RL policy successfully transforms a directional signal into a limit order trading strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The policies acquired by RL outperform our baseline trading algorithm, which places marketable limit orders to trade into positions and passive limit orders to exit positions, both in terms of mean return and Sharpe ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We investigate the effect of different levels of noise in the alpha signal on the RL performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Unsurprisingly, more accurate signals lead to higher trading returns but we also find that RL provides a similar added benefit to trading performance across all noise levels investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The task of converting high-frequency forecasts into tradeable and profitable strategies is difficult to solve as transaction costs, due to high portfolio turnover, can have a prohibitively large impact on the bottom line profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We suggest that RL can be a useful tool to perform this translational role and learn optimal strategies for a specific signal and market combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' We have shown that tailoring strategies in this way can significantly improve performance, and eliminates the need for manually fine-tuning execution strategies for different markets and signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For practical applications, multiple different signals could even be combined into a single observation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' That way the problem of integrating different forecasts into a single coherent trading strategy could be directly integrated into the RL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' While we here show an interesting use-case of RL in limit order book markets, we also want to motivate the need for further research in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' There are many years of high-frequency market data available, which ought to be utilised to make further progress in LOB-based tasks and improve RL in noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' This, together with the newest type of neural network architectures, such as attention-based transformers [29, 30], enables learning tasks in LOB environments directly from raw data with even better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' For the task we have considered in this paper, future research could enlarge the action space, allowing for placement of limit orders deeper into the book and larger orders sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Allowing for larger sizes however would require a realistic model of market impact, considering the reaction of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Double Q-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 23, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' [29] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' [30] Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Generating long sequences with sparse transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='10509, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' [31] Eric Liang, Richard Liaw, Robert Nishihara, Philipp Moritz, Roy Fox, Ken Goldberg, Joseph Gonzalez, Michael Jordan, and Ion Stoica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Rllib: Abstractions for distributed reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' In International Conference on Machine Learning, pages 3053–3062.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 7 Appendix We use the RLlib library [31] for a reference implementation of the APEX algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Table 2 shows a selection of relevant parameters we used for RL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Paramter Value timesteps_total 300e6 framework torch num_gpus 1 num_workers 42 batch_mode truncate_episode gamma .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='99 lr_schedule [[0,2e-5], [1e6, 5e-6]] buffer_size 2e6 learning_starts 5000 train_batch_size 50 rollout_fragment_length 50 target_network_update_freq 5000 n_step 3 prioritized_replay False Table 2: Selected RL parameters for APEX algorithm using RLlib [31] library for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Figure 4 shows confusion matrices interpreting the oracle signal scores as probabilities over the three classes: down, stationary, and up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' The predicted class is thus the one with the highest score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' Figure 4: Confusion matrices of the artificial oracle signal for three noise levels, from low to high noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content=' 10 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='1 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='3 a=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9FAT4oBgHgl3EQfxh78/content/2301.08688v1.pdf'} +page_content='6 0.' metadata={'source': 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a/MtFQT4oBgHgl3EQfVjao/content/tmp_files/2301.13301v1.pdf.txt b/MtFQT4oBgHgl3EQfVjao/content/tmp_files/2301.13301v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c515c7805e9d01c4ee3460333cad9cbc17629bb2 --- /dev/null +++ b/MtFQT4oBgHgl3EQfVjao/content/tmp_files/2301.13301v1.pdf.txt @@ -0,0 +1,2388 @@ +New experimental constraint on the 185W(n, γ)186W cross section +A. C. Larsen,1, ∗ G. M. Tveten,1, 2, † T. Renstrøm,1, 2 H. Utsunomiya,3, 4 E. Algin,5 T. Ari-izumi,3 +K. O. Ay,6 F. L. Bello Garrote,1 L. Crespo Campo,1 F. Furmyr,1 S. Goriely,7 A. G¨orgen,1 +M. Guttormsen,1 V. W. Ingeberg,1 B. V. Kheswa,8, 9 I. K. B. Kullmann,10 T. Laplace,11 E. Lima,1 +M. Markova,1 J. E. Midtbø,1 S. Miyamoto,12 A. H. Mjøs,1 V. Modamio,1 M. Ozgur,6 F. Pogliano,1 +S. Riemer-Sørensen,13, 14 E. Sahin,1 S. Shen,13 S. Siem,1 A. Spyrou,15, 16, 17 and M. Wiedeking8, 18 +1Department of Physics, University of Oslo, N-0316 Oslo, Norway +2Expert Analytics AS, N-0179 Oslo, Norway +3Department of Physics, Konan University, Okamoto 8-9-1, Higashinada, Kobe 658-8501, Japan +4Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai 201210, China +5Department of Metallurgical and Materials Engineering, Pamukkale University, 20160 Denizli, Turkey +6Department of Physics, Eskisehir Osmangazi University, 26480 Eskisehir, Turkey +7Institut d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles, +Campus de la Plaine, CP-226, 1050 Brussels, Belgium +8iThemba LABS, P.O. Box 722, 7129 Somerset West, South Africa +9Department of Applied Physics and Engineering Mathematics, +University of Johannesburg, Johannesburg, 2028, South Africa +10Institute d’Astronomie et d’Astrophysique, Universit´e Libre de Bruxelles, Belgium +11Department of Nuclear Engineering, University of California, Berkeley, 94720, USA +12Laboratory of Advanced Science and Technology for Industry, +University of Hyogo, 3-1-2 Kouto, Kamigori, Ako-gun, Hyogo 678-1205, Japan +13Institute of Theoretical Astrophysics, University of Oslo, N-0316 Oslo, Norway +14Department of Mathematics and Cybernetics, SINTEF Digital, N-0314 Oslo, Norway +15Physics Department, Michigan State University, East Lansing, Michigan 48824, USA +16National Superconducting Cyclotron Laboratory, +Michigan State University, East Lansing, Michigan 48824, USA +17Joint Institute for Nuclear Astrophysics Center for the Evolution of the Elements, +University of Notre Dame, Notre Dame, Indiana 46556, USA +18School of Physics, University of the Witwatersrand, Johannesburg 2050, South Africa +(Dated: February 1, 2023) +In this work, we present new data on the 182,183,184W(γ, n) cross sections, utilizing a quasi- +monochromatic photon beam produced at the NewSUBARU synchrotron radiation facility. Further, +we have extracted the nuclear level density and γ-ray strength function of 186W from data on +the 186W(α, α′γ)186W reaction measured at the Oslo Cyclotron Laboratory. Combining previous +measurements on the 186W(γ, n) cross section with our new 182,183,184W(γ, n) and (α, α′γ)186W +data sets, we have deduced the 186W γ-ray strength function in the range of 1 < Eγ < 6 MeV and +7 < Eγ < 14 MeV. +Our data are used to extract the level density and γ-ray strength functions needed as input to the +nuclear-reaction code TALYS, providing an indirect, experimental constraint for the 185W(n, γ)186W +cross section and reaction rate. Compared to the recommended Maxwellian-averaged cross section +(MACS) in the KADoNiS-1.0 data base, our results are on average lower for the relevant energy +range kBT ∈ [5, 100] keV, and we provide a smaller uncertainty for the MACS. The theoretical +values of Bao et al. +and the cross section experimentally constrained on photoneutron data of +Sonnabend et al. are significantly higher than our result. The lower value by Mohr et al. is in very +good agreement with our deduced MACS. Our new results could have implications for the s-process +and in particular the predicted s-process production of 186,187Os nuclei. +I. +INTRODUCTION +Neutron-capture reactions are known to be the main +producers of elements heavier than iron in our Uni- +verse [1, 2]. +The rapid (r) and the slow (s) neutron- +capture processes are traditionally believed to account +for almost 100% of the Solar-system heavy-element abun- +∗ a.c.larsen@fys.uio.no +† gry@xal.no +dances [3, 4]. The r process takes place in an environment +with an extremely high neutron density typicallly larger +than 1024 neutrons/cm3, which produces very neutron- +rich nuclei within a short time window (≈1s). In contrast, +the s process is, as the name implies, a slow process; +the neutron density is comparatively low (∼ 106 − 108 +neutrons/cm3 in asymptotic giant branch stars [5]) and +it can take from days to thousands of years between each +neutron-capture reaction. +Consequently, the s-process +“path” in the nuclear chart remains close to the valley of +stability, as the β-decay rates are typically much faster +than the (n, γ) rates when an unstable nucleus is reached. +arXiv:2301.13301v1 [nucl-ex] 30 Jan 2023 + +2 +108 +109 +110 +111 +112 +113 +114 +115 +116 +Neutron number +73 +74 +75 +76 +77 +78 +Proton number +W +182 +W +183 +W +184 +W +185 +W +186 +W +187 +Re +185 +Re +186 +Re +187 +Re +188 +Os +186 +Os +187 +Os +188 +Os +189 +Os +190 +Os +191 +Os +192 +FIG. 1. (Color online) Schematic illustration of the nuclear chart in the W-Re-Os region. The black arrows indicate (n, γ) +reactions on stable or near-stable isotopes, the blue dashed arrows show the possible (n, γ) branch on the long-lived W, Re and +Os isotopes, while the pink arrows display the β− decay branch. +However, this is not true for some particular nuclei +along the s-process path. At the branch points [6] the +β-decay rate is comparable to the (n, γ) rate, so that +there is a non-negligible possibility for the nucleus to ei- +ther undergo β-decay or capture another neutron. On +the one hand, such branch points could complicate the +s-process nucleosynthesis calculation significantly; on the +other hand, they may provide valuable information about +the neutron density and/or temperature at the astro- +physical site for which the s process operates [7–9]. +In this work, we focus on the branch-point nucleus +185W, with a laboratory half-life of 75.1(3) days [10]. +This nucleus is of interest for the Re/Os cosmochronol- +ogy first discussed by Clayton [11]. The main idea behind +the Re/Os cosmochronology is the following: the mat- +ter from which the Solar system was formed, contained +a given amount of 187Re and 187Os. Further, 187Re is +usually assigned a pure r-process origin, while 187Os is +produced only in the s process. As 187Re has a very long +half-life of more than 4·1010 years [12], Clayton suggested +to use the solar-system amount of 187Re and 187Os as a +“clock”, which would display the time span for which +nucleosynthesis events produced various elements up to +the time of the formation of our Solar system. Provided +that the 187Os amount stemming from the s process can +be reliably calculated, the extra amount of 187Os origi- +nates from the 187Re decay. Thus, at least in principle, +the abundances of the parent/child pair 187Re/187Os can +be used as a cosmochronometer, although not without +complications [13–15]. As discussed in Refs. [7, 14–16], +the branchings at 185W and 186Re (see Fig. 1) could well +have a non-negligible impact on this cosmochronometer. +Moreover, several authors [7–9, 17] have discussed the +185W and 186Re branchings as a “neutron dosimeter” for +the effective s-process neutron density; this application +again depends on the radiative neutron-capture cross sec- +tions of 185W and 186Re. No direct measurement of the +neutron-capture cross section is possible on these target +nuclei, and only constraints on the electromagnetic de- +cay of the compound system have been obtained through +photoneutron experiments at relatively high photon en- +ergies [8, 9]. +Here +we +present +new +photoneutron +data +on +182,183,184W that complete the (γ, n) measurements +on the W isotopes (Sec. II). Moreover, in Sec. III, +we present the +186W(α, α′γ) data taken at the Oslo +Cyclotron Laboratory, and the data analysis with the +resulting level density and γ strength function of 186W. +Using our new data to constrain the input to the +nuclear reaction code TALYS-1.9 [18] we estimate the +185W(n, γ)186W Maxwellian-averaged cross section and +reaction rate, and compare to previous measurements +and evaluations in Sec. IV B. Finally, we give a summary +and outlook in Sec. V. +II. +THE (γ, n) EXPERIMENTS +A. +Experimental details +The photo-neutron measurements on 182,183,184W took +place at the NewSUBARU synchrotron radiation facil- +ity. +Figure 2 shows a schematic illustration of the γ- +ray beam line and experimental setup. Beams of γ rays +were produced through laser Compton scattering (LCS) +of 1064 nm photons in head-on collisions with relativistic + +3 +Collision point Collision point +P1 (Nd laser) P2 (CO2 laser) +Collimator +C1 +C2 +Lasers: Nd(w, 2w), CO2 +8.9 m 7.5 m +18.5 m +GACKO +g - ray +Gamma hutch-2 +Gamma +hutch-1 +FIG. 2. (Color online) A schematic illustration of the experimental set up at NewSUBARU used in the (γ, n) cross-section +measurements. +electrons at the most-efficient collision point P1. The γ +beams were collimated using the Pb collimators C1 and +C2, each 10 cm long, with 3 mm and 2 mm apertures, +respectively. The beam profile on target nearly follows +the geometrical aperture of the collimator C2 with re- +spect to the collision point P1, thus avoiding any interac- +tion between beam and other materials than the target. +Throughout the experiment, the laser was periodically +on for 80 ms and off for 20 ms, in order to measure +background neutrons and γ-rays. +In this experiment, +the beams produced had an energy resolution ranging +from 0.6 MeV to 0.9 MeV (full-width at half maximum, +FWHM). +The electrons were injected from a linear accelerator +into the NewSUBARU storage ring with an initial energy +of 974 MeV, then subsequently decelerated to nominal +energies ranging from 608 MeV to 849 MeV, providing +LCS γ-ray beams of energies up to 13 MeV and down to +the neutron separation energies of the W isotopes (thus +varied for each individual case). +The maximum γ-ray +energy of the beams was increased in steps of 0.25 MeV. +The electron beam energy has been calibrated with the +accuracy on the order of 10−5 [19]. +The energy is re- +produced in every injection of an electron beam from a +linear accelerator to the storage ring. The reproducibil- +ity of the electron energy is assured in the deceleration +down to 0.5 GeV by an automated control of the electron +beam-optics parameters. +The energy profiles of the produced γ-ray beams were +measured with a 3.5in.×4.0in. LaBr3(Ce) (LaBr3) de- +tector. The measured LaBr3 spectra were reproduced by +a Geant4 code [20–23] that incorporated the kinematics +of the LCS process, including the beam emittance and +the interactions between the LCS beam and the LaBr3 +detector. +In this way we were routinely able to sim- +ulate the energy profile of the incoming γ beams with +the maximum energies accurately determined by the cal- +ibrated electron beam energy by best reproducing the +LaBr3 spectra [24, 25]. +The W targets were made from isotopically enriched +tungsten as metallic powder. The material was pressed +together and enclosed in an aluminium cylinder with a +thin cap. The targets had areal densities of 0.7421 g/cm2 +(182W), 0.754 g/cm2 (183W), and 1.7925 g/cm2 (184W). +Due to the presence of the Al cap, we limited the γ-ray +beam energy maximum 13 MeV to avoid getting contam- +inant neutrons from 27Al (Sn=13.056 MeV). +To measure the emitted neutrons, a high-efficiency +4π detector was used, consisting of 20 3He proportional +counters, arranged in three concentric rings and embed- +ded in a 36 × 36 × 50 cm3 polyethylene neutron modera- +tor [26]. The ring ratio technique, originally developed by +Berman and Fultz [27], was used to determine the aver- +age energy of the neutrons from the (γ, n) reactions. The +efficiency of the neutron detector varies with the average +neutron energy. The efficiency was measured with a cal- +ibrated 252Cf source with the emission rate of 2.27 · 104 +s−1 with 2.2% uncertainty, and the energy dependence +was determined by Monte Carlo simulations [28]. The +efficiency of the neutron detector was simulated using +isotropically distributed, mono-energetic neutrons. Once +the neutron detection efficiency for a given beam energy +has been determined, we were able to deduce the number +of (γ, n) reactions that took place during each run. +The LCS γ-ray flux was monitored by a 8in.×12in. +NaI(Tl) (NaI) detector during neutron measurement runs +with 100% detection efficiency for the beam energies used +in this experiment. +The number of incoming γ rays +per measurement was determined using the pile-up and +Poisson-fitting technique described in Refs. [29, 30]. +B. +Analysis +The measured photo-neutron cross section for an in- +coming beam with maximum γ energy Emax is given by + +4 +FIG. 3. (Color online) The simulated energy profiles for the +γ beams used. The distributions (integrated over all Eγ) are +normalized to unity. +the convoluted cross section, +σEmax +exp += +� Emax +Sn +DEmax(Eγ)σ(Eγ)dEγ = +Nn +NtNγξϵng . +(1) +Here, +DEmax +is the normalized energy distribution +( +� Emax +Sn +DEmaxdEγ = 1) of the γ-ray beam obtained from +Geant4 simulations. Examples of the simulated γ-beam +profiles, DEmax, are shown in Fig. 3. Furthermore, σ(Eγ) +is the true photo-neutron cross section as a function of +energy. The quantity Nn represents the number of neu- +trons detected, Nt gives the number of target nuclei per +unit area, Nγ is the number of γ rays incident on tar- +get, ϵn represents the neutron detection efficiency, and +finally ξ = (1 − e−µt)/(µt) gives a correction factor for +self-attenuation in the target. The factor g represents the +fraction of the γ flux above Sn. +We have determined the convoluted cross sections +σEmax +exp +given by Eq. (1) for γ beams with maximum en- +ergies in the range Sn ≤ Emax ≤ 13 MeV. The convo- +luted cross sections σEmax +exp +are not connected to a specific +Eγ, and we choose to plot them as a function of Emax. +The convoluted cross sections of the three W isotopes, +which are often called monochromatic cross sections, are +shown in Fig. 4. The error bars in Fig. 4 represent the +total uncertainty in the quantities comprising Eq. (1), +and consists of ∼ 3.2% from the efficiency determination +of the neutron detector, ∼ 1% from the pile-up method +that gives the number of γ rays, and the statistical un- +certainty in the number of detected neutrons [30]. The +statistical uncertainty ranges between ∼ 5.0 % close to +neutron threshold and 4.4 % for the highest maximum +γ-ray beam energies. The systematic error is dominated +by the uncertainty from the pile-up method and from the +simulated efficiency of the neutron detector. For the to- +tal uncertainty, we have added these uncorrelated errors +in quadrature. +By approximating the integral in Eq. (1) with a sum for +FIG. 4. +(Color online) Monochromatic cross sections of +182,183,184W. The error bars contain statistical uncertainties +from the number of detected neutrons, the uncertainty in the +efficiency of the neutron detector and the uncertainly in the +pile-up method used to determine the integrated γ-flux on +target. +each γ-beam profile, we are able to express the unfolding +problem as a set of linear equations +σf = Dσ, +(2) +where σf is the cross section folded with the beam pro- +file D. The indexes i and j of the matrix element Dij +corresponds to Emax and Eγ, respectively. +The set of +equations is given by +� +� +� +� +� +σ1 +σ2 +... +σN +� +� +� +� +� +f += +� +� +� +� +� +D11 +D12 +· · · · · · D1M +D21 +D22 +· · · · · · D2M +... +... +... +... +... +DN1 DN2 · · · · · · DNM +� +� +� +� +� +� +� +� +� +� +� +� +� +σ1 +σ2 +... +... +σM +� +� +� +� +� +� +� +� +. (3) +Each row of D corresponds to a Geant4 simulated γ beam +profile belonging to a specific measurement characterized +by Emax (see Fig. 3 for a visual representation of some of +the rows in the response matrix D). It is clear that D is +highly asymmetrical. +The number of γ-ray beam energies used to study the +cross section is much lower than the bin size (10 keV) +of the simulated beam profiles above Sn. As the system +of linear equations in Eq. (3) is under-determined, the +true σ vector cannot be extracted by matrix inversion. +In order to find σ, we utilize a folding iteration method. +The main features of this method are as follows [31]: +1) As a starting point, we choose for the 0th iteration, +a constant trial function σ0. This initial vector is +multiplied with D, and we get the 0th folded vector +σ0 +f = Dσ0. +2) The next trial input function, σ1, can be estab- +lished by adding the difference of the experimen- + +0.025 +0.02 +Relative intensity +0.015 +0.01 +0.005 +6 +8 +10 +12 +14 +E, [MeV]350 +W-182. before deconvolution +300 +W-183. before deconvolution +250 +W-184. before deconvolution +[mb] +200 +ouo +150 +b +100 +50 +8 +10 +11 +12 +13 +9 +x [MeV]5 +tally measured spectrum, σexp, and the folded spec- +trum, σ0 +f , to σ0. In order to be able to add the +folded and the input vector together, we first per- +form a Piecewise Cubic Hermite Interpolating Poly- +nomial (pchip) interpolation on the folded vector so +that the two vectors have equal dimensions. Our +new input vector is: +σ1 = σ0 + (σexp − σ0 +f ). +(4) +3) The steps 1) and 2) are iterated i times giving +σi +f = Dσi +(5) +σi+1 = σi + (σexp − σi +f) +(6) +until convergence is achieved. +This means that +σi+1 +f +≈ σexp within the statistical errors. In order +to quantitatively check convergence, we calculate +the reduced χ2 of σi+1 +f +and σexp after each iter- +ation. +Approximately four iterations are usually +enough for convergence, which is defined when the +reduced χ2 value approaches ≈ 1. +We stopped iterating when the χ2 became lower than +unity. In principle, the iteration could continue until the +reduced χ2 approaches zero, but that results in large un- +realistic fluctuations in σi due to over-fitting to the mea- +sured points σexp. +We estimate the total uncertainty in the unfolded cross +sections by calculating an upper limit of the monochro- +matic cross sections from Fig. 4 by adding and subtract- +ing the errors to the measured cross section values. These +upper and lower limits are then unfolded separately, re- +sulting in the unfolded cross sections shown in Fig. 5. +In Fig. 5, the unfolded cross sections for 182,183,184W +are evaluated at the maximum energies of the incom- +ing γ beams. +The error bars represent the statistical +errors and the systematic error due to the uncertainty in +the absolute efficiency calibration of the neutron detec- +tor. The results are compared to data on 182,184W from +Goryachev et al. [32], and the agreement is overall quite +reasonable although some local discrepancies can be ob- +served. +These discrepancies are sometimes not within +the given uncertainties, and could be due to unknown +systematic errors. +III. +THE OSLO EXPERIMENT +A. +Experimental details +The 186W(α, α′γ) inelastic-scattering experiment was +performed at the Oslo Cyclotron Laboratory. A fully- +ionized 30-MeV α beam was delivered by the MC-35 +Scanditronix cyclotron and directed to the 186W target. +The radio frequency was set to 23.76 MHz, giving a beam +burst every 42.09 ns. The experiment was run for about +FIG. 5. (Color online) Cross sections of 182,183,184W obtained +after deconvolution. Also shown are cross sections of 182,184W +from Goryachev et al. [32]. +eight days with typical beam intensities of 1.5 − 2.2 enA. +The target was mounted on a 24-µm carbon backing, and +the target thickness was 0.31 mg/cm2 with enrichment +> 98% in 186W. +To detect the outgoing charged particles, we used the +Silicon Ring (SiRi) [33] placed in backward angles with +respect to the beam direction. SiRi is a ∆E-E telescope +array consisting of eight 1550-µm thick back (E) detec- +tors, each of which has a 130-µm thick front (∆E) de- +tector divided in eight strips. A 10.5-µm thick Al foil +was placed in front of SiRi to reduce the amount δ elec- +trons from the target. SiRi covers about 6% of 4π and +the strips have an angular resolution of about 2◦, where +the center of the strip is at 126 − 140◦ (in steps of 2◦); +measured from the center of the front detector (at 133◦), +the distance of SiRi from the center of the target was 5 +cm. +The ∆E-E telescopes allow for separating different +charged-particle species. Figure 6a shows the measured +protons, deuterons, tritons, and α particles for a strip at +130◦. To select the 186W(α, α′) events, a gate was set on +the “banana” corresponding to the α particles. To cal- +ibrate the SiRi front and back detectors, we used range +calculations for our setup with the Qkinz code [34], see +Fig. 6b. +The resolution of the α particles was measured to +be 330–360 keV FWHM for the peak of the elastically- +scattered α particles. The relatively poor resolution is +mainly due to a rather elongated beam spot on the tar- +get (≈ 3–4 mm in diameter in the vertical direction, and +≈ 1 mm in the horizontal direction). The master-gate +signal for the data acquisition system was a logical signal +of 2µs generated when an E detector gave a signal above +threshold, which was set to ≈ 200 mV. +Using the CACTUS array [35], we measured γ rays +in coincidence with the inelastic scattered α particles. +In the configuration used for this experiment, CAC- +TUS consisted of 26 NaI(Tl) crystals of cylindrical shape + +500 +T +W-182, NewSUBARU +450 +W-183,NewSUBARU +400 +W-184, NewSUBARU +W-184, Goryachev et al. +350 +W-182, Goryachev et al. +Q +[qw] (u'l)o +300 +250 +200 +150 +100 +50 +0 +8 +10 +6 +12 +14 +16 +E, [MeV]6 +FIG. 6. +(Color online) (a) Particle-identification spectrum for one of the front strips at 130◦ with its corresponding back +detector (∆E–E banana plot); (b) a zoom on the α-particle banana with the Qkinz calculations used for calibration (crosses). +(5in.× 5in.). All crystals were collimated with lead col- +limators and had 2-mm thick Cu shields in front to at- +tenuate X-rays. The NaI(Tl) detectors were mounted on +the spherical CACTUS frame, so that the front end of +each crystal was positioned 22 cm from the center of the +target. The efficiency of CACTUS (for 26 NaI(Tl) detec- +tors) is 14.1(2)% as measured with a 60Co source, and +with a resolution of ≈ 6.8% FWHM for Eγ = 1.33 MeV. +Using analog electronics, we obtained a lower threshold +of about 350 keV for the NaI(Tl) detectors. +The CACTUS detectors were calibrated in energy by +gating on the protons in SiRi. +As the target had a +significant contamination of carbon (from the backing) +and oxygen, we used peaks in the proton spectrum from +the 12C(α, pγ)15N and 16O(α, pγ)19F reactions to further +identify γ rays for calibration. In particular, we used the +5.269-MeV transition from the 5/2+ first-excited level +in 15N together with the 1.868-MeV transition from the +13/2+ level at Ex = 4.648 MeV in 19F. Then we cross- +checked the obtained calibration with the 1235-keV and +2583-keV lines of 19F, in addition to the 511-keV γ ray +from positron annihilation. +To obtain α–γ coincident events, we applied a gate +on the time-to-digital converter (TDC) spectra for +the prompt peak, and subtracting randomly correlated +events. The start of the TDCs is given by the master +gate, and the stop signal is generated from the NaI(Tl) +detectors (each NaI(Tl) has an individual TDC), with a +built-in delay from the Mesytec shapers of ≈ 400 ns. The +range of the TDCs was 1.2 µs. The gate on the prompt +peak was set to ∆t = 0 ± 20 ns, while the gate for the +background subtraction was set to ∆t = 135 ± 20 ns. +Using the reaction kinematics, we determined the ini- +tial excitation energy of the residual nucleus from the +deposited energy of the α particles in SiRi. Applying the +time gates for the γ rays, we obtained excitation-energy +tagged, background-subtracted γ-ray spectra as shown in +Fig. 7a. +The γ-ray spectra needed to be corrected for the CAC- +TUS detector response. For this purpose, we applied the +iterative unfolding method of Ref. [36] available in the +Oslo-method software package [37]. This method takes +the raw γ-ray spectrum as a starting point for the un- +folded (“true”) spectrum. This trial spectrum is folded +with the known detector response, and then compared +with the raw spectrum. By taking the difference between +the folded spectrum and the raw spectrum, a new, im- +proved trial spectrum is made. This process is repeated +until the folded spectrum is approximately equal to the +raw spectrum, within the experimental uncertainties. To +preserve the experimental statistical fluctuations, and +not introduce artificial, spurious ones, the Compton sub- +traction method is also applied. This takes advantage of +the fact that the Compton distribution is very smooth. +For more details, see Ref. [36]. The unfolded γ-ray spec- +tra for each Ex bin are shown in Fig. 7b. +After unfolding, the first-generation γ rays were ex- +tracted from the data by applying an iterative subtrac- +tion method [38]. The first-generation γ rays are the ones +that are emitted first in the decay cascades, and their dis- +tribution represents the branching ratios for the various +γ transitions at a given Ex bin. The principle behind +the subtraction method is as follows. For a given Ex bin, +say, at Ex = 4 MeV, the unfolded spectrum contains all +the γ rays from all the possible decay cascades originat- +ing from the levels populated in that Ex bin. If we now +consider the Ex bins below Ex = 4 MeV, they will con- +tain all the same γ rays as the Ex = 4 MeV bin, except +the first-generation γs at Ex = 4 MeV. This is true if +the Ex bins have the same decay cascades whether the +levels in the bin were populated directly through the nu- +clear reaction, or if they were populated from γ decay of + +104 +103 +14000 +TTTT +[ke] +9500 +(a) θ = 130° +(b) ++ +Qkinz calculations +detector [ +12000 +9000 +103 +二 +10000 +8500 + deposited in △E +102 +8000 +8000 +20 +102 +7500 +11 +6000 +7000 +Energy +10 +4000 +6500 +10 +6000 +2000 +5500 +0 +5000 +10000 +15000 +20000 +25000 +10000 +12000 +14000 +16000 +18000 +20000 +22000 +Energy deposited in E detector [keV] +Energy deposited in E detector [keV]7 +0 +1 +2 +3 +4 +5 +6 +7 +8 +1 +2 +3 +4 +5 +6 +7 +8 + [MeV] +x +E +(a) +0 +1 +2 +3 +4 +5 +6 +7 +8 + [MeV] +γ +E +(b) +0 +1 +2 +3 +4 +5 +6 +7 +8 +1 +10 +2 +10 +1 +10 +2 +10 +(c) +max +x +E +min +x +E +min +γ +E +FIG. 7. (Color online) Excitation-energy vs. γ-ray energy matrices of 186W. (a) Background-subtracted data; (b) unfolded +γ-ray spectra; (c) first-generation γ-ray spectra. The lines indicate the limits set for the further analysis. +above-lying levels. We refer the reader to Ref. [39] for a +more in-depth discussion on the assumptions behind the +first-generation method. The first-generation γ spectra +are displayed in Fig. 7c. +B. +Extraction of level density and γ-ray +transmission coefficient +We now exploit the fact that the first-generation γ +spectra represent the (relative) branching ratios for a +given initial excitation-energy bin, and that we have +many such branching ratios available for a large Ex re- +gion. In the spirit of Fermi’s Golden Rule [40, 41], where +the decay rate is proportional to the level density at the +final excitation energy and the reduced transition proba- +bility for decay between a given initial and final level, we +use the following ansatz [42]: +P(Eγ, Ex) ∝ ρ(Ex − Eγ) · T (Eγ), +(7) +where P(Eγ, Ex) is the matrix of first-generation γ rays +(Fig. 7c), ρ(Ex − Eγ) is the level density at the excita- +tion energy where the γ transition “lands” and T (Eγ) is +the γ-ray transmission coefficient. Note that T (Eγ) is +only a function of Eγ, which means that the Brink-Axel +hypothesis [43, 44] is invoked. Brink stated that +“...we assume that the energy dependence of +the photo effect is independent of the detailed +structure of the initial state so that, if it were +possible to perform the photo effect on an ex- +cited state, the cross section for absorption +of a photon of energy E would still have an +energy dependence given by (15).” +where “(15)” is referring to the equation describing +the Giant Dipole Resonance (GDR) with a Lorentzian +function that only depends on the γ-transition energy. +Brink’s original formulation (as well as Axel’s application +of Brink’s hypothesis) concerned only E1 transitions, and +there is a wealth of recent works in the literature dis- +cussing the validity and/or violation of the hypothesis; +see, e.g., Refs. [45–53]. +A necessary condition for the Oslo method is that the +Brink hypothesis is at least approximately true for the +specific excitation-energy region used for extracting the +level density and γ-ray transmission coefficient. We have +performed tests of this assumption for the application in +the Oslo method in Ref. [39]. When the Brink hypothesis +is applicable, we can fit the data of the first-generation γ +rays to obtain a reliable estimate of the level density and +the γ-ray transmission coefficient through an iterative +optimization using a least-squares fit: +χ2 +red = +1 +Nfree +Emax +x� +Ei=Emin +x +Ei +� +Eγ=Emin +γ +[P(Eγ, Ei) − Pth(Eγ, Ei)]2 +[∆P(Eγ, Ei)]2 +. +(8) +Here, P(Eγ, Ei) is the experimental matrix of first- +generation γ rays where each row is normalized to unity: +Ei +� +Ei=Emin +γ +P(Eγ, Ei) = 1, +(9) +and +∆P(Eγ, Ei) +is +the +uncertainties +in +the +first- +generation matrix (including statistical errors and an es- +timate for systematic uncertainties due to unfolding and +the first-generation method, see Ref. [42]). +Moreover, +Nfree is the number of degrees of freedom and Pth(Eγ, Ei) +is the approximation for the theoretical first-generation +matrix [42]: +Pth(Eγ, Ei) = +ρ(Ei − Eγ)T (Eγ) +�Ei +Eγ=Emin +γ +ρ(Ei − Eγ)T (Eγ) +. +(10) + +8 +0 +1 +2 +3 +4 +5 +6 +7 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +Probability distribution + = 4.14 MeV +x +E +(a) + first-gen. data + +T + x +ρ + +0 +1 +2 +3 +4 +5 +6 +7 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +Probability distribution + = 5.26 MeV +x +E +(d) +0 +1 +2 +3 +4 +5 +6 +7 + = 4.59 MeV +x +E +(b) +0 +1 +2 +3 +4 +5 +6 +7 + [MeV] +γ +E + energy +γ +Probability distribution + = 5.70 MeV +x +E +(e) +0 +1 +2 +3 +4 +5 +6 +7 + = 4.81 MeV +x +E +(c) +0 +1 +2 +3 +4 +5 +6 +7 + = 6.16 MeV +x +E +(f) +FIG. 8. (Color online) Experimental first-generation spectra (black crosses) compared to the predicted ones using the extracted +level density and γ-transmission coefficient (blue line) for various excitation-energy bins (224-keV wide). +The number of degrees of freedom, Nfree, is given by +Nfree = Nch(P) − Nch(ρ) − Nch(T ). For the present data +set, we have used Emin +γ += 0.90 MeV, Emin +x += 4.0 MeV, +and Emax +x += 7.2 MeV as shown in Fig. 7c. Note that +the neutron separation energy Sn of 186W is 7.1920(12) +MeV [54], and as we have no way of discriminating +against neutrons, the Oslo method is usually limited to a +maximum excitation energy (close to) Sn. With bin size +of 224 keV, and the limits applied as shown in Fig. 7c, +we have the number of pixels in the first-generation ma- +trix Nch(P) = 330, while the number of elements in the +vectors of ρ and T is Nch(ρ) = Nch(T ) = 39, giving +Nfree = 252. It is important to note that the number +of data points in the first-generation matrix, Nch(P), is +much bigger than the number of points to be estimated, +which is 2 × 39 points; this is why the method usually +converges very well. When convergence is reached, the +extracted ρ(Ex − Eγ) and T (Eγ) are the ones that best +describe the experimental P(Eγ, Ei) matrix. +For this +case, we obtain χ2 +red = 0.85 after 20 iterations. +As a visual illustration of the fit, Fig. 8 shows some of +the experimental first-generation spectra together with +the spectra obtained for Pth. Overall, the agreement is +quite good, although we remark that the experimental +errors are rather large. Note that the fit is performed on +all the first-generation spectra (for 15 excitation-energy +bins), and so the fit is still well constrained. +Schiller et al. showed [42] that the χ2 minimization +obtains a unique solution for the relative variation of +neighboring points in the functions ρ and T ; however, +an equally good fit to the experimental P matrix is given +by the transformation +˜ρ(Ei − Eγ) = A exp[α(Ei − Eγ)] ρ(Ei − Eγ), +(11) +˜T (Eγ) = B exp(αEγ)T (Eγ). +(12) +Here, α is the common slope adjustment of ρ and T , +while A and B gives the absolute scaling of ρ and T , +respectively. These parameters must be determined from +external data, as described in the following sections. +C. +Normalization of level density +To normalize the level density by determining the α +and A parameters, we make use of discrete levels [54] +at low Ex and data on s-wave neutron resonance spac- +ings [55] at the neutron separation energy Sn. The av- +erage s-wave neutron resonance spacing D0 = 9.3(16) +eV [55] represents the spacing of levels with Jπ = 1−, 2− +as the target nucleus 185W has ground-state spin/parity +Iπ +t = +3 +2 +−. To obtain the total level density at Sn, we +need to apply a model for the spin distribution, in par- +ticular the spin cutoff parameter σJ(Ex). Here, we use +as a starting point the model of von Egidy and Bu- +curescu [56, 57] employing the rigid-body moment of in- +ertia. +However, as shown by Uhrenholt et al. [58], at +excitation energies around 7−8 MeV for heavy nuclei, a +full rigid-body moment of inertia might not be reached +yet: in Fig. 10 of Ref. [58], the effective moment of inertia +is ≈ 85% of the rigid-body moment of inertia at Ex ≈ 8 +MeV. We take this as the reference value for which we +will vary the spin cutoff parameter to obtain an estimate +for the systematic uncertainty connected to the spin dis- + +9 +tribution, with the effective moment of inertia ranging +from 70%−100% of the rigid-body moment of inertia: +σ2 +J(Sn) = η 0.0146A5/3 1 + +� +1 + 4a(Sn − E1) +2a +, +(13) +where η is the reduction factor set to 0.85(15), A is the +mass number of the nucleus (here 186), a is the level- +density parameter and E1 is an excitation-energy shift +taken from the global systematics of von Egidy and Bu- +curescu [56, 57] calculated with the robin.c code in the +Oslo-method software package (see Table I). This gives +us a range of values for the estimated ρ(Sn), which is +then calculated as [39, 42] +ρ(Sn) = +2σ2 +J +D0 +� +(It + 1)e−(It+1)2/2σ2 +J + Ite−I2 +t /2σ2 +J�, +(14) +assuming an equal parity distribution for all spins at +the neutron separation energy. Uncertainties in the D0 +value and the spin cutoff parameter are propagated (for a +derivation, see Appendix A). All the applied parameters +are given in Table I. +Moreover, due to the argument in the level density +function being Ei − Eγ, we get an upper limit for the +extracted level density given by Emax +x +−Emin +γ +. Therefore, +we need to make an extrapolation from our data points +up to ρ(Sn). Here, we use the constant-temperature (CT) +model of Ericson [59]: +ρCT(Ex) = 1 +T exp Ex − E0 +T +, +(15) +where T denotes the nuclear “temperature” and E0 is a +shift; both parameters are usually obtained from fits to +discrete data and to neutron resonance spacings. +The +parameters used for 186W are shown in Table I. +From the Oslo-method software, statistical uncertain- +ties and an estimate of systematic errors due to the un- +folding procedure and the first-generation method are +calculated as described in Ref. [42]. +We also include +systematic errors from the normalization procedure, ac- +counting for the uncertainty in the experimental D0 value +as well as the uncertainty in the moment of inertia and +thus the spin cutoff parameter as described above. We es- +timate the uncertainty (approximately one standard de- +viation) including all these factors as +δρ = ρrec +��δD0 +D0 +�2 ++ +�δσJ +σJ +�2 ++ +�∆ρrec +ρrec +�2 +, +(16) +where ρrec is the central value (“recommended” nor- +malization), and ∆ρrec represents statistical uncertain- +ties and systematic errors from unfolding and the first- +generation method. The resulting normalized level den- +sity is shown in Fig. 9. +D. +Normalization of γ-ray strength +Having the normalized level density at hand, we +proceed to normalizing the γ-ray transmission coeffi- +0 +1 +2 +3 +4 +5 +6 +7 + [MeV] +x +E +Excitation energy +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +] +-1 +) [MeV +x +E +( +ρ +Level density +W +186 + Oslo data, + stat.+sys. +σ + 1 + Known levels + CT interpolation +) +n +S +( +ρ + Estimated +FIG. 9. (Color online) Normalized level density of 186W. The +discrete levels [54] are binned with the same bin size as our +data (224 keV/channel). The dashed line shows the CT-model +interpolation between our data and ρ(Sn). The black error +bars represent statistical uncertainties from the experiment +and systematic errors connected to the unfolding procedure +and the first-generation method. The blue band includes also +systematic errors from the normalization procedure (see text). +0 +1 +2 +3 +4 +5 +6 +7 +8 + [MeV] +γ +E +-ray energy +γ +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +Transmission coeff. (arb. units) +W Oslo data, this work +186 + + low-energy extrapolation + high-energy extrapolation +FIG. 10. Gamma-ray transmission coefficient of 186W before +normalization. The arrows indicate the fit regions used for de- +termining the extrapolations (see text). The gray data points +are not considered further in the analysis due to very low +statistics in the first-generation matrix for these γ energies. +cient T (Eγ) by determining the scaling parameter B in +Eq. (12). Here we make use of the relation between the +average, total radiative width ⟨Γγ0⟩ deduced from s-wave +neutron resonances, the level density and the transmis- + +10 +TABLE I. Parameters used for the normalization of the level density and γ-ray transmission coefficient. Note that the E0 +parameter is adjusted to make ρCT(Sn) match with ρ(Sn) = 26.5·105 MeV−1. The uncertainty in ⟨Γγ0⟩ from Mughabghab [55] +is given as 5 meV; however, this uncertainty seems too small based on the experimental errors in the radiative width for other +W isotopes, and we have chosen a more conservative uncertainty in line with the experimental errors of 182,183,184,186W. +Sn +Iπ +t +D0 +σ2 +J(Sn) +a +E1 +ρ(Sn) +T +E0 +⟨Γγ0⟩ +σ2 +d +Ed +(MeV) +(eV) +(MeV−1) (MeV) 105 (MeV−1) (MeV) (MeV) (meV) +(MeV) +7.192 3/2− 9.3(16) +47(8) +19.38 +0.28 +26.5(64) +0.51(1) -0.0077 60+13 +−9 +7.3(13) 0.86(19) +sion coefficient [39, 60]: +⟨Γγ0⟩ = BD0 +4π +� Sn +Eγ=0 +dEγT (Eγ)ρ(Sn − Eγ)× +1 +� +J=−1 +[g(Sn − Eγ, It − 1/2 + J) + g(Sn − Eγ, It + 1/2 + J)] , +(17) +where g is the spin distribution [61, 62]: +g(Ex, J) ≃ 2J + 1 +2σ2 +J +exp +� +−(J + 1/2)2/2σ2 +J +� +. +(18) +As we need the spin distribution for the excitation-energy +range Ex ∈ [0, Sn], we make use of the spin cutoff param- +eter in the general form [63] +σ2 +J(Ex) = σ2 +d + Ex − Ed +Sn − Ed +� +σ2 +J(Sn) − σ2 +d +� +, +(19) +which is motivated also from microscopic calculations +(e.g., shell-model calculations [64] and the work of Uhren- +holt et al. [58]). Here, σ2 +d represents the spin cutoff pa- +rameter at the low excitation energy Ed, where the lev- +els are still resolved and with firm spin/parity assign- +ments [54], see Table I. +We need to estimate the γ-ray transmission coefficient +for Eγ < Emin +γ +, i.e., where we do not have experimen- +tal data, in order to calculate the integral in Eq. (17). +Therefore, we extrapolate with a fit to the low-energy +data points using the functional form E3 +γ exp(p1Eγ +p2), +where p1 and p2 are free parameters1. +Moreover, the +statistics is very low at high γ-ray energies, and so we +make use of an extrapolation here as well, here using a +simple exponential, exp(p3Eγ + p4), where p3 and p4 are +again free parameters. The fit regions and the extrapo- +lation functions are shown in Fig. 10. The data points in +gray color (Eγ > 6 MeV) are from a region in the first- +generation matrix with very low statistics (see Fig. 7c), +and we therefore choose to exclude those data points from +the further analysis. +To obtain the γ-ray strength function, we use the fact +that γ decay at high excitation energies is largely domi- +nated by dipole transitions (see, e.g., Refs. [67–69]). As +1 This functional form is motivated by shell-model calculations of +the low-energy γ strength, e.g. Refs. [65, 66]. +0 +1 +2 +3 +4 +5 +6 + [MeV] +γ +-ray energy E +γ +8 +− +10 +7 +− +10 +6 +− +10 +] +-3 +) [MeV +γ +-ray strength function f(E +γ +W +186 + Oslo data, + stat.+sys. +σ + 1 +FIG. 11. +(Color online) Gamma-ray strength function of +186W. The black error bars represent statistical uncertain- +ties from the experiment and systematic errors connected to +the unfolding procedure and the first-generation method. The +blue band includes also systematic errors from the normaliza- +tion procedure (see text). +our experimental data in principle contain transitions of +both electric and magnetic character, we get the total +dipole strength function f(Eγ) through +f(Eγ) = T (Eγ) +2πE3γ +. +(20) +In accordance with the approach for the level density, we +estimate the uncertainty in the γ-ray strength function +through +δf = frec +��δD0 +D0 +�2 ++ +�δσJ +σJ +�2 ++ +�δΓγ0 +Γγ0 +�2 ++ +�∆frec +frec +�2 +, +(21) +where ∆frec is again the central value (“recommended” +normalization), and ∆frec represents statistical uncer- +tainties and systematic errors from unfolding and the +first-generation method. The resulting, normalized γ-ray +strength function is shown in Fig. 11. + +11 +IV. +RESULTS AND DISCUSSION +A. +Comparison to other data and models +The level-density data are compared to various mod- +els available in the TALYS-1.9 code [18], see Fig. 12. +The models are: ldmodel 1, the composite formula of +Gilbert and Cameron [70]; ldmodel 2, the back-shifted +Fermi gas model [71]; ldmodel 3, the generalized super- +fluid model [72]; ldmodel 4, calculated within the Hartree- +Fock-BCS approach [73]; ldmodel 5, the combinatorial- +plus-Hartree-Fock-Bogoliubov approach [74]; +and ld- +model 6, the combinatorial model combined with a +temperature-dependent Hartree-Fock-Bogoliubov calcu- +lation [75]. +0 +1 +2 +3 +4 +5 +6 +7 + [MeV] +x +E +Excitation energy +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 +7 +10 +] +-1 +) [MeV +x +E +( +ρ +Level density +W +186 + Oslo data, + stat.+sys. +σ + 1 + Known levels + ldmodel 1 + ldmodel 2 + ldmodel 3 + ldmodel 4 + ldmodel 5 + ldmodel 6 +FIG. 12. (Color online) Comparison of the level-density data +from this work with models included in the TALYS code (see +text). +From a first look, none of the models seem to be in good +agreement with the data, and we remark that the TALYS +level densities have not been normalized to the D0 value +from Ref. [55]. In adition, we take notice of two impor- +tant issues: (i) the spin cutoff parameter we have used in +our normalization procedure might not be representative +of the corresponding spin distribution in the TALYS mod- +els; (ii) our data can be re-normalized more coherently +for each model by adopting its energy-dependence to ex- +trapolate between the highest energy point and ρ(Sn), as +was done e.g. in Ref. [76]. Nevertheless, it is clear that +the overall shape of our data points are significantly dif- +ferent from several of the level-density models. We also +remark that the slope of our level-density data points is +directly linked to the slope of the γ-strength function as +given in Eq. (12). If we were to renormalize our level +density to the TALYS models, this would inevitably lead +to a change in slope in the γ-strength function as well. +We now compare our γ-strength data from the (γ, n) +measurements and the OCL experiment to external data +found in the literature, as shown in Fig. 13a. We observe +a good agreement with the E1 strength extracted from +primary γ rays following neutron capture by Kopecky et +al [78], which brings further support to the absolute nor- +malization procedure. +Moreover, we compare our new +photoneutron data to several data sets found in the liter- +ature, where the photoneutron cross section σγn is con- +verted into dipole strength using the relation of Axel [79]: +fγn(Eγ) = +1 +3π2ℏ2c2 +σγn(Eγ) +Eγ +, +(22) +where σγn is in units of mb, Eγ in MeV, and the factor +1/(3π2ℏ2c2) = 8.674 · 10−8 mb−1MeV−2. Overall, there +is good agreement between the various data sets for the +W isotopes. +In Fig. 13a, we also compare the data with avail- +able models in TALYS: strength 1, the Generalized +Lorentzian [67]; strength 2, the Standard Lorentzian +(Brink-Axel model) [43, 44]; +strength 3, the Quasi- +Particle Random Phase Approximation (QRPA) on top +of a Hartree-Fock-plus-BCS calculation [80]; strength 4, +the QRPA on top of a Hartree-Fock-Bogoliubov (HFB) +calculations [81]; +strength 5, the Hybrid model [82] +with parameters from global systematics [18]; strength +6, QRPA as in Ref. [81] but on top of a temperature- +dependent HFB calculation [75]; and finally strength 7, +a relativistic mean-field calculation plus a continuum +QRPA calculation [83]. +Out of these models, strength +4 and strength 6 match reasonably well the present Oslo +data, but not the (γ, n) data. In general, the models are +deviating significantly from each other and from either +the Oslo data or the (γ, n) data. +To obtain a model description that can reproduce our +data reasonably well over the entire energy range, we +take a pragmatic approach and exploit phenomenologi- +cal models for the dipole strength. For the main part +of the E1 strength which is dominated by the Giant +Dipole Resonance (GDR), we apply the Hybrid model +of Goriely [82]: +f Hyb +E1 (Eγ, Tf) = +1 +3π2ℏ2c2 +EγσrΓrΓ(Eγ, Tf) +(E2γ − E2r)2 + E2γΓrΓ(Eγ, Tf), +(23) +where σr is the peak cross section, Er the centroid, and +Γr the width of the GDR. Further, the γ-energy and +temperature dependent width Γ(Eγ, Tf) is given by +Γ(Eγ, Tf) = 0.7 · Γr +E2 +γ + 4π2T 2 +f +EγEr +. +(24) +The temperature of the final levels, Tf, is here considered +as a constant, in line with the Brink-Axel hypothesis. We +also include extra E1 strength (labeled “E1 pygmy” in +Fig. 13b) to make a smooth connection between our data +and the (γ, n) data. +Finally, we also add a magnetic- +dipole component (marked “M1 spin-flip” in Fig. 13b). + +12 +0 +2 +4 +6 +8 +10 +12 +14 + [MeV] +γ +E +-ray energy +γ +8 +− +10 +7 +− +10 +6 +− +10 +] +-3 +) [MeV +γ +E +(f +-ray strength function +γ +(a) +W Oslo data, this work +186 + + stat.+sys. +σ + 1 +,n), this work +γ +W( +182 + +,n), this work +γ +W( +183 + +,n), this work +γ +W( +184 + +,n), Berman et al. +γ +W( +186 + +,n), Mohr et al. +γ +W( +186 + + +et al. +, Kopecky +E1 +f +W +184 + + +et al. +, Kopecky +M1 +f +W +184 + + strength1 + strength2 + strength3 + strength4 + strength5 + strength6 + strength7 +0 +2 +4 +6 +8 +10 +12 +14 + [MeV] +γ +E +-ray energy +γ +8 +− +10 +7 +− +10 +6 +− +10 + E1 hybrid + E1 pygmy + M1 spin-flip + total fit function +(b) +FIG. 13. (Color online) (a) Comparison of γ-strength data from this work with data from the literature (Berman et al. [77], +Mohr et al. [9], and Kopecky et al. [78]), and to models included in the TALYS code (see text); (b) Fit to the γ-ray strength +function data of 186W and the 184W data of Kopecky et al. [78]) (see text). +TABLE II. Parameters found from the model fits of ftot to the γ-strength data (see text). The uncertainties given are from +the fit only. Note that EM1 and ΓM1 are fixed. +Norm. +Er +Γr +σr +EPyg +ΓPyg +σPyg +Tf +EM1 +ΓM1 +σM1 +(MeV) (MeV) (mb) +(MeV) (MeV) (mb) +(MeV) (MeV) (MeV) (mb) +Rec. +12.9(1) 4.1(1) 382(2) 6.3(1) 2.6(2) 7.2(3) 0.43(3) 7.2 +2.5 +4.4(4) +For both the E1 pygmy and the M1 spin-flip contribu- +tions, we apply a resonance-like form using a Standard +Lorentzian: +fPyg,M1(Eγ) = +1 +3π2ℏ2c2 +σPyg,M1Γ2 +Pyg,M1Eγ +(E2γ − E2 +Pyg,M1)2 + Γ2 +Pyg,M1E2γ +(25) +where σPyg,M1, ΓPyg,M1, and EPyg,M1 are the peak cross +section, width, and centroid for the pygmy (Pyg) and +the spin-flip (M1) resonance, respectively. The total fit +function is then given by +ftot(Eγ) = f Hyb +E1 (Eγ, Tf = const.) + fPyg(Eγ) + fM1(Eγ). +(26) +For the fit, we first constrain the Hybrid component by +fitting only the Hybrid model to the GDR data (Mohr et +al. [9] and Berman et al. [77]) in the range Eγ = 7.7−14.5 +MeV. We choose to fix the Tf parameter to the one used +for the extrapolation of the level density (see Sec. III C) +to ease the fit, as Tf is largely determined from the γ- +strength function below neutron threshold. From this fit, +we determine the GDR parameters σr, Er, and Γr, to be +used as start values for the next fit including the data for +γ energies below neutron threshold as well. +For the spin-flip part, we use a fixed centroid EM1 +taken from systematics [63], and a fixed width of ΓM1 of +2.5 MeV. The peak cross section σM1 is then found from a +fit to the M1 data of 184W from Kopecky et al. [78]. Then +we make a fit using the full energy range Eγ = 1.0 − 14.5 +MeV, with only the spin-flip parameters fixed, and with +the first fit of the GDR data as starting values. In the fit, +we include the present OCL data of 186W, the E1 data +from Kopecky et al. [78] on 184W, and the GDR data +from Mohr et al. [9] and Berman et al. [77]. The result- +ing fit is shown in Fig. 13b, and the parameters are listed +in Table II. As this model fit will be used to calculate the +(n, γ) cross section and reactivity in the following sec- +tion, we repeat the fit for all the different normalizations +(varying D0, Γγ0, σJ and taking into account ∆f). All +fits are performed within the ROOT software tool [84] +using the Minuit package. +The resulting fit function gives a reasonable description +of the strength function data, although we note a poten- +tial issue in that the region between Eγ = 6 − 8 MeV +contains practically no data points for 186W. Moreover, +the 184W data points from primary transitions following +neutron capture typically have large fluctuations. Hence, +it is very difficult to assess the actual parameters for the +E1 pygmy, and the deduced parameters given in Table II + +13 +should be used with caution. +We also remark that the data points at the lowest γ +energies, Eγ < 1.5 MeV, might indicate some low-energy +increase in the γ-strength function, as first observed in +iron isotopes [85]. +However, in contrast to clear cases +like 56Fe [68, 69, 85], it is hard to conclude here as there +are only a few data points that might show an increas- +ing trend. We therefore choose not to include an extra +“upbend” component in the fit. +B. +Maxwellian-averaged cross section and reaction +rate +Using our level-density data and γ-strength function +data, we now calculate the Maxwellian-averaged cross +section (MACS) with the TALYS code, which is based on +the statistical model of Wolfenstein [86] and Hauser and +Feshbach [87]. The resulting MACS is shown in Fig. 14, +where we also show the TALYS MACS with default in- +puts (strength 1, ldmodel 1, a global optical-model po- +tential, and no upbend), and the variation of the MACS +as the different level-density and γ-strength models are +used. We have tested using the semi-microscopic optical- +model potential of Bauge et al. [88] for comparison with +the one of Koning and Delaroche [89]. +As seen from +Fig. 14 (dashed line versus dashed-dotted line), there +is only a minor difference between the two for neutron +energies around kBT = 30 keV, and overall the semi- +microscopic potential gives a lower MACS. Nevertheless, +the presented uncertainty band on our experimentally- +constrained MACS includes the variation between the +two different optical models in the lower uncertainty, in +addition to uncertainties from D0, Γγ0, and σJ. +In Fig. 14, we compare our result with the KADoNiS +database [90], and find agreement within the error bars, +although the KADoNiS values are overall larger than our +central values. We remark that the KADoNiS values are +from a weighted average of MACS constrained by pho- +tonuclear data above Sn, while our results include infor- +mation on both the level density as well as the γ-strength +function below Sn. +We have multiplied the KADoNiS +MACS values with their corresponding stellar enhance- +ment factor (SEF) as given in Ref. [90] for 185W(n, γ). +Furthermore, our estimated uncertainty band is smaller +than the KADoNiS uncertainties, Our result at kBT = 30 +keV, 508+76 +−106 mb, agrees well within error bars with the +MACS from Mohr et al. [9], 553(60) mb. On the other +hand, the evaluation of Bao et al. [91] of 703(113) mb, +and the measurement of Sonnabend et al. [8], 687(110) +mb, are both larger than our estimate, although still +within the estimated uncertainties. We note that none +of these values are directly measured, as Bao et al. gives +a purely theoretical prediction, while the MACS value +from Sonnabend et al. is constrained on (γ, n) data above +Sn. In comparison with the TALYS estimates using the +default input as well as the resulting MACS when vary- +ing the level-density and γ-strength models, our deduced +20 +40 +60 +80 +100 + [keV] +T +B +k +Energy +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +MACS [mb] + exp. constrained + KADoNiS-1.0 + Bao et al. (2000) + Sonnabend et al. (2003) + Mohr et al. (2004) + TALYS default + Variations, NLD +SF +γ + Variations, + Variation, n-OMP +FIG. 14. +(Color online) Maxwellian-averaged cross section +for the 185W(n, γ) reaction. The shaded band indicates the +present data-constrained MACS. The thick, azure dashed- +dotted line shows the TALYS result using default input, the +thin, azure dashed lines show the TALYS MACS when vary- +ing the level-density models, and the thin, cyan lines show +the variation due to different γ-strength models. +The dot- +ted line shows the deviation from the default when using the +optical-model potential of Bauge et al. [88]. +1 +− +10 +1 +Temperature [GK] +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +6 +10 +× +] +-1 + mol +-1 + s +3 + [cm +〉 +v +σ +〈 +A +N +exp. constrained + KADoNiS-1.0 + TALYS default + Variations, NLD +SF +γ + Variations, + Variation, n-OMP +FIG. 15. (Color online) Reaction rate for the 185W(n, γ) reac- +tion. The shaded band indicates the present data-constrained +result. See also the caption of Fig. 14. +MACS is in between the extremes. +In Fig. 15, we show the corresponding reaction rate +(stellar reactivity) deduced from our data compared to +the KADoNiS rate, the TALYS default and the variations +using different model inputs. +Again we find that the +KADoNiS values are overall higher than our estimated +rate, in particular for temperatures below 0.3 GK. + +14 +To address possible implications for the s process +and the Re/Os cosmochronometer in a reliable way, the +branch points at 186Re and 191Os should also be consid- +ered in realistic stellar models for thermally-pulsing AGB +stars. The 191Os MACS has been estimated by a similar +procedure as in this work by Kullmann et al. [92]. The +186Re MACS remains to be experimentally constrained +in the same way; the 186W(α, dγ)187Re data from this +same experiment is currently being analyzed. With this +experimentally-constrained MACS also at hand, we in- +tend to perform a consistent study of the s process in +this mass region. +V. +SUMMARY AND OUTLOOK +In this work, we have performed photoneutron cross +section measurements on the 182,183,184W isotopes. This +completes the photoneutron measurements on the stable +W isotopic chain. Furthermore, we have presented data +on the 186W(α, α′γ) reaction, and used the extracted +level density and γ-ray strength function to provide an +experimentally constrained (n, γ) cross section for the +branch-point nucleus 185W. +In comparison with other data and the recommended +MACS from the KADoNiS data base, we find that our +estimated MACS and reaction rate are lower than most +of the other available values, except for the result of Mohr +et al. Our reaction rate could possibly impact the s pro- +cess in this mass region, in particular the deduced neu- +tron density and the calculation of the 186Os abundance. +When the 186Re MACS also becomes available, we intend +to perform a systematic study of the s-process conditions +in the W-Re-Os region in the near future. +ACKNOWLEDGMENTS +The authors would like to thank J. C. M¨uller, +P. A. Sobas, and J. C. Wikne at the Oslo Cyclotron Labo- +ratory for operating the cyclotron and providing excellent +experimental conditions. We sincerely thank T. W. Ha- +gen, S. J. Rose and F. Zeiser for helping with the OCL +experiment, Y.-W. Lui for helping with the NewSUB- +ARU experiments, and S. N. Liddick for inspiring dis- +cussions. +A. C. L. gratefully acknowledges funding of +this research by the European Research Council through +ERC-STG-2014 under grant agreement no. 637686, and +from the Research Council of Norway, project grant no. +316116. S. G. acknowledges the support from the F.R.S.- +FNRS. This work was supported in part by the National +Science Foundation under Grant No. +OISE-1927130 +(IReNA). The photoneutron cross section measurement +was performed as part of the IAEA CRP on “Updating +the Photonuclear Data Library and generating a Refer- +ence Database for Photon Strength Functions” (F41032). +A. G., V. W. I., and S. S. gratefully acknowledge fi- +nancial support from the Research Council of Norway, +project grant no. 325714. This work is in part based on +the research supported partly by the National Research +Foundation of South Africa (Grant Number: 118846). +Appendix A: Uncertainty in ρ(Sn) +To estimate the total NLD at the neutron separation +energy using Eq. (14), we propagate errors from the D0 +value and the spin cutoff parameter σJ(Sn) assuming +that they are independent variables, which is a justified +assumption. Thus, we get that +�δρ(Sn) +ρ(Sn) +�2 += +�δD0 +D0 +�2 ++ +�δξ(σJ(Sn)) +ξ(σJ(Sn)) +�2 +, +(A1) +where ξ represents the function containing the depen- +dency on the spin cutoff parameter σJ at the neutron +separation energy Sn: +ξ(σJ) = +2σ2 +J +Ite−I2 +t /2σ2 +J + (It + 1)e−(It+1)2/2σ2 +J . +(A2) +Now we take the derivative of ξ with respect to σJ and +obtain: +δξ +δσJ += +4σJ +� +Ite−I2 +t /2σ2 +J + (It + 1)e−(It+1)2/2σ2 +J +� +− +2 +σJ +� +I3 +t e−I2 +t /2σ2 +J + (It + 1)3e−(It+1)2/2σ2 +J +� +� +Ite−I2 +t /2σ2 +J + (It + 1)e−(It+1)2/2σ2 +J�2 . +(A3) +For convenience, we now define the auxilliary functions +z1 ≡ I3 +t e−I2 +t /2σ2 +J + (It + 1)3e−(It+1)2/2σ2 +J, +z2 ≡ Ite−I2 +t /2σ2 +J + (It + 1)e−(It+1)2/2σ2 +J. +Using these and dividing Eq. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' N-0316 Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Norway 2Expert Analytics AS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' N-0179 Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Norway 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Konan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Okamoto 8-9-1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Higashinada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Kobe 658-8501,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Japan 4Shanghai Advanced Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Shanghai 201210,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' China 5Department of Metallurgical and Materials Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Pamukkale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 20160 Denizli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Turkey 6Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Eskisehir Osmangazi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 26480 Eskisehir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Turkey 7Institut d’Astronomie et d’Astrophysique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Universit´e Libre de Bruxelles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Campus de la Plaine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' CP-226,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 1050 Brussels,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Belgium 8iThemba LABS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Box 722,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7129 Somerset West,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' South Africa 9Department of Applied Physics and Engineering Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' University of Johannesburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Johannesburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 2028,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' South Africa 10Institute d’Astronomie et d’Astrophysique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Universit´e Libre de Bruxelles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Belgium 11Department of Nuclear Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' USA 12Laboratory of Advanced Science and Technology for Industry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' University of Hyogo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 3-1-2 Kouto,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Kamigori,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Ako-gun,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Hyogo 678-1205,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Japan 13Institute of Theoretical Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' University of Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' N-0316 Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Norway 14Department of Mathematics and Cybernetics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' SINTEF Digital,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' N-0314 Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Norway 15Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Michigan State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' East Lansing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Michigan 48824,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' USA 16National Superconducting Cyclotron Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Michigan State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' East Lansing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Michigan 48824,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' USA 17Joint Institute for Nuclear Astrophysics Center for the Evolution of the Elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Indiana 46556,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' USA 18School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' University of the Witwatersrand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Johannesburg 2050,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' South Africa (Dated: February 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 2023) In this work,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' we present new data on the 182,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='183,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='184W(γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' n) cross sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' utilizing a quasi- monochromatic photon beam produced at the NewSUBARU synchrotron radiation facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Further, we have extracted the nuclear level density and γ-ray strength function of 186W from data on the 186W(α, α′γ)186W reaction measured at the Oslo Cyclotron Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Combining previous measurements on the 186W(γ, n) cross section with our new 182,183,184W(γ, n) and (α, α′γ)186W data sets, we have deduced the 186W γ-ray strength function in the range of 1 < Eγ < 6 MeV and 7 < Eγ < 14 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Our data are used to extract the level density and γ-ray strength functions needed as input to the nuclear-reaction code TALYS, providing an indirect, experimental constraint for the 185W(n, γ)186W cross section and reaction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Compared to the recommended Maxwellian-averaged cross section (MACS) in the KADoNiS-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0 data base, our results are on average lower for the relevant energy range kBT ∈ [5, 100] keV, and we provide a smaller uncertainty for the MACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The theoretical values of Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' and the cross section experimentally constrained on photoneutron data of Sonnabend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' are significantly higher than our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The lower value by Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' is in very good agreement with our deduced MACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Our new results could have implications for the s-process and in particular the predicted s-process production of 186,187Os nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' INTRODUCTION Neutron-capture reactions are known to be the main producers of elements heavier than iron in our Uni- verse [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The rapid (r) and the slow (s) neutron- capture processes are traditionally believed to account for almost 100% of the Solar-system heavy-element abun- ∗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='larsen@fys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='uio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='no † gry@xal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='no dances [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The r process takes place in an environment with an extremely high neutron density typicallly larger than 1024 neutrons/cm3, which produces very neutron- rich nuclei within a short time window (≈1s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In contrast, the s process is, as the name implies, a slow process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' the neutron density is comparatively low (∼ 106 − 108 neutrons/cm3 in asymptotic giant branch stars [5]) and it can take from days to thousands of years between each neutron-capture reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Consequently, the s-process “path” in the nuclear chart remains close to the valley of stability, as the β-decay rates are typically much faster than the (n, γ) rates when an unstable nucleus is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='13301v1 [nucl-ex] 30 Jan 2023 2 108 109 110 111 112 113 114 115 116 Neutron number 73 74 75 76 77 78 Proton number W 182 W 183 W 184 W 185 W 186 W 187 Re 185 Re 186 Re 187 Re 188 Os 186 Os 187 Os 188 Os 189 Os 190 Os 191 Os 192 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Schematic illustration of the nuclear chart in the W-Re-Os region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The black arrows indicate (n, γ) reactions on stable or near-stable isotopes, the blue dashed arrows show the possible (n, γ) branch on the long-lived W, Re and Os isotopes, while the pink arrows display the β− decay branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' However, this is not true for some particular nuclei along the s-process path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' At the branch points [6] the β-decay rate is comparable to the (n, γ) rate, so that there is a non-negligible possibility for the nucleus to ei- ther undergo β-decay or capture another neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' On the one hand, such branch points could complicate the s-process nucleosynthesis calculation significantly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' on the other hand, they may provide valuable information about the neutron density and/or temperature at the astro- physical site for which the s process operates [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In this work, we focus on the branch-point nucleus 185W, with a laboratory half-life of 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='1(3) days [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This nucleus is of interest for the Re/Os cosmochronol- ogy first discussed by Clayton [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The main idea behind the Re/Os cosmochronology is the following: the mat- ter from which the Solar system was formed, contained a given amount of 187Re and 187Os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Further, 187Re is usually assigned a pure r-process origin, while 187Os is produced only in the s process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As 187Re has a very long half-life of more than 4·1010 years [12], Clayton suggested to use the solar-system amount of 187Re and 187Os as a “clock”, which would display the time span for which nucleosynthesis events produced various elements up to the time of the formation of our Solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Provided that the 187Os amount stemming from the s process can be reliably calculated, the extra amount of 187Os origi- nates from the 187Re decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Thus, at least in principle, the abundances of the parent/child pair 187Re/187Os can be used as a cosmochronometer, although not without complications [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [7, 14–16], the branchings at 185W and 186Re (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 1) could well have a non-negligible impact on this cosmochronometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, several authors [7–9, 17] have discussed the 185W and 186Re branchings as a “neutron dosimeter” for the effective s-process neutron density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' this application again depends on the radiative neutron-capture cross sec- tions of 185W and 186Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' No direct measurement of the neutron-capture cross section is possible on these target nuclei, and only constraints on the electromagnetic de- cay of the compound system have been obtained through photoneutron experiments at relatively high photon en- ergies [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Here we present new photoneutron data on 182,183,184W that complete the (γ, n) measurements on the W isotopes (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' III, we present the 186W(α, α′γ) data taken at the Oslo Cyclotron Laboratory, and the data analysis with the resulting level density and γ strength function of 186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Using our new data to constrain the input to the nuclear reaction code TALYS-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9 [18] we estimate the 185W(n, γ)186W Maxwellian-averaged cross section and reaction rate, and compare to previous measurements and evaluations in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Finally, we give a summary and outlook in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' THE (γ, n) EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Experimental details The photo-neutron measurements on 182,183,184W took place at the NewSUBARU synchrotron radiation facil- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Figure 2 shows a schematic illustration of the γ- ray beam line and experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Beams of γ rays were produced through laser Compton scattering (LCS) of 1064 nm photons in head-on collisions with relativistic 3 Collision point Collision point P1 (Nd laser) P2 (CO2 laser) Collimator C1 C2 Lasers: Nd(w, 2w), CO2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9 m 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 m 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 m GACKO g - ray Gamma hutch-2 Gamma hutch-1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) A schematic illustration of the experimental set up at NewSUBARU used in the (γ, n) cross-section measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' electrons at the most-efficient collision point P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The γ beams were collimated using the Pb collimators C1 and C2, each 10 cm long, with 3 mm and 2 mm apertures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The beam profile on target nearly follows the geometrical aperture of the collimator C2 with re- spect to the collision point P1, thus avoiding any interac- tion between beam and other materials than the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Throughout the experiment, the laser was periodically on for 80 ms and off for 20 ms, in order to measure background neutrons and γ-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In this experiment, the beams produced had an energy resolution ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='6 MeV to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9 MeV (full-width at half maximum, FWHM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The electrons were injected from a linear accelerator into the NewSUBARU storage ring with an initial energy of 974 MeV, then subsequently decelerated to nominal energies ranging from 608 MeV to 849 MeV, providing LCS γ-ray beams of energies up to 13 MeV and down to the neutron separation energies of the W isotopes (thus varied for each individual case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The maximum γ-ray energy of the beams was increased in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='25 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The electron beam energy has been calibrated with the accuracy on the order of 10−5 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The energy is re- produced in every injection of an electron beam from a linear accelerator to the storage ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The reproducibil- ity of the electron energy is assured in the deceleration down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 GeV by an automated control of the electron beam-optics parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The energy profiles of the produced γ-ray beams were measured with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' LaBr3(Ce) (LaBr3) de- tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The measured LaBr3 spectra were reproduced by a Geant4 code [20–23] that incorporated the kinematics of the LCS process, including the beam emittance and the interactions between the LCS beam and the LaBr3 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In this way we were routinely able to sim- ulate the energy profile of the incoming γ beams with the maximum energies accurately determined by the cal- ibrated electron beam energy by best reproducing the LaBr3 spectra [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The W targets were made from isotopically enriched tungsten as metallic powder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The material was pressed together and enclosed in an aluminium cylinder with a thin cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The targets had areal densities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='7421 g/cm2 (182W), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='754 g/cm2 (183W), and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='7925 g/cm2 (184W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Due to the presence of the Al cap, we limited the γ-ray beam energy maximum 13 MeV to avoid getting contam- inant neutrons from 27Al (Sn=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='056 MeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To measure the emitted neutrons, a high-efficiency 4π detector was used, consisting of 20 3He proportional counters, arranged in three concentric rings and embed- ded in a 36 × 36 × 50 cm3 polyethylene neutron modera- tor [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The ring ratio technique, originally developed by Berman and Fultz [27], was used to determine the aver- age energy of the neutrons from the (γ, n) reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The efficiency of the neutron detector varies with the average neutron energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The efficiency was measured with a cal- ibrated 252Cf source with the emission rate of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='27 · 104 s−1 with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2% uncertainty, and the energy dependence was determined by Monte Carlo simulations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The efficiency of the neutron detector was simulated using isotropically distributed, mono-energetic neutrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Once the neutron detection efficiency for a given beam energy has been determined, we were able to deduce the number of (γ, n) reactions that took place during each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The LCS γ-ray flux was monitored by a 8in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='×12in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' NaI(Tl) (NaI) detector during neutron measurement runs with 100% detection efficiency for the beam energies used in this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The number of incoming γ rays per measurement was determined using the pile-up and Poisson-fitting technique described in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Analysis The measured photo-neutron cross section for an in- coming beam with maximum γ energy Emax is given by 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) The simulated energy profiles for the γ beams used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The distributions (integrated over all Eγ) are normalized to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' the convoluted cross section, σEmax exp = � Emax Sn DEmax(Eγ)σ(Eγ)dEγ = Nn NtNγξϵng .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (1) Here, DEmax is the normalized energy distribution ( � Emax Sn DEmaxdEγ = 1) of the γ-ray beam obtained from Geant4 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Examples of the simulated γ-beam profiles, DEmax, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Furthermore, σ(Eγ) is the true photo-neutron cross section as a function of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The quantity Nn represents the number of neu- trons detected, Nt gives the number of target nuclei per unit area, Nγ is the number of γ rays incident on tar- get, ϵn represents the neutron detection efficiency, and finally ξ = (1 − e−µt)/(µt) gives a correction factor for self-attenuation in the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The factor g represents the fraction of the γ flux above Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We have determined the convoluted cross sections σEmax exp given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (1) for γ beams with maximum en- ergies in the range Sn ≤ Emax ≤ 13 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The convo- luted cross sections σEmax exp are not connected to a specific Eγ, and we choose to plot them as a function of Emax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The convoluted cross sections of the three W isotopes, which are often called monochromatic cross sections, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The error bars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 4 represent the total uncertainty in the quantities comprising Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (1), and consists of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2% from the efficiency determination of the neutron detector, ∼ 1% from the pile-up method that gives the number of γ rays, and the statistical un- certainty in the number of detected neutrons [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The statistical uncertainty ranges between ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0 % close to neutron threshold and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='4 % for the highest maximum γ-ray beam energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The systematic error is dominated by the uncertainty from the pile-up method and from the simulated efficiency of the neutron detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For the to- tal uncertainty, we have added these uncorrelated errors in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' By approximating the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (1) with a sum for FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Monochromatic cross sections of 182,183,184W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The error bars contain statistical uncertainties from the number of detected neutrons, the uncertainty in the efficiency of the neutron detector and the uncertainly in the pile-up method used to determine the integrated γ-flux on target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' each γ-beam profile, we are able to express the unfolding problem as a set of linear equations σf = Dσ, (2) where σf is the cross section folded with the beam pro- file D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The indexes i and j of the matrix element Dij corresponds to Emax and Eγ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The set of equations is given by � � � � � σ1 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' σN � � � � � f = � � � � � D11 D12 · · · · · D1M D21 D22 · · · · · D2M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' DN1 DN2 · · · · · · DNM � � � � � � � � � � � � � σ1 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' σM � � � � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (3) Each row of D corresponds to a Geant4 simulated γ beam profile belonging to a specific measurement characterized by Emax (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 3 for a visual representation of some of the rows in the response matrix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' It is clear that D is highly asymmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The number of γ-ray beam energies used to study the cross section is much lower than the bin size (10 keV) of the simulated beam profiles above Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As the system of linear equations in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (3) is under-determined, the true σ vector cannot be extracted by matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In order to find σ, we utilize a folding iteration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The main features of this method are as follows [31]: 1) As a starting point, we choose for the 0th iteration, a constant trial function σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This initial vector is multiplied with D, and we get the 0th folded vector σ0 f = Dσ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 2) The next trial input function, σ1, can be estab- lished by adding the difference of the experimen- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='02 Relative intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='005 6 8 10 12 14 E, [MeV]350 W-182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' before deconvolution 300 W-183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' before deconvolution 250 W-184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' before deconvolution [mb] 200 ouo 150 b 100 50 8 10 11 12 13 9 x [MeV]5 tally measured spectrum, σexp, and the folded spec- trum, σ0 f , to σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In order to be able to add the folded and the input vector together, we first per- form a Piecewise Cubic Hermite Interpolating Poly- nomial (pchip) interpolation on the folded vector so that the two vectors have equal dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Our new input vector is: σ1 = σ0 + (σexp − σ0 f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (4) 3) The steps 1) and 2) are iterated i times giving σi f = Dσi (5) σi+1 = σi + (σexp − σi f) (6) until convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This means that σi+1 f ≈ σexp within the statistical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In order to quantitatively check convergence, we calculate the reduced χ2 of σi+1 f and σexp after each iter- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Approximately four iterations are usually enough for convergence, which is defined when the reduced χ2 value approaches ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We stopped iterating when the χ2 became lower than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In principle, the iteration could continue until the reduced χ2 approaches zero, but that results in large un- realistic fluctuations in σi due to over-fitting to the mea- sured points σexp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We estimate the total uncertainty in the unfolded cross sections by calculating an upper limit of the monochro- matic cross sections from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 4 by adding and subtract- ing the errors to the measured cross section values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' These upper and lower limits are then unfolded separately, re- sulting in the unfolded cross sections shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 5, the unfolded cross sections for 182,183,184W are evaluated at the maximum energies of the incom- ing γ beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The error bars represent the statistical errors and the systematic error due to the uncertainty in the absolute efficiency calibration of the neutron detec- tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The results are compared to data on 182,184W from Goryachev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [32], and the agreement is overall quite reasonable although some local discrepancies can be ob- served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' These discrepancies are sometimes not within the given uncertainties, and could be due to unknown systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' THE OSLO EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Experimental details The 186W(α, α′γ) inelastic-scattering experiment was performed at the Oslo Cyclotron Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A fully- ionized 30-MeV α beam was delivered by the MC-35 Scanditronix cyclotron and directed to the 186W target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The radio frequency was set to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='76 MHz, giving a beam burst every 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='09 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The experiment was run for about FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Cross sections of 182,183,184W obtained after deconvolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Also shown are cross sections of 182,184W from Goryachev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' eight days with typical beam intensities of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2 enA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The target was mounted on a 24-µm carbon backing, and the target thickness was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='31 mg/cm2 with enrichment > 98% in 186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To detect the outgoing charged particles, we used the Silicon Ring (SiRi) [33] placed in backward angles with respect to the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' SiRi is a ∆E-E telescope array consisting of eight 1550-µm thick back (E) detec- tors, each of which has a 130-µm thick front (∆E) de- tector divided in eight strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5-µm thick Al foil was placed in front of SiRi to reduce the amount δ elec- trons from the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' SiRi covers about 6% of 4π and the strips have an angular resolution of about 2◦, where the center of the strip is at 126 − 140◦ (in steps of 2◦);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' measured from the center of the front detector (at 133◦), the distance of SiRi from the center of the target was 5 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The ∆E-E telescopes allow for separating different charged-particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Figure 6a shows the measured protons, deuterons, tritons, and α particles for a strip at 130◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To select the 186W(α, α′) events, a gate was set on the “banana” corresponding to the α particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To cal- ibrate the SiRi front and back detectors, we used range calculations for our setup with the Qkinz code [34], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The resolution of the α particles was measured to be 330–360 keV FWHM for the peak of the elastically- scattered α particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The relatively poor resolution is mainly due to a rather elongated beam spot on the tar- get (≈ 3–4 mm in diameter in the vertical direction, and ≈ 1 mm in the horizontal direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The master-gate signal for the data acquisition system was a logical signal of 2µs generated when an E detector gave a signal above threshold, which was set to ≈ 200 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Using the CACTUS array [35], we measured γ rays in coincidence with the inelastic scattered α particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In the configuration used for this experiment, CAC- TUS consisted of 26 NaI(Tl) crystals of cylindrical shape 500 T W-182, NewSUBARU 450 W-183,NewSUBARU 400 W-184, NewSUBARU W-184, Goryachev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 350 W-182, Goryachev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=" Q [qw] (u'l)o 300 250 200 150 100 50 0 8 10 6 12 14 16 E, [MeV]6 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) (a) Particle-identification spectrum for one of the front strips at 130◦ with its corresponding back detector (∆E–E banana plot);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (b) a zoom on the α-particle banana with the Qkinz calculations used for calibration (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (5in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='× 5in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' All crystals were collimated with lead col- limators and had 2-mm thick Cu shields in front to at- tenuate X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The NaI(Tl) detectors were mounted on the spherical CACTUS frame, so that the front end of each crystal was positioned 22 cm from the center of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The efficiency of CACTUS (for 26 NaI(Tl) detec- tors) is 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='1(2)% as measured with a 60Co source, and with a resolution of ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='8% FWHM for Eγ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='33 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Using analog electronics, we obtained a lower threshold of about 350 keV for the NaI(Tl) detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The CACTUS detectors were calibrated in energy by gating on the protons in SiRi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As the target had a significant contamination of carbon (from the backing) and oxygen, we used peaks in the proton spectrum from the 12C(α, pγ)15N and 16O(α, pγ)19F reactions to further identify γ rays for calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In particular, we used the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='269-MeV transition from the 5/2+ first-excited level in 15N together with the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='868-MeV transition from the 13/2+ level at Ex = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='648 MeV in 19F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Then we cross- checked the obtained calibration with the 1235-keV and 2583-keV lines of 19F, in addition to the 511-keV γ ray from positron annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To obtain α–γ coincident events, we applied a gate on the time-to-digital converter (TDC) spectra for the prompt peak, and subtracting randomly correlated events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The start of the TDCs is given by the master gate, and the stop signal is generated from the NaI(Tl) detectors (each NaI(Tl) has an individual TDC), with a built-in delay from the Mesytec shapers of ≈ 400 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The range of the TDCs was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The gate on the prompt peak was set to ∆t = 0 ± 20 ns, while the gate for the background subtraction was set to ∆t = 135 ± 20 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Using the reaction kinematics, we determined the ini- tial excitation energy of the residual nucleus from the deposited energy of the α particles in SiRi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Applying the time gates for the γ rays, we obtained excitation-energy tagged, background-subtracted γ-ray spectra as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The γ-ray spectra needed to be corrected for the CAC- TUS detector response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For this purpose, we applied the iterative unfolding method of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [36] available in the Oslo-method software package [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This method takes the raw γ-ray spectrum as a starting point for the un- folded (“true”) spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This trial spectrum is folded with the known detector response, and then compared with the raw spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' By taking the difference between the folded spectrum and the raw spectrum, a new, im- proved trial spectrum is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This process is repeated until the folded spectrum is approximately equal to the raw spectrum, within the experimental uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To preserve the experimental statistical fluctuations, and not introduce artificial, spurious ones, the Compton sub- traction method is also applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This takes advantage of the fact that the Compton distribution is very smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For more details, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The unfolded γ-ray spec- tra for each Ex bin are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' After unfolding, the first-generation γ rays were ex- tracted from the data by applying an iterative subtrac- tion method [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The first-generation γ rays are the ones that are emitted first in the decay cascades, and their dis- tribution represents the branching ratios for the various γ transitions at a given Ex bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The principle behind the subtraction method is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For a given Ex bin, say, at Ex = 4 MeV, the unfolded spectrum contains all the γ rays from all the possible decay cascades originat- ing from the levels populated in that Ex bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' If we now consider the Ex bins below Ex = 4 MeV, they will con- tain all the same γ rays as the Ex = 4 MeV bin, except the first-generation γs at Ex = 4 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This is true if the Ex bins have the same decay cascades whether the levels in the bin were populated directly through the nu- clear reaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' or if they were populated from γ decay of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='14000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='TTTT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='[ke] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='(a) θ = 130° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='Qkinz calculations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='detector [ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='12000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='二 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='10000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='8500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='deposited in △E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='102 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Excitation-energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' γ-ray energy matrices of 186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (a) Background-subtracted data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (b) unfolded γ-ray spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (c) first-generation γ-ray spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The lines indicate the limits set for the further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' above-lying levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [39] for a more in-depth discussion on the assumptions behind the first-generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The first-generation γ spectra are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Extraction of level density and γ-ray transmission coefficient We now exploit the fact that the first-generation γ spectra represent the (relative) branching ratios for a given initial excitation-energy bin, and that we have many such branching ratios available for a large Ex re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In the spirit of Fermi’s Golden Rule [40, 41], where the decay rate is proportional to the level density at the final excitation energy and the reduced transition proba- bility for decay between a given initial and final level, we use the following ansatz [42]: P(Eγ, Ex) ∝ ρ(Ex − Eγ) · T (Eγ), (7) where P(Eγ, Ex) is the matrix of first-generation γ rays (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7c), ρ(Ex − Eγ) is the level density at the excita- tion energy where the γ transition “lands” and T (Eγ) is the γ-ray transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Note that T (Eγ) is only a function of Eγ, which means that the Brink-Axel hypothesis [43, 44] is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Brink stated that “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='we assume that the energy dependence of the photo effect is independent of the detailed structure of the initial state so that, if it were possible to perform the photo effect on an ex- cited state, the cross section for absorption of a photon of energy E would still have an energy dependence given by (15).” where “(15)” is referring to the equation describing the Giant Dipole Resonance (GDR) with a Lorentzian function that only depends on the γ-transition energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Brink’s original formulation (as well as Axel’s application of Brink’s hypothesis) concerned only E1 transitions, and there is a wealth of recent works in the literature dis- cussing the validity and/or violation of the hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [45–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A necessary condition for the Oslo method is that the Brink hypothesis is at least approximately true for the specific excitation-energy region used for extracting the level density and γ-ray transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We have performed tests of this assumption for the application in the Oslo method in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' When the Brink hypothesis is applicable, we can fit the data of the first-generation γ rays to obtain a reliable estimate of the level density and the γ-ray transmission coefficient through an iterative optimization using a least-squares fit: χ2 red = 1 Nfree Emax x� Ei=Emin x Ei � Eγ=Emin γ [P(Eγ, Ei) − Pth(Eγ, Ei)]2 [∆P(Eγ, Ei)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (8) Here, P(Eγ, Ei) is the experimental matrix of first- generation γ rays where each row is normalized to unity: Ei � Ei=Emin γ P(Eγ, Ei) = 1, (9) and ∆P(Eγ, Ei) is the uncertainties in the first- generation matrix (including statistical errors and an es- timate for systematic uncertainties due to unfolding and the first-generation method, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, Nfree is the number of degrees of freedom and Pth(Eγ, Ei) is the approximation for the theoretical first-generation matrix [42]: Pth(Eγ, Ei) = ρ(Ei − Eγ)T (Eγ) �Ei Eγ=Emin γ ρ(Ei − Eγ)T (Eγ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (10) 8 0 1 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='14 Probability distribution = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='14 MeV x E (a) first-gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' data T x ρ 0 1 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='14 Probability distribution = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='26 MeV x E (d) 0 1 2 3 4 5 6 7 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='59 MeV x E (b) 0 1 2 3 4 5 6 7 [MeV] γ E energy γ Probability distribution = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='70 MeV x E (e) 0 1 2 3 4 5 6 7 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='81 MeV x E (c) 0 1 2 3 4 5 6 7 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='16 MeV x E (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Experimental first-generation spectra (black crosses) compared to the predicted ones using the extracted level density and γ-transmission coefficient (blue line) for various excitation-energy bins (224-keV wide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The number of degrees of freedom, Nfree, is given by Nfree = Nch(P) − Nch(ρ) − Nch(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For the present data set, we have used Emin γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='90 MeV, Emin x = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0 MeV, and Emax x = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2 MeV as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Note that the neutron separation energy Sn of 186W is 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='1920(12) MeV [54], and as we have no way of discriminating against neutrons, the Oslo method is usually limited to a maximum excitation energy (close to) Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' With bin size of 224 keV, and the limits applied as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7c, we have the number of pixels in the first-generation ma- trix Nch(P) = 330, while the number of elements in the vectors of ρ and T is Nch(ρ) = Nch(T ) = 39, giving Nfree = 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' It is important to note that the number of data points in the first-generation matrix, Nch(P), is much bigger than the number of points to be estimated, which is 2 × 39 points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' this is why the method usually converges very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' When convergence is reached, the extracted ρ(Ex − Eγ) and T (Eγ) are the ones that best describe the experimental P(Eγ, Ei) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For this case, we obtain χ2 red = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='85 after 20 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As a visual illustration of the fit, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 8 shows some of the experimental first-generation spectra together with the spectra obtained for Pth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Overall, the agreement is quite good, although we remark that the experimental errors are rather large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Note that the fit is performed on all the first-generation spectra (for 15 excitation-energy bins), and so the fit is still well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Schiller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' showed [42] that the χ2 minimization obtains a unique solution for the relative variation of neighboring points in the functions ρ and T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' however, an equally good fit to the experimental P matrix is given by the transformation ˜ρ(Ei − Eγ) = A exp[α(Ei − Eγ)] ρ(Ei − Eγ), (11) ˜T (Eγ) = B exp(αEγ)T (Eγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (12) Here, α is the common slope adjustment of ρ and T , while A and B gives the absolute scaling of ρ and T , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' These parameters must be determined from external data, as described in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Normalization of level density To normalize the level density by determining the α and A parameters, we make use of discrete levels [54] at low Ex and data on s-wave neutron resonance spac- ings [55] at the neutron separation energy Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The av- erage s-wave neutron resonance spacing D0 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='3(16) eV [55] represents the spacing of levels with Jπ = 1−, 2− as the target nucleus 185W has ground-state spin/parity Iπ t = 3 2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To obtain the total level density at Sn, we need to apply a model for the spin distribution, in par- ticular the spin cutoff parameter σJ(Ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Here, we use as a starting point the model of von Egidy and Bu- curescu [56, 57] employing the rigid-body moment of in- ertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' However, as shown by Uhrenholt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [58], at excitation energies around 7−8 MeV for heavy nuclei, a full rigid-body moment of inertia might not be reached yet: in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 10 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [58], the effective moment of inertia is ≈ 85% of the rigid-body moment of inertia at Ex ≈ 8 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We take this as the reference value for which we will vary the spin cutoff parameter to obtain an estimate for the systematic uncertainty connected to the spin dis- 9 tribution, with the effective moment of inertia ranging from 70%−100% of the rigid-body moment of inertia: σ2 J(Sn) = η 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0146A5/3 1 + � 1 + 4a(Sn − E1) 2a , (13) where η is the reduction factor set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='85(15), A is the mass number of the nucleus (here 186), a is the level- density parameter and E1 is an excitation-energy shift taken from the global systematics of von Egidy and Bu- curescu [56, 57] calculated with the robin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='c code in the Oslo-method software package (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This gives us a range of values for the estimated ρ(Sn), which is then calculated as [39, 42] ρ(Sn) = 2σ2 J D0 � (It + 1)e−(It+1)2/2σ2 J + Ite−I2 t /2σ2 J�, (14) assuming an equal parity distribution for all spins at the neutron separation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Uncertainties in the D0 value and the spin cutoff parameter are propagated (for a derivation, see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' All the applied parameters are given in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, due to the argument in the level density function being Ei − Eγ, we get an upper limit for the extracted level density given by Emax x −Emin γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Therefore, we need to make an extrapolation from our data points up to ρ(Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Here, we use the constant-temperature (CT) model of Ericson [59]: ρCT(Ex) = 1 T exp Ex − E0 T , (15) where T denotes the nuclear “temperature” and E0 is a shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' both parameters are usually obtained from fits to discrete data and to neutron resonance spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The parameters used for 186W are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' From the Oslo-method software, statistical uncertain- ties and an estimate of systematic errors due to the un- folding procedure and the first-generation method are calculated as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We also include systematic errors from the normalization procedure, ac- counting for the uncertainty in the experimental D0 value as well as the uncertainty in the moment of inertia and thus the spin cutoff parameter as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We es- timate the uncertainty (approximately one standard de- viation) including all these factors as δρ = ρrec ��δD0 D0 �2 + �δσJ σJ �2 + �∆ρrec ρrec �2 , (16) where ρrec is the central value (“recommended” nor- malization), and ∆ρrec represents statistical uncertain- ties and systematic errors from unfolding and the first- generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The resulting normalized level den- sity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Normalization of γ-ray strength Having the normalized level density at hand, we proceed to normalizing the γ-ray transmission coeffi- 0 1 2 3 4 5 6 7 [MeV] x E Excitation energy 1 10 2 10 3 10 4 10 5 10 6 10 7 10 ] 1 ) [MeV x E ( ρ Level density W 186 Oslo data, stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='+sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' σ 1 Known levels CT interpolation ) n S ( ρ Estimated FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Normalized level density of 186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The discrete levels [54] are binned with the same bin size as our data (224 keV/channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The dashed line shows the CT-model interpolation between our data and ρ(Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The black error bars represent statistical uncertainties from the experiment and systematic errors connected to the unfolding procedure and the first-generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The blue band includes also systematic errors from the normalization procedure (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 8 [MeV] γ E ray energy γ 10 2 10 3 10 4 10 5 10 6 10 Transmission coeff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' units) W Oslo data, this work 186 low-energy extrapolation high-energy extrapolation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Gamma-ray transmission coefficient of 186W before normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The arrows indicate the fit regions used for de- termining the extrapolations (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The gray data points are not considered further in the analysis due to very low statistics in the first-generation matrix for these γ energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' cient T (Eγ) by determining the scaling parameter B in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Here we make use of the relation between the average, total radiative width ⟨Γγ0⟩ deduced from s-wave neutron resonances, the level density and the transmis- 10 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Parameters used for the normalization of the level density and γ-ray transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Note that the E0 parameter is adjusted to make ρCT(Sn) match with ρ(Sn) = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5·105 MeV−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The uncertainty in ⟨Γγ0⟩ from Mughabghab [55] is given as 5 meV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' however, this uncertainty seems too small based on the experimental errors in the radiative width for other W isotopes, and we have chosen a more conservative uncertainty in line with the experimental errors of 182,183,184,186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Sn Iπ t D0 σ2 J(Sn) a E1 ρ(Sn) T E0 ⟨Γγ0⟩ σ2 d Ed (MeV) (eV) (MeV−1) (MeV) 105 (MeV−1) (MeV) (MeV) (meV) (MeV) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='192 3/2− 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='3(16) 47(8) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='28 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5(64) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='51(1) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0077 60+13 −9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='3(13) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='86(19) sion coefficient [39, 60]: ⟨Γγ0⟩ = BD0 4π � Sn Eγ=0 dEγT (Eγ)ρ(Sn − Eγ)× 1 � J=−1 [g(Sn − Eγ, It − 1/2 + J) + g(Sn − Eγ, It + 1/2 + J)] , (17) where g is the spin distribution [61, 62]: g(Ex, J) ≃ 2J + 1 2σ2 J exp � −(J + 1/2)2/2σ2 J � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (18) As we need the spin distribution for the excitation-energy range Ex ∈ [0, Sn], we make use of the spin cutoff param- eter in the general form [63] σ2 J(Ex) = σ2 d + Ex − Ed Sn − Ed � σ2 J(Sn) − σ2 d � , (19) which is motivated also from microscopic calculations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=', shell-model calculations [64] and the work of Uhren- holt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Here, σ2 d represents the spin cutoff pa- rameter at the low excitation energy Ed, where the lev- els are still resolved and with firm spin/parity assign- ments [54], see Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We need to estimate the γ-ray transmission coefficient for Eγ < Emin γ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=', where we do not have experimen- tal data, in order to calculate the integral in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Therefore, we extrapolate with a fit to the low-energy data points using the functional form E3 γ exp(p1Eγ +p2), where p1 and p2 are free parameters1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, the statistics is very low at high γ-ray energies, and so we make use of an extrapolation here as well, here using a simple exponential, exp(p3Eγ + p4), where p3 and p4 are again free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The fit regions and the extrapo- lation functions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The data points in gray color (Eγ > 6 MeV) are from a region in the first- generation matrix with very low statistics (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 7c), and we therefore choose to exclude those data points from the further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To obtain the γ-ray strength function, we use the fact that γ decay at high excitation energies is largely domi- nated by dipole transitions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [67–69]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As 1 This functional form is motivated by shell-model calculations of the low-energy γ strength, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 0 1 2 3 4 5 6 [MeV] γ ray energy E γ 8 − 10 7 − 10 6 − 10 ] 3 ) [MeV γ ray strength function f(E γ W 186 Oslo data, stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='+sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' σ 1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Gamma-ray strength function of 186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The black error bars represent statistical uncertain- ties from the experiment and systematic errors connected to the unfolding procedure and the first-generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The blue band includes also systematic errors from the normaliza- tion procedure (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' our experimental data in principle contain transitions of both electric and magnetic character, we get the total dipole strength function f(Eγ) through f(Eγ) = T (Eγ) 2πE3γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (20) In accordance with the approach for the level density, we estimate the uncertainty in the γ-ray strength function through δf = frec ��δD0 D0 �2 + �δσJ σJ �2 + �δΓγ0 Γγ0 �2 + �∆frec frec �2 , (21) where ∆frec is again the central value (“recommended” normalization), and ∆frec represents statistical uncer- tainties and systematic errors from unfolding and the first-generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The resulting, normalized γ-ray strength function is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 11 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' RESULTS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Comparison to other data and models The level-density data are compared to various mod- els available in the TALYS-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9 code [18], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The models are: ldmodel 1, the composite formula of Gilbert and Cameron [70];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' ldmodel 2, the back-shifted Fermi gas model [71];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' ldmodel 3, the generalized super- fluid model [72];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' ldmodel 4, calculated within the Hartree- Fock-BCS approach [73];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' ldmodel 5, the combinatorial- plus-Hartree-Fock-Bogoliubov approach [74];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' and ld- model 6, the combinatorial model combined with a temperature-dependent Hartree-Fock-Bogoliubov calcu- lation [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 [MeV] x E Excitation energy 1 10 2 10 3 10 4 10 5 10 6 10 7 10 ] 1 ) [MeV x E ( ρ Level density W 186 Oslo data, stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='+sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' σ 1 Known levels ldmodel 1 ldmodel 2 ldmodel 3 ldmodel 4 ldmodel 5 ldmodel 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Comparison of the level-density data from this work with models included in the TALYS code (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' From a first look, none of the models seem to be in good agreement with the data, and we remark that the TALYS level densities have not been normalized to the D0 value from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In adition, we take notice of two impor- tant issues: (i) the spin cutoff parameter we have used in our normalization procedure might not be representative of the corresponding spin distribution in the TALYS mod- els;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (ii) our data can be re-normalized more coherently for each model by adopting its energy-dependence to ex- trapolate between the highest energy point and ρ(Sn), as was done e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Nevertheless, it is clear that the overall shape of our data points are significantly dif- ferent from several of the level-density models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We also remark that the slope of our level-density data points is directly linked to the slope of the γ-strength function as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' If we were to renormalize our level density to the TALYS models, this would inevitably lead to a change in slope in the γ-strength function as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We now compare our γ-strength data from the (γ, n) measurements and the OCL experiment to external data found in the literature, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We observe a good agreement with the E1 strength extracted from primary γ rays following neutron capture by Kopecky et al [78], which brings further support to the absolute nor- malization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, we compare our new photoneutron data to several data sets found in the liter- ature, where the photoneutron cross section σγn is con- verted into dipole strength using the relation of Axel [79]: fγn(Eγ) = 1 3π2ℏ2c2 σγn(Eγ) Eγ , (22) where σγn is in units of mb, Eγ in MeV, and the factor 1/(3π2ℏ2c2) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='674 · 10−8 mb−1MeV−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Overall, there is good agreement between the various data sets for the W isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 13a, we also compare the data with avail- able models in TALYS: strength 1, the Generalized Lorentzian [67];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' strength 2, the Standard Lorentzian (Brink-Axel model) [43, 44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' strength 3, the Quasi- Particle Random Phase Approximation (QRPA) on top of a Hartree-Fock-plus-BCS calculation [80];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' strength 4, the QRPA on top of a Hartree-Fock-Bogoliubov (HFB) calculations [81];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' strength 5, the Hybrid model [82] with parameters from global systematics [18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' strength 6, QRPA as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [81] but on top of a temperature- dependent HFB calculation [75];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' and finally strength 7, a relativistic mean-field calculation plus a continuum QRPA calculation [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Out of these models, strength 4 and strength 6 match reasonably well the present Oslo data, but not the (γ, n) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In general, the models are deviating significantly from each other and from either the Oslo data or the (γ, n) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' To obtain a model description that can reproduce our data reasonably well over the entire energy range, we take a pragmatic approach and exploit phenomenologi- cal models for the dipole strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For the main part of the E1 strength which is dominated by the Giant Dipole Resonance (GDR), we apply the Hybrid model of Goriely [82]: f Hyb E1 (Eγ, Tf) = 1 3π2ℏ2c2 EγσrΓrΓ(Eγ, Tf) (E2γ − E2r)2 + E2γΓrΓ(Eγ, Tf), (23) where σr is the peak cross section, Er the centroid, and Γr the width of the GDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Further, the γ-energy and temperature dependent width Γ(Eγ, Tf) is given by Γ(Eγ, Tf) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='7 · Γr E2 γ + 4π2T 2 f EγEr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (24) The temperature of the final levels, Tf, is here considered as a constant, in line with the Brink-Axel hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We also include extra E1 strength (labeled “E1 pygmy” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 13b) to make a smooth connection between our data and the (γ, n) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Finally, we also add a magnetic- dipole component (marked “M1 spin-flip” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 13b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 12 0 2 4 6 8 10 12 14 [MeV] γ E ray energy γ 8 − 10 7 − 10 6 − 10 ] 3 ) [MeV γ E (f ray strength function γ (a) W Oslo data, this work 186 stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='+sys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' σ 1 ,n), this work γ W( 182 ,n), this work γ W( 183 ,n), this work γ W( 184 ,n), Berman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' γ W( 186 ,n), Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' γ W( 186 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' , Kopecky E1 f W 184 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' , Kopecky M1 f W 184 strength1 strength2 strength3 strength4 strength5 strength6 strength7 0 2 4 6 8 10 12 14 [MeV] γ E ray energy γ 8 − 10 7 − 10 6 − 10 E1 hybrid E1 pygmy M1 spin-flip total fit function (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) (a) Comparison of γ-strength data from this work with data from the literature (Berman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [77], Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [9], and Kopecky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [78]), and to models included in the TALYS code (see text);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (b) Fit to the γ-ray strength function data of 186W and the 184W data of Kopecky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [78]) (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Parameters found from the model fits of ftot to the γ-strength data (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The uncertainties given are from the fit only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Note that EM1 and ΓM1 are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Er Γr σr EPyg ΓPyg σPyg Tf EM1 ΓM1 σM1 (MeV) (MeV) (mb) (MeV) (MeV) (mb) (MeV) (MeV) (MeV) (mb) Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='9(1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='1(1) 382(2) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='3(1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='6(2) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='43(3) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='4(4) For both the E1 pygmy and the M1 spin-flip contribu- tions, we apply a resonance-like form using a Standard Lorentzian: fPyg,M1(Eγ) = 1 3π2ℏ2c2 σPyg,M1Γ2 Pyg,M1Eγ (E2γ − E2 Pyg,M1)2 + Γ2 Pyg,M1E2γ (25) where σPyg,M1, ΓPyg,M1, and EPyg,M1 are the peak cross section, width, and centroid for the pygmy (Pyg) and the spin-flip (M1) resonance, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The total fit function is then given by ftot(Eγ) = f Hyb E1 (Eγ, Tf = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=') + fPyg(Eγ) + fM1(Eγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (26) For the fit, we first constrain the Hybrid component by fitting only the Hybrid model to the GDR data (Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [9] and Berman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [77]) in the range Eγ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='7−14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We choose to fix the Tf parameter to the one used for the extrapolation of the level density (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' III C) to ease the fit, as Tf is largely determined from the γ- strength function below neutron threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' From this fit, we determine the GDR parameters σr, Er, and Γr, to be used as start values for the next fit including the data for γ energies below neutron threshold as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' For the spin-flip part, we use a fixed centroid EM1 taken from systematics [63], and a fixed width of ΓM1 of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The peak cross section σM1 is then found from a fit to the M1 data of 184W from Kopecky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Then we make a fit using the full energy range Eγ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0 − 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 MeV, with only the spin-flip parameters fixed, and with the first fit of the GDR data as starting values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In the fit, we include the present OCL data of 186W, the E1 data from Kopecky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [78] on 184W, and the GDR data from Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [9] and Berman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The result- ing fit is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 13b, and the parameters are listed in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As this model fit will be used to calculate the (n, γ) cross section and reactivity in the following sec- tion, we repeat the fit for all the different normalizations (varying D0, Γγ0, σJ and taking into account ∆f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' All fits are performed within the ROOT software tool [84] using the Minuit package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The resulting fit function gives a reasonable description of the strength function data, although we note a poten- tial issue in that the region between Eγ = 6 − 8 MeV contains practically no data points for 186W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Moreover, the 184W data points from primary transitions following neutron capture typically have large fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Hence, it is very difficult to assess the actual parameters for the E1 pygmy, and the deduced parameters given in Table II 13 should be used with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We also remark that the data points at the lowest γ energies, Eγ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='5 MeV, might indicate some low-energy increase in the γ-strength function, as first observed in iron isotopes [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' However, in contrast to clear cases like 56Fe [68, 69, 85], it is hard to conclude here as there are only a few data points that might show an increas- ing trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We therefore choose not to include an extra “upbend” component in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Maxwellian-averaged cross section and reaction rate Using our level-density data and γ-strength function data, we now calculate the Maxwellian-averaged cross section (MACS) with the TALYS code, which is based on the statistical model of Wolfenstein [86] and Hauser and Feshbach [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The resulting MACS is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 14, where we also show the TALYS MACS with default in- puts (strength 1, ldmodel 1, a global optical-model po- tential, and no upbend), and the variation of the MACS as the different level-density and γ-strength models are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We have tested using the semi-microscopic optical- model potential of Bauge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [88] for comparison with the one of Koning and Delaroche [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' As seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 14 (dashed line versus dashed-dotted line), there is only a minor difference between the two for neutron energies around kBT = 30 keV, and overall the semi- microscopic potential gives a lower MACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Nevertheless, the presented uncertainty band on our experimentally- constrained MACS includes the variation between the two different optical models in the lower uncertainty, in addition to uncertainties from D0, Γγ0, and σJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 14, we compare our result with the KADoNiS database [90], and find agreement within the error bars, although the KADoNiS values are overall larger than our central values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We remark that the KADoNiS values are from a weighted average of MACS constrained by pho- tonuclear data above Sn, while our results include infor- mation on both the level density as well as the γ-strength function below Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We have multiplied the KADoNiS MACS values with their corresponding stellar enhance- ment factor (SEF) as given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [90] for 185W(n, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Furthermore, our estimated uncertainty band is smaller than the KADoNiS uncertainties, Our result at kBT = 30 keV, 508+76 −106 mb, agrees well within error bars with the MACS from Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [9], 553(60) mb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' On the other hand, the evaluation of Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [91] of 703(113) mb, and the measurement of Sonnabend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [8], 687(110) mb, are both larger than our estimate, although still within the estimated uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We note that none of these values are directly measured, as Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' gives a purely theoretical prediction, while the MACS value from Sonnabend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' is constrained on (γ, n) data above Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In comparison with the TALYS estimates using the default input as well as the resulting MACS when vary- ing the level-density and γ-strength models, our deduced 20 40 60 80 100 [keV] T B k Energy 0 200 400 600 800 1000 1200 1400 1600 MACS [mb] exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' constrained KADoNiS-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0 Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (2000) Sonnabend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (2003) Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (2004) TALYS default Variations, NLD SF γ Variations, Variation, n-OMP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Maxwellian-averaged cross section for the 185W(n, γ) reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The shaded band indicates the present data-constrained MACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The thick, azure dashed- dotted line shows the TALYS result using default input, the thin, azure dashed lines show the TALYS MACS when vary- ing the level-density models, and the thin, cyan lines show the variation due to different γ-strength models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The dot- ted line shows the deviation from the default when using the optical-model potential of Bauge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 1 − 10 1 Temperature [GK] 0 20 40 60 80 100 120 140 160 180 200 220 6 10 × ] 1 mol 1 s 3 [cm 〉 v σ 〈 A N exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' constrained KADoNiS-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='0 TALYS default Variations, NLD SF γ Variations, Variation, n-OMP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (Color online) Reaction rate for the 185W(n, γ) reac- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The shaded band indicates the present data-constrained result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' See also the caption of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' MACS is in between the extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 15, we show the corresponding reaction rate (stellar reactivity) deduced from our data compared to the KADoNiS rate, the TALYS default and the variations using different model inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Again we find that the KADoNiS values are overall higher than our estimated rate, in particular for temperatures below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='3 GK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 14 To address possible implications for the s process and the Re/Os cosmochronometer in a reliable way, the branch points at 186Re and 191Os should also be consid- ered in realistic stellar models for thermally-pulsing AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The 191Os MACS has been estimated by a similar procedure as in this work by Kullmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The 186Re MACS remains to be experimentally constrained in the same way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' the 186W(α, dγ)187Re data from this same experiment is currently being analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' With this experimentally-constrained MACS also at hand, we in- tend to perform a consistent study of the s process in this mass region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' SUMMARY AND OUTLOOK In this work, we have performed photoneutron cross section measurements on the 182,183,184W isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This completes the photoneutron measurements on the stable W isotopic chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Furthermore, we have presented data on the 186W(α, α′γ) reaction, and used the extracted level density and γ-ray strength function to provide an experimentally constrained (n, γ) cross section for the branch-point nucleus 185W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' In comparison with other data and the recommended MACS from the KADoNiS data base, we find that our estimated MACS and reaction rate are lower than most of the other available values, except for the result of Mohr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Our reaction rate could possibly impact the s pro- cess in this mass region, in particular the deduced neu- tron density and the calculation of the 186Os abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' When the 186Re MACS also becomes available, we intend to perform a systematic study of the s-process conditions in the W-Re-Os region in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors would like to thank J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' M¨uller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Sobas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Wikne at the Oslo Cyclotron Labo- ratory for operating the cyclotron and providing excellent experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' We sincerely thank T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Ha- gen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Rose and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Zeiser for helping with the OCL experiment, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Lui for helping with the NewSUB- ARU experiments, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Liddick for inspiring dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' gratefully acknowledges funding of this research by the European Research Council through ERC-STG-2014 under grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 637686, and from the Research Council of Norway, project grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 316116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' acknowledges the support from the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='- FNRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This work was supported in part by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' OISE-1927130 (IReNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' The photoneutron cross section measurement was performed as part of the IAEA CRP on “Updating the Photonuclear Data Library and generating a Refer- ence Database for Photon Strength Functions” (F41032).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=', V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' gratefully acknowledge fi- nancial support from the Research Council of Norway, project grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 325714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' This work is in part based on the research supported partly by the National Research Foundation of South Africa (Grant Number: 118846).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Appendix A: Uncertainty in ρ(Sn) To estimate the total NLD at the neutron separation energy using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (14), we propagate errors from the D0 value and the spin cutoff parameter σJ(Sn) assuming that they are independent variables, which is a justified assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Thus, we get that �δρ(Sn) ρ(Sn) �2 = �δD0 D0 �2 + �δξ(σJ(Sn)) ξ(σJ(Sn)) �2 , (A1) where ξ represents the function containing the depen- dency on the spin cutoff parameter σJ at the neutron separation energy Sn: ξ(σJ) = 2σ2 J Ite−I2 t /2σ2 J + (It + 1)e−(It+1)2/2σ2 J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (A2) Now we take the derivative of ξ with respect to σJ and obtain: δξ δσJ = 4σJ � Ite−I2 t /2σ2 J + (It + 1)e−(It+1)2/2σ2 J � − 2 σJ � I3 t e−I2 t /2σ2 J + (It + 1)3e−(It+1)2/2σ2 J � � Ite−I2 t /2σ2 J + (It + 1)e−(It+1)2/2σ2 J�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (A3) For convenience, we now define the auxilliary functions z1 ≡ I3 t e−I2 t /2σ2 J + (It + 1)3e−(It+1)2/2σ2 J, z2 ≡ Ite−I2 t /2σ2 J + (It + 1)e−(It+1)2/2σ2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Using these and dividing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (A3) on the function ξ(σJ), we get δξ ξδσJ = 2 σJ − z1 σ3 Jz2 = 2 σJ � 1 − 1 2σ2 J z1 z2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (A4) Finally, we obtain �δξ ξ �2 = �2δσJ σJ �2 � 1 − 1 2σ2 J z1 z2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' (A5) 15 This is what is implemented in the code d2rho in the Oslo software package [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 62, 9 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Arnould, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Goriely, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Takahashi, Physics Re- ports 450, 97 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [4] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Langanke, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Mart´ınez-Pinedo, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='- K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Thielemann, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Renstrøm, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Utsunomiya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Goriely, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Filipescu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Gheorghe, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Tesileanu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Glodariu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Shima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Takahisa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Miyamoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Lui, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Hi- laire, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Akimune, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Yama- gata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Okamoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Harada, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Kitatani, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Shima, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Horikawa, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Stopani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Symochko, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Fan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Renstrøm, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Lui, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Midtbø, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' P´eru, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Shima, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Siem, and O.' 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Donaldson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Fujita, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Fujita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Hashimoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Hatanaka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Ito, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Krugmann, B.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Ponomarev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Richter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Shima, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Yamamoto, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Zweidinger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Kamerdzhiev, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Brown, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Avdeenkov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Ari-izumi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' G´’orgen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 111, 232504 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' [66] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Brown and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Larsen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Million, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Nyhus, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Renstrøm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Rose, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Ruud, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Siem, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Tornyi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Tveten, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Voinov, and M.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Guttormsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Midtbø, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Sahin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Siem, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Tveten, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Zeiser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} +page_content=' C 99, 065806 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFQT4oBgHgl3EQfVjao/content/2301.13301v1.pdf'} diff --git a/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf b/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c7465b5571e913c5f0fb66e60719bb79281ba2d3 Binary files /dev/null and b/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf differ diff --git a/NtFPT4oBgHgl3EQfmDWp/content/tmp_files/2301.13124v1.pdf.txt b/NtFPT4oBgHgl3EQfmDWp/content/tmp_files/2301.13124v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f500ba9884fd0c65ef672017f387ccb4df7116f --- /dev/null +++ b/NtFPT4oBgHgl3EQfmDWp/content/tmp_files/2301.13124v1.pdf.txt @@ -0,0 +1,173 @@ +arXiv:2301.13124v1 [cs.CY] 30 Jan 2023 +MED1stMR: Mixed Reality to Enhance the Training of Medical First +Responders for Challenging Contexts +HELMUT SCHROM-FEIERTAG, GEORG REGAL, and MARKUS MURTINGER, AIT Austrian Insti- +tute of Technology, Vienna +1 +INTRODUCTION +Mass-casualty incidents with a large number of injured persons caused by human-made or by natural disasters are +increasing globally. In such situations, medical first responders (MFRs) need to perform diagnosis, basic life support, +or other first aid to help stabilize victims and keep them alive to wait for the arrival of further support. Situational +awareness and effective coping with acute stressors are essential [5] to enable first responders to take appropriate +action that saves lives. Such tasks are particularly challenging for first responders, that lack the special training to act +optimally in these situations. Increasingly severe consequences of natural disasters and terrorist threats will expand +the occurrence probability of such stressful and demanding situations and require the development and deployment +of innovative technological solutions adapted to the (cross-sectoral) needs of first responders. +In that context, the adage “practice makes perfect” is well-fitting to situational training. Lectures, books, videos, etc. +are no substitute for hands-on experiences, and humans often learn more from their mistakes than from their successes. +Unfortunately, it is difficult to provide such training for large-scale emergency medicine or in dangerous conditions. +Current training of medical first responders often happens through live exercises: medics practice stabilisation and +wound care via moulage, a training exercise where live persons are given highly realistic “fake” wounds. The drawback +of such training is the large effort needed to create such live training exercises (a large number of ‘victim actors’ needed, +availability of infrastructure, etc.) and the lack of realistic treatments. Hence, such training or exercises are not executed +very often and sometimes also fail to create “real” stressful and demanding environments. +Virtual Reality (VR) has already been demonstrated in several domains to be a serious alternative, and in some areas +also a significant improvement to conventional learning and training. Especially for the challenges in the training +of MFRs, it can be highly useful for practising and learning domains where the context of the training is not easily +available. VR training offers controlled, easy-to-create environments that can be created and trained repeatedly under +the same conditions. This repetition makes it possible to master a new skill or process. Like in real-life training, trainees +are transformed into active users who need to be physically and mentally engaged to evaluate the situation, take +appropriate measures and act accordingly. +There are two types of VR medical training systems realised up to date. Systems centred on teaching direct physical +skills and procedures e.g. surgery (e.g. [7], [10], [14], [12]). Those usually employ highly sophisticated hardware user +interfaces providing realistic haptic feedback or/and mimicking real devices. On the other end of the spectrum, there +are VR systems helping the trainee to develop psychological skills required in real-world scenarios (e.g. [15], [11]) and +in particular decision training. +However, these approaches are currently rather disjunct and do not allow to train decision-making under stress +together with physical skill scenarios. Also, other medical tasks or preclinical routines in patient care and treatment +are not yet covered by haptic solutions. Training medical skills are about vision and haptics for tangible interaction, +and if a simulation has only one of those two, it will provide only half of the experience. +1 + +Schrom-Feiertag, Regal and Murtinger +2 +VISION OF THE PROJECT MED1STMR +As an advanced alternative to VR, Mixed Reality (MR) environments have the potential to augment current VR training +by providing a dynamic simulation of an environment and hands-on practice on injured victims. +There are several interpretations of MR, one is that the real environment is augmented by digital objects, another +is that physical objects are integrated into the VR. In our vision, MR is to be considered as the latter interpretation +with a fully digital environment, where the user sees a fully digital environment without looking at the real world, but +this digital environment is connected to real physical objects. This VR-based mixed reality is also called augmented +virtuality (AV) according to [13]. +Building on this interpretation of MR, the main aim of MED1stMR is to develop a new generation of MR training +with haptic feedback for enhanced realism. To this end, we will pursue the following pioneering concepts: +2.1 +Integration of high-fidelity patient simulation manikins for enhanced realism +Evidence shows that the use of VR is useful when the training domain is complex and difficult to master (cf. [9], [2]) +and when the audio-visual features assisted by haptic feedback of the training environment are crucial to the overall +training success (cf. [1]). This makes virtual environments the solution for practising and learning domains where the +context of the training is not easily available or replicable due to security and safety issues (e.g. [6]). It allows creating +easily a diverse range of training scenarios tailored to the training goals and needs (single user vs. teams, from single +to many people injured in large incidents, influence of psychological and contextual factors). +Through the integration of high-fidelity patient simulation manikins and medical equipment into the MR experience, +MED1stMR offers a much richer sensory experience. This MR training environment allows trainees to immerse into +virtual scenarios and be able to feel and perceive actual movements of the limbs, head, and face through tactile and +visual interaction as they are actuated. Furthermore, it enables systematic manipulation of a large set of potential +influence factors in order to optimise training effects. This will bring virtual training closer to reality and enable both +scenario training and medical training in the same MR training environment. +2.2 +Biosignal feedback loop and smart scenario control to enhance effectiveness of MR training +The wireless integration of wearables in MR training environments is emerging (e.g., [8]) and particularly in the train- +ing of highly demanding skills such as piloting/aviation, medical surgeries (e.g., [4]) or first responders (e.g., [3]). +In order to better support, assist and personalise MFRs training, we will integrate wearable technology for moni- +toring trainees’ physiological data. Smart electronic devices can detect and transmit information regarding biosignals, +informing on trainees’ physiological status. Monitoring these signals will provide the detection of physical and psy- +chological strain and stress during training. +This will provide information for the debriefing sessions and can be used for real-time scenario control through the +trainer (manual control) or automatically by the training system through artificial intelligence-based adaptive smart +scenarios. The data on trainee state and behaviour can then be used to constitute a feedback loop for personalising +and adapting training to the trainees’ needs. Such a system can automatically adjust the scenarios according to the +stress level of the trainee, for example, low stress increases the difficulty of the scenario and allows longer and more +complex scenarios to be trained without the intervention of a trainer. +2 + +MED1stMR: Mixed Reality to Enhance Training of Medical First Responder +3 +OUR MOTIVATION +The development of such a training system for MFRs requires research, expertise, and knowledge in the areas of +medical research, biosensors, and wearable technologies, human factors research, psychology, physiological research, +technology experience, user research, VR/MR, and medical training simulation development to answer all the questions +that arise in order to develop an optimal training system for MFRs: +• How can haptic feedback for training medical skills on victims be provided for MFRs? +• Which scenarios and use cases are most suitable for MR training and deliver the greatest benefits? +• How can effective MR training scenarios be developed? +• How effective are such training approaches, how good is the learning progress and how does it compare to +real-world training? +• How should a MR training curriculum be designed and merged with existing training curricula? +• What about the costs for the training system and does the benefit justify the effort? +4 +OUR CONTRIBUTION TO THE WORKSHOP +MED1stMR develop a MR training system based on a combination of VR environment and the integration of VR- +enabled manikins. With this new training environment, MED1stMR delivers a training platform for collaborative multi- +user training to train the medical skills of MFRs as well the decision-making abilities in disaster situations. In the project, +the training is designed for teams of up to four people to enable emergency teams to train together. The technological +basis is the Refense trainings platform (www.refense.com) that allows up to 10 users on an area of 11 x 20 meters to +immerse themselves into a realistic common shared scenario, see each other in real time with full-body VR tracking. A +trainer can also be included as an invisible observer in the training. Every movement and voice spoken are recorded for +debriefing of team collaboration and actions taken. The inclusion of manikins as tangible objects as learning support +provides a more realistic experience and enables novel possibilities for hands-on tasks. The basis will build the ADAM- +X manikin (https://medical-x.com/product/adam-x/) and will be advanced to a fully functional touch-enabled human +manikin designed for practising skills in trauma emergency situation. +The goal of the MED1stMR training solution is to train the situation awareness and the procedure in the first and +second triage. The realisation of a biosignal feedback loop with body sensors allows to monitor trainee (stress, anxiety, +etc.) states and behaviours of MFRs during training and will make this data available for scenario control. For this +purpose, heart rate variability is measured as one of the most reliable indicators of stress. The trainer can adapt training +to the personal needs of trainees and provides a new way of interaction between trainer and trainee and requires an +appropriate user interface for support. +A key point in MED1stMR is to examine the effectiveness of training for the different roles. To increase the effective- +ness of the training, the simple repetition of the training scenarios as well as the recording of all movements, activities +and communication during the training for the debriefing play an important role. The presentation of our project is in- +tended to provide an insight into our approach and to open up an exchange of experiences with other people, projects, +research, and developments and also to get an impression of how the topics are received, what others are doing in this +field and where further and interesting research topics lie in this area. +We can contribute existing knowledge to the workshop and discuss with the other participants’ challenges and help +to set up a future research agenda for collaborative multi-user VR training. +3 + +Schrom-Feiertag, Regal and Murtinger +ACKNOWLEDGEMENTS +The project MED1stMR has received funding from the European Union’s Horizon 2020 Research and Innovation Pro- +gramme under grant agreement No 101021775. The content reflects only the MED1stMR consortium’s view. Research +Executive Agency and European Commission is not liable for any use that may be made of the information contained +herein. +REFERENCES +[1] Andrea F Abate, Mariano Guida, Paolo Leoncini, Michele Nappi, and Stefano Ricciardi. 2009. A haptic-based approach to virtual training for +aerospace industry. Journal of Visual Languages & Computing 20, 5 (2009), 318–325. +[2] Cyril Bossard, Gilles Kermarrec, Cédric Buche, and Jacques Tisseau. 2008. Transfer of learning in virtual environments: a new challenge? Virtual +Reality 12, 3 (2008), 151–161. +[3] Meredith Carroll, Mitchell Ruble, Mark Dranias,Summer Rebensky, Maria Chaparro, Joanna Chiang, and Brent Winslow. 2020. Automatic detection +of learner engagement using machine learning and wearable sensors. Journal of Behavioral and Brain Science 10, 3 (2020), 165–178. +[4] Jonathan Currie, Raymond R Bond, Paul McCullagh, Pauline Black, Dewar D Finlay, Stephen Gallagher, Peter Kearney, Aaron Peace, Danail Stoy- +anov, Colin D Bicknell, et al. 2019. Wearable technology-based metrics for predicting operator performance during cardiac catheterisation. Inter- +national journal of computer assisted radiology and surgery 14, 4 (2019), 645–657. +[5] Marie Ottilie Frenkel, Laura Giessing, Sebastian Egger-Lampl, Vana Hutter, Raoul RD Oudejans, Lisanne Kleygrewe, Emma Jaspaert, and Henning +Plessner. 2021. The impact of the COVID-19 pandemic on European police officers: Stress, demands, and coping resources. Journal of Criminal +justice 72 (2021), 101756. +[6] Andrzej Grabowski and Jarosław Jankowski. 2015. Virtual reality-based pilot training for underground coal miners. 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Surgical endoscopy 32, 6 (2018), 2958–2967. +[13] Paul Milgram, Haruo Takemura, Akira Utsumi, and Fumio Kishino. 1995. Augmented reality: A class of displays on the reality-virtuality continuum. +In Telemanipulator and telepresence technologies, Vol. 2351. International Society for Optics and Photonics, 282–292. +[14] Felix Nickel, Julia A Brzoska, Matthias Gondan, Henriette M Rangnick, Jackson Chu, Hannes G Kenngott, Georg R Linke, Martina Kadmon, Lars +Fischer, and Beat P Müller-Stich. 2015. Virtual reality training versus blended learning of laparoscopic cholecystectomy: a randomized controlled +trial with laparoscopic novices. Medicine 94, 20 (2015). +[15] Federica Pallavicini, Luca Argenton, Nicola Toniazzi, Luciana Aceti, and Fabrizia Mantovani. 2016. Virtual reality applications for stress manage- +ment training in the military. Aerospace medicine and human performance 87, 12 (2016), 1021–1030. +4 + diff --git a/NtFPT4oBgHgl3EQfmDWp/content/tmp_files/load_file.txt b/NtFPT4oBgHgl3EQfmDWp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f9dcb92206dd34b86a922d082eb191c4a7832f6 --- /dev/null +++ b/NtFPT4oBgHgl3EQfmDWp/content/tmp_files/load_file.txt @@ -0,0 +1,171 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf,len=170 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='13124v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='CY] 30 Jan 2023 MED1stMR: Mixed Reality to Enhance the Training of Medical First Responders for Challenging Contexts HELMUT SCHROM-FEIERTAG, GEORG REGAL, and MARKUS MURTINGER, AIT Austrian Insti- tute of Technology, Vienna 1 INTRODUCTION Mass-casualty incidents with a large number of injured persons caused by human-made or by natural disasters are increasing globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' In such situations, medical first responders (MFRs) need to perform diagnosis, basic life support, or other first aid to help stabilize victims and keep them alive to wait for the arrival of further support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Situational awareness and effective coping with acute stressors are essential [5] to enable first responders to take appropriate action that saves lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Such tasks are particularly challenging for first responders, that lack the special training to act optimally in these situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Increasingly severe consequences of natural disasters and terrorist threats will expand the occurrence probability of such stressful and demanding situations and require the development and deployment of innovative technological solutions adapted to the (cross-sectoral) needs of first responders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' In that context, the adage “practice makes perfect” is well-fitting to situational training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Lectures, books, videos, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' are no substitute for hands-on experiences, and humans often learn more from their mistakes than from their successes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Unfortunately, it is difficult to provide such training for large-scale emergency medicine or in dangerous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Current training of medical first responders often happens through live exercises: medics practice stabilisation and wound care via moulage, a training exercise where live persons are given highly realistic “fake” wounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The drawback of such training is the large effort needed to create such live training exercises (a large number of ‘victim actors’ needed, availability of infrastructure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=') and the lack of realistic treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Hence, such training or exercises are not executed very often and sometimes also fail to create “real” stressful and demanding environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Virtual Reality (VR) has already been demonstrated in several domains to be a serious alternative, and in some areas also a significant improvement to conventional learning and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Especially for the challenges in the training of MFRs, it can be highly useful for practising and learning domains where the context of the training is not easily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' VR training offers controlled, easy-to-create environments that can be created and trained repeatedly under the same conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' This repetition makes it possible to master a new skill or process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Like in real-life training, trainees are transformed into active users who need to be physically and mentally engaged to evaluate the situation, take appropriate measures and act accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' There are two types of VR medical training systems realised up to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Systems centred on teaching direct physical skills and procedures e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' surgery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' [7], [10], [14], [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Those usually employ highly sophisticated hardware user interfaces providing realistic haptic feedback or/and mimicking real devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' On the other end of the spectrum, there are VR systems helping the trainee to develop psychological skills required in real-world scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' [15], [11]) and in particular decision training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' However, these approaches are currently rather disjunct and do not allow to train decision-making under stress together with physical skill scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Also, other medical tasks or preclinical routines in patient care and treatment are not yet covered by haptic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Training medical skills are about vision and haptics for tangible interaction, and if a simulation has only one of those two, it will provide only half of the experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' 1 Schrom-Feiertag, Regal and Murtinger 2 VISION OF THE PROJECT MED1STMR As an advanced alternative to VR, Mixed Reality (MR) environments have the potential to augment current VR training by providing a dynamic simulation of an environment and hands-on practice on injured victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' There are several interpretations of MR, one is that the real environment is augmented by digital objects, another is that physical objects are integrated into the VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' In our vision, MR is to be considered as the latter interpretation with a fully digital environment, where the user sees a fully digital environment without looking at the real world, but this digital environment is connected to real physical objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' This VR-based mixed reality is also called augmented virtuality (AV) according to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Building on this interpretation of MR, the main aim of MED1stMR is to develop a new generation of MR training with haptic feedback for enhanced realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' To this end, we will pursue the following pioneering concepts: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='1 Integration of high-fidelity patient simulation manikins for enhanced realism Evidence shows that the use of VR is useful when the training domain is complex and difficult to master (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' [9], [2]) and when the audio-visual features assisted by haptic feedback of the training environment are crucial to the overall training success (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' This makes virtual environments the solution for practising and learning domains where the context of the training is not easily available or replicable due to security and safety issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' It allows creating easily a diverse range of training scenarios tailored to the training goals and needs (single user vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' teams, from single to many people injured in large incidents, influence of psychological and contextual factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Through the integration of high-fidelity patient simulation manikins and medical equipment into the MR experience, MED1stMR offers a much richer sensory experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' This MR training environment allows trainees to immerse into virtual scenarios and be able to feel and perceive actual movements of the limbs, head, and face through tactile and visual interaction as they are actuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Furthermore, it enables systematic manipulation of a large set of potential influence factors in order to optimise training effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' This will bring virtual training closer to reality and enable both scenario training and medical training in the same MR training environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='2 Biosignal feedback loop and smart scenario control to enhance effectiveness of MR training The wireless integration of wearables in MR training environments is emerging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=', [8]) and particularly in the train- ing of highly demanding skills such as piloting/aviation, medical surgeries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=', [4]) or first responders (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=', [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' In order to better support, assist and personalise MFRs training, we will integrate wearable technology for moni- toring trainees’ physiological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Smart electronic devices can detect and transmit information regarding biosignals, informing on trainees’ physiological status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Monitoring these signals will provide the detection of physical and psy- chological strain and stress during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' This will provide information for the debriefing sessions and can be used for real-time scenario control through the trainer (manual control) or automatically by the training system through artificial intelligence-based adaptive smart scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The data on trainee state and behaviour can then be used to constitute a feedback loop for personalising and adapting training to the trainees’ needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Such a system can automatically adjust the scenarios according to the stress level of the trainee, for example, low stress increases the difficulty of the scenario and allows longer and more complex scenarios to be trained without the intervention of a trainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' 2 MED1stMR: Mixed Reality to Enhance Training of Medical First Responder 3 OUR MOTIVATION The development of such a training system for MFRs requires research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' expertise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' and knowledge in the areas of medical research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' biosensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' and wearable technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' human factors research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' psychology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' physiological research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' technology experience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' user research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' VR/MR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' and medical training simulation development to answer all the questions that arise in order to develop an optimal training system for MFRs: How can haptic feedback for training medical skills on victims be provided for MFRs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Which scenarios and use cases are most suitable for MR training and deliver the greatest benefits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' How can effective MR training scenarios be developed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' How effective are such training approaches, how good is the learning progress and how does it compare to real-world training?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' How should a MR training curriculum be designed and merged with existing training curricula?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' What about the costs for the training system and does the benefit justify the effort?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' 4 OUR CONTRIBUTION TO THE WORKSHOP MED1stMR develop a MR training system based on a combination of VR environment and the integration of VR- enabled manikins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' With this new training environment, MED1stMR delivers a training platform for collaborative multi- user training to train the medical skills of MFRs as well the decision-making abilities in disaster situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' In the project, the training is designed for teams of up to four people to enable emergency teams to train together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The technological basis is the Refense trainings platform (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='refense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='com) that allows up to 10 users on an area of 11 x 20 meters to immerse themselves into a realistic common shared scenario, see each other in real time with full-body VR tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' A trainer can also be included as an invisible observer in the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Every movement and voice spoken are recorded for debriefing of team collaboration and actions taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The inclusion of manikins as tangible objects as learning support provides a more realistic experience and enables novel possibilities for hands-on tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The basis will build the ADAM- X manikin (https://medical-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content='com/product/adam-x/) and will be advanced to a fully functional touch-enabled human manikin designed for practising skills in trauma emergency situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The goal of the MED1stMR training solution is to train the situation awareness and the procedure in the first and second triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The realisation of a biosignal feedback loop with body sensors allows to monitor trainee (stress, anxiety, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=') states and behaviours of MFRs during training and will make this data available for scenario control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' For this purpose, heart rate variability is measured as one of the most reliable indicators of stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The trainer can adapt training to the personal needs of trainees and provides a new way of interaction between trainer and trainee and requires an appropriate user interface for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' A key point in MED1stMR is to examine the effectiveness of training for the different roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' To increase the effective- ness of the training, the simple repetition of the training scenarios as well as the recording of all movements, activities and communication during the training for the debriefing play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The presentation of our project is in- tended to provide an insight into our approach and to open up an exchange of experiences with other people, projects, research, and developments and also to get an impression of how the topics are received, what others are doing in this field and where further and interesting research topics lie in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' We can contribute existing knowledge to the workshop and discuss with the other participants’ challenges and help to set up a future research agenda for collaborative multi-user VR training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' 3 Schrom-Feiertag, Regal and Murtinger ACKNOWLEDGEMENTS The project MED1stMR has received funding from the European Union’s Horizon 2020 Research and Innovation Pro- gramme under grant agreement No 101021775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' The content reflects only the MED1stMR consortium’s view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' Research Executive Agency 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} +page_content=' 4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFPT4oBgHgl3EQfmDWp/content/2301.13124v1.pdf'} diff --git a/PdFIT4oBgHgl3EQfeytv/content/tmp_files/2301.11276v1.pdf.txt b/PdFIT4oBgHgl3EQfeytv/content/tmp_files/2301.11276v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8bd8c2c0499f2766e47edf8d094535e8942c4cf1 --- /dev/null +++ b/PdFIT4oBgHgl3EQfeytv/content/tmp_files/2301.11276v1.pdf.txt @@ -0,0 +1,634 @@ +BayesSpeech: A Bayesian Transformer Network for +Automatic Speech Recognition +Will Rieger +Master of Science in Computer Science +Department of Computer Science +The University of Texas at Austin +Abstract +Recent developments using End-to-End Deep Learning models have +been shown to have near or better performance than state of the art +Recurrent Neural Networks (RNNs) on Automatic Speech Recognition +tasks. These models tend to be lighter weight and require less training +time than traditional RNN-based approaches. However, these models take +frequentist approach to weight training. In theory, network weights are +drawn from a latent, intractable probability distribution. We introduce +BayesSpeech for end-to-end Automatic Speech Recognition. BayesSpeech +is a Bayesian Transformer Network where these intractable posteriors are +learned through variational inference and the local reparameterization +trick without recurrence. We show how the introduction of variance in +the weights leads to faster training time and near state-of-the-art perfor- +mance on LibriSpeech-960. +1. Introduction +In the majority of neural networks, randomness is usually introduced through perturbation +of the input or randomly removing nodes from the network (Hinton et al., 2012). There +has been great success using these methods across a variety of domains including Automatic +Speech Recognition (Park et al., 2019). +Models continue to evolve. +However and data +augmentation methods rarely take large leaps in terms of the features they can help express. +Newer models are generally larger and larger and require incredible amounts of compute to +properly train. We especially see this in the field of Automatic Speech Recognition. Newer +models such as Jasper (Li et al., 2019), the Conformer (Gulati et al., 2020), LAS (Chan et +al., 2016), and the Transformer (Vaswani et al., 2017) all require training for multiple days +across multiple GPUs. Creating deeper models can certainly help attain better performance +on the domain task. But what if we approach the model differently and try to leverage their +probabilistic nature? +1 +arXiv:2301.11276v1 [eess.AS] 16 Jan 2023 + +Most Neural Network models take a frequentist approach to model training. As you in- +troduce non-linearities and apply gradient methods to solving these optimization problems, +we become less and less likely to know we have reached a true minima. +In theory, our +true weights are drawn from an intractable prior distribution. If we approach the problem +through a Bayesian lens, we can better contextualize our model’s output and weights on the +input data. Using variational inference techniques, we can design a network whose weights +are drawn from a learnable, tractable posterior. We present BayesSpeech; a Bayesian Trans- +former Network for End-to-End Automatic Speech recognition where feed forward layers are +contextualized with probability distributions. +2. Background +2.1 Automatic Speech Recognition +Automatic Speech Recognition models have been evolving rapidly in recent years. Models +can either be sub-domain specific and focus on speech representation (Mohamed et al., 2012) +(Lee et al., 2009) (Conneau et al., 2020) (Devlin et al., 2018) (Schneider et al., 2019) (Chung +et al., 2019) (Maas, 2013) (Baevski et al., 2020) attention (Vaswani et al., 2017) (Chorowski +et al., 2015) (Conneau et al., 2020) or be end-to-end and incorporate the aforementioned +components into one model and jointly train them. +2.1.1 Connectionist Transporal Classification +In order to jointly train end-to-end model including alignment, and encoding/decoding the +input/output sequence, Connectionist Transporal (CTC) Loss can be used to better manage +the alignments (Graves et al., 2006). Alignment of the input and output sequence becomes +especially challenging in speech recognition tasks as the input sequence is generally longer +than the output sequence. CTC Loss aids this process by penalizing models based on the +joint probability of the current token in the sequence and all other tokens predicted. +For decoders that only output set-length sequences, we can further augment the CTC loss +by utilizing traditional Cross Entropy loss on the predictions (Hori et al., 2017). By enabling +a joint CTC and Cross Entropy (CE) loss function we not only penalize characters based on +their sequence but also on their absolute positioning in the output. This is covered further +in Section 3.2.2. +2.1.2 Models for End-to-End Speech Recognition +End-to-End Speech recognition models combine all of the individual aspects of Automatic +Speech Recognition into one model that is trained jointly. Traditional models rely on RNNs, +2 + +LSTMs, and generally recurrences for defining the output sequence (Liu et al., 2022)(Chan +et al., 2016). These models are cumbersome to train and parallelize and create additional +operational hurdles in properly tuning. +Recently, new models involving convolutions and linear outputs have been used within +Encoder-Decoder frameworks for end-to-end speech recognition tasks. These models such as +Jasper (Li et al., 2019), Conformer (Gulati et al., 2020), and Transformer (Dong et al., 2018) +are huge models with one billion or more parameters relying on feed-forward architectures. +The three models were all trained for multiple days on multiple GPUs and required incred- +ible compute power. The Speech-Transformer model (Dong et al., 2018) tried to address +these issues by creating a thinner model to yield similar performance on the WSJ dataset. +However, compared to its larger counterparts, it did not have the same performance charac- +teristics. Although it did lend hope that smaller models could be trained to compete with +their larger counterparts. +2.2 Bayesian Methods +Bayesian models have begun to show further promise in multiple fields such as Image Recog- +nition (Blundell et al., 2015), attention mechanisms (Zhang et al., 2021) (Fan et al., 2020), +and auto-encoders (Kingma & Welling, 2013). In a Bayesian approach, networks weights +are samples from an intractable distribution which we can estimate over training iterations +through Variational Inference. +2.2.1 Variational Inference +Variational Inference (VI) is the estimation of an intractable distribution through minimiz- +ing the Kullback-Lieblier divergence (DKL) between a sample and some true distribution. +Different works have shown that these estimation methods have value when applied to a +Bayesian Neural Network (Graves, 2011) (Kingma & Welling, 2013). There are two different +approaches to VI that largely depend on the what the true distribution is believed to be. If +the true prior can be any distribution Monte-Carlo sampling is the only option for estimating +the gradient from DKL. When using MC sampling, a network is sampled multiple times for +the same input and the gradients are averaged together across the number of samples. +If the prior is generally assumed to be Gaussian, the KL Divergence can be explicitly calcu- +lated and the Local Reparameterization Trick (Kingma et al., 2015) can be used for finding +the gradient with just one sample. This is discussed further in Section 3.1.4. +3 + +2.2.2 Bayes by Backprop +Blundell et al. +introduced the Bayes by Backprop algorithm for jointly learning the in- +tractable distribution as well as a domain problem in a Bayesian Neural Network. They +introduce the joint loss function in two parts: +1. Weighting the KL Divergence of the model against the epoch +2. Creating an Evidence Lower Bound (ELBO) on the loss function for the purpose of +training +We discuss the weighting of the KL Divergence loss term over time in Section 3.2.1. The +introduction of ELBO loss serves as the method for tuning an efficient approximator for +the Maximum Likelihood given an a-posteriori inference of the parameters. Because we are +always sampling from an intractable distribution, the KL divergence term can be thought +of as a regularization constant on the network. Over time, the KL divergence impact on +loss will encourage the approximate posterior to be close to the true prior. +While not +readily apparent, ELBO loss is implicitly involved in the loss function described in Section +3.2.3. +3. Research & Methods +As introduced above, sampling network weights (Bayesian approach) rather than explicitly +defining them (frequentist approach) has been shown to have increased performance and +faster convergence times. Our goal is to produce a network, leveraging Bayesian layers, to +compete with state-of-the-art models and require less training time. BayesSpeech, is largely +based on the no-recurrence model, the Transformer (Vaswani et al., 2017). While we leverage +the model’s general architecture, we introduce modified Encoder and Decoder layers with +Bayesian, Pointwise Feed Forward sub-layers. In this section, we explore the core components +of the model, the model’s architecture, and a new training methodology for an ensemble loss +function. +3.1 Core Components +3.1.1 Attention Mechanisms +Part of Vaswani et al.’s Transformer (Vaswani et al., 2017) network was the introduction +of Scaled Dot-Product attention and, further, Multi-Headed Attention. The goal of these +mechanisms is to generate a temporal-rich representation of the inputs by attending to +different positions within the input sequence. An attention function maps a query (or set +of queries) in a matrix, Q, and a set of key-value pairs in matrices, K, V , to the input +4 + +sequence. +Scaled Dot-Product Attention first takes the softmax of the matrix multiplication of Q, K. +Then it matrix multiplies that value with V and normalizes by √dk (the dimension of the +keys, K) (Equation 1). The normalization by the key size is used to prevent the softmax +function from suffering the vanishing gradient problem. +Attention(Q, K, V ) = Softmax(QKT +√dk +)V +(1) +Scaled Dot-Product attention only performs a single attention function at a time. Multi-Head +Attention addresses this by linearly projecting the queries, keys, and values h times to each of +the input dimensions (dq, dk, dv, respectively). Then Scaled Dot-Product attention is applied +to these newly projected inputs in order to attend across the h different ”heads” (Equation +2). +Each headi is equal to Attention(QW Q +i , KW K +i , V W V +i ). +This way the model jointly +attends different input representations across the different subspaces introduced through the +projection. +MultiHead(Q, K, V ) = Concat(head1, ..., headh)W O +(2) +3.1.2 Sinusoidal Positional Encoding +Because the model does not have recurrence or convolutions, in order to make use of the +temporal attentions from the Multi-Head modules, position information about the position of +the tokens in the sequence must be introduced (Vaswani et al., 2017). Positional Encodings +are used in the both the Encoder and Decoder modules to align each module’s outputs and +allow them to be summed. The model leverages Sinusoidal embeddings (Equation 3) where +the frequency corresponds to the token position (pos) and the dimension (i). Vaswani et al. +hypothesize that using this encoding function will make it easy for the model to learn the +attention weights for relative positions and adapt for longer sequences. +PositionalEmbedding(pos,i) = +� +� +� +sin(pos/10000 +2i +dmodel ) +if i < dmodel +2 +cos(pos/10000 +2i +dmodel ) +if i ≥ dmodel +2 +(3) +3.1.4 Proposal: Bayesian, Positionwise Feed Forward Layer +Our primary proposal, is the Bayesian, Positionwise Feed Forward Layer (Figure 1). Rather +than use two linear layers with dropout, we substitute the input layer with a Bayesian Linear +Layer suplemented with the Local Reparameterization Trick (Bayes Linear LRT) (Kingma +5 + +et al., 2015) and remove the dropout. Other approaches to producing Bayesian components +rely on sampling the same input sequence multiple times from a Monte-Carlo process in order +to define an average gradient for modeling the intractable posterior (Blundell et al., 2015). +In addition to being computationally expensive, this can also lead to high variance in the +gradients increasing training time. The Local Reparameterization Trick addresses this by +assuming that both the prior and posterior are Gaussian. Using only a single sample, the KL +divergence between the estimators can be explicitly solved for. This new estimator is efficient +(as it has less computational complexity) and reduces the variance in the gradient. +For each Bayesian Layer in Figure 1, let din, dout be the input and output dimensions of the +layer, respectiveley. We define matrices Wµ ∈ RdoutXdin and Wρ ∈ RdoutXdin representing the +mean and variance scalar for each weight in the network. Similarly we have bias vectors for +the output of Wµ and Wρ defined as bµ ∈ Rdout and bρ ∈ Rdout, respectively. Finally, at each +iteration we sample a term from a standard Gaussian, ϵ ∼ N(0, 1). +In order to calculate the output of the layer, we define a function for explicitly calculating +the KL divergence when the prior and posterior are both Gaussian (Equation 4). The prior +is represented by p and posterior represented by q. A sum is taken over all elements in the +input matrices. +KLD(µp, σp, µq, σq) = 1 +2 +� +(2 log(σp +σq +) − (1 + (σp +σq +)2) + (µp − µq +σp +)2) +(4) +In the forward pass of the algorithm, we first must use our variance parameter ρ for estimating +our standard deviation of weight and biases (Equation 5). The same applies for the bias +Figure 1: Bayesian, Positionwise Feed Forward Layer Diagram +6 + +Activation (GELU +Bayes Linear LRT +Linearvector b (e.g. we arrive at a bσ using bρ). +Wσ = log(1 + eWρ) +(5) +Next we sample from our Gaussian and introduce variance in the weights and biases for the +input sequence X (Equation 6, Equation 7). +Wout = XW T +µ + +� +(W 2 +σ)TX ∗ ϵ +(6) +bout = XbT +µ + bσ ∗ ϵ +(7) +Then we calculate the KL divergence between our estimated posteriors and true priors for +the weights and biases: WKL = KLD(0, 1, Wµ, Wσ) and bKL = KLD(0, 0.1, bµ, bσ). Finally, +the forward computation sets a global KL divergence term (KL = WKL + bKL) and returns +Wout+bout. The KL divergence term is used in the join loss function for tuning our variational +posterior. +3.1.4 Encoder Feature Extraction +Although the original Transformer architecture does not involve any convolutions, recent +work in the Image Recognition domain ((Simonyan & Zisserman, 2014)) has lent itself useful +for sequence-to-sequence ASR tasks (Hori et al., 2017). The modified VGG Network from +(Hori et al., 2017) is used in the Encoder to further enhance in the input feature set drawing +ideas from unsupervised speech representation tasks as seen in (Chung et al., 2019), (Lee et +al., 2009), and (Mohamed et al., 2012). The output from this initial convolutional layer is +passed to the encoder layers in the final network. +3.1.5 BayesSpeech Model +Putting this together, we arrive at our final model architecture (Figure 2). The model passes +the input through an Encoder (Figure 2a) and then passes the encoder output through a +Decoder (Figure 2b). In our model, we use 12 encoder block layers (de = 12) and 6 decoder +block layers (dd = 6). These 18 inner layers contain a Mutli-Head attention block as well as +a Bayesian Position-wise Feed Forward block. The model has a dimension of 512 and a feed +forward dimension of 2148. +3.2 Model Training +In order to train our transformer model, we utilize a variation of the Bayes-By-Backprop +algorithm (Blundell et al., 2015) with a joint Connectionist Temporal Classification and +7 + +(a) BayesSpeech Encoder +(b) BayesSpeech Decoder +Figure 2: BayesSpeech Encoder and Decoder Diagrams +Cross Entropy loss function (Joint CTC, CrossEntropy Loss). The two-stage training is +meant to: +1. further tune the sampling mechanics for the variational posterior distribution the +weights are drawn from +2. and learn the temporal alignments and classification loss of the output tokens. +We have found that trying to optimize each component separately leads to over-fitting in +one of the domains of this problem. If we choose a large step size and seek to minimize +the aggregate KL divergence across the Bayesian layers, we cannot further learn the align- +ments. And if we choose a small step-size and learn the alignments, we introduce too much +randomness in the output for our results to be meaningful. Therefore we introduce a scal- +ing function, similar to the one in Bayes-by-Backprop for managing the tradeoff over epoch +iterations (Minibatch Weighting). Our model was trained on the LibriSpeech-960 dataset +(Panayotov et al., 2015). The utterances in the dataset were converted to Mel Spectrogram +form with 80 channels a width of 20ms and a stride of 10ms. +3.2.1 Tuning Variational Posterior +The Bayesian part of our model tries to fit a variational posterior distribution (qθ) to a true +intractable posterior (p) for each of the weights in the network. In order to do so, we reduce +the problem of fitting qθ to that of a Minimum Description Length (MDL) problem (Hinton & +van Camp, 1993a) (Rissanen, 1978) (Hinton & van Camp, 1993b). While we have introduced +the explicit calculation of the Kullback-Leibler divergence between our variational posterior +and true prior above (DKL(qθ||p)), it is important to conceptualize the divergence as the +8 + +VGG Extractor +Linear +Norm +Sinusoidal Encoding +Dropout +Layer Norm +Multi-Head Attention +x de +Encoder Layer(s) +Bayesian Position-wise +Feed Forward +Norm +LinearEmbedding +Sinusoidal Encoding +Layer Norm +Multi-Head Attention +Dropout +Layer Norm +C +xdd +Decoder Layer(s) +Multi-Head Attention +Layer Norm +Norm +Bayesian Position-wise +Linear +Feed ForwardMDL problem. +The Minimum Description Length principal is that the best model for a given dataset bal- +ances the tradeoff between describing the model and describing the misfit between the model +and the data (Hinton & van Camp, 1993b). The KL divergence criteria we use has the goal +of keeping weights simple by penalizing the amount of information they contain. Ultimately, +this methodology will lead to a better separation between prediction accuracy and model +complexity and is explicitly differentiable (Graves, 2011). The variational loss function used +has two parts: +1. Error Loss - the expected value of negative log probability in samples from qθ(β) (where +β are the model’s parameters) +2. Complexity Loss - the KL divergence between the tractable, variational posterior and +the parameterized prior, DKL(qθ(β)||pα). +In each batch, we seek to gently tune our model’s variational posterior (qθ) to continue +random sampling but isolate different weights that have different levels of kurtosis. Due to +the minibatch weighting, discussed in a later section, we see a consistent decline in the joint +loss value dominated by the KL divergence term (blue, Figure 3b). Blundell et al. also show +that using this relative kurtosis can create thinner models with an explicit scheme for weight +pruning. Weights that are more leptokurtic are kept while platykurtic ones are discarded. +While this is beyond the scope of this paper, it would present and interesting future research +case for the model presented. +(a) CTC Loss Over Training Iterations +(b) Joint (Blue) and Scaled (Red) Loss Over Time +Figure 3: Loss Functions over Training Iterations +9 + +CTCLossOverEpochs +CTC LoSS +CTCLoss(Smooth) +4.0 +3.5 +3.0 +2.5 +2.0 +1.5 +0 +1000 +2000 +3000 +4000 +Training IterationComparisonofScaledandJointLoss +1e7 +1e7 +7 +7.57 +6 +7.56 +Scaled Loss (Red) +Joint Loss (Blue) +5 +4 +7.55 +3 +7.54 +2 +1 +7.53 +0 +1000 +2000 +3000 +4000 +idx3.2.2 Joint CTC, CrossEntropy Loss +In order to penalize the model for alignment of the input sequence to the output tokens, we +utilize a joint Connectionist Transporal Classification (Graves et al., 2006) and Cross Entropy +Loss function. The goal of this two term loss function is to manage a gradient through the +alignment of tokens in the feature input (CTC) as well as the actual classification loss of +the aligned output and the true tokens. The loss functions weights the two as so: L(X) = +0.3 ∗ CTC(X) + 0.7 ∗ CE(X). We do this in order to help smooth out the gradient while +maintaining the proper loss to back-propagate through the network. Due to the adversarial +nature of the Bayesian outputs, we find that this joint loss descends rapidly then continues +to descend without adjustment to the original learning rate (Figure 3a). In our training, we +held the learning rate fixed at 10−6. +3.2.3 Minibatch Weighting +Blundell et al. found that earlier epochs have a greater importance on tuning of the varia- +tional posterior than later ones. We adopt a similar methodology where we weight the KL +divergence term according to the epoch (e) and number of epochs (ne) (Equation). +MinibatchWeight(e, ne) = 2ne−e +2ne − e +(8) +To better aid training over time, we choose an epoch indexer where the epoch index is +integer divided by 10. When the training loop runs for multiple hours, this helps keep the +KL divergence more heavily weighted at first. We then weight the KL divergence term by +the minibatch weight term (Equation 9). The KLdiv term is the sum of all KL divergences +over the Bayesian layers. +L(X, e, ne) = MinibatchWeight(e, ne) ∗ KLdiv + 0.3 ∗ CTC(X) + 0.7 ∗ CE(X) +(9) +4. Results +We split our model into two variants: one that outputs a character sequence and one that +outputs tokenized word-pieces from a Sentencepiece language model with vocab size of 1000 +(Kudo & Richardson, 2018). We trained each model variant on a single A-100 GPU through +Google Colab for 8 hours with a batchsize of 24. +As shown in Table 1, our model performs nearly as well as the state of the art ASR models. +Our Bayes speech model reaches respectable Word Error Rates with and without a language +model on the LibriSpeech dataset. The Bayes Model as well was trained for just 8 hours on +10 + +Model +WER (w/o LM) +WER (w/ LM) +test-clean +test-other +test-clean +test-other +LAS (Chan et al., 2016) +2.89% +6.98% +2.33% +5.17% +Transformer (Vaswani et al., 2017) +2.4% +5.6% +2.0% +4.6% +Conformer (Gulati et al., 2020) +2.1% +5.0% +2.0% +4.3% +BayesSpeech +4.5% +6.5% +4.0% +5.7% +Table 1: WER Results on LibriSpeech dataset +a single GPU. For instance, the Conformer model was trained over the course of multiple +days on multiple GPUs (8). During evaluation, we use beam search with a beam width of +10 over the set of possible decoded sequences. This appears to be the standard decoding +methodology giving the probabilistic output of the model’s decoder. +When the input sequence passes through our Bayesian feed forward layers, we believe this +creates an adversarial input stream. Rather than artificially augment the input Mel Spec- +trogram inputs (Park et al., 2019), these layers produce a probabilistic feature encoding of +the input. We believe that this general adversarial training technique allows our model to +converge faster with less training time and resources. The randomness introduced in the +model also helps better contextualize outputs. As we continue to tune the variational poste- +rior over the weights, I imagine we would see a dramatic increase in performance. Because +our model yielded reasonable results after 8 hours, we stopped training but future work may +investigate if increased training could further improve our performance. There may also be a +benefit to equally weighting the variational component and the CTC loss component of the +global loss function. Similarly, in future work it may be useful to explore systematic model +pruning as presented in Blundell et al. +5. Conclusion +Currently, best in class Automatic Speech Recognition solutions require multiple days of +training on multiple GPUs. These models also take a frequentist approach to weight training. +In this work, we present BayesSpeech; a Bayesian Transformer Network for learning an +intractable posterior distribution over which weights are drawn in feed forward layers. We +believe this probabilistic encoding of the input feature set creates a better representation of +the input Mel Spectrogram. This mechanic in conjunction with a joint loss function yields +near state-of-the-art results on the LibriSpeech dataset. +11 + +References +Baevski, A., Zhou, H., Mohamed, A., & Auli, M. 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Bayesian attention belief networks. arXiv. +Retrieved from https://arxiv.org/abs/2106.05251 +doi: 10.48550/ARXIV.2106 +.05251 +14 + diff --git a/PdFIT4oBgHgl3EQfeytv/content/tmp_files/load_file.txt b/PdFIT4oBgHgl3EQfeytv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ef4af17b306842f570278681cf9fa4e70bf2e19 --- /dev/null +++ b/PdFIT4oBgHgl3EQfeytv/content/tmp_files/load_file.txt @@ -0,0 +1,672 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf,len=671 +page_content='BayesSpeech: A Bayesian Transformer Network for Automatic Speech Recognition Will Rieger Master of Science in Computer Science Department of Computer Science The University of Texas at Austin Abstract Recent developments using End-to-End Deep Learning models have been shown to have near or better performance than state of the art Recurrent Neural Networks (RNNs) on Automatic Speech Recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' These models tend to be lighter weight and require less training time than traditional RNN-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' However, these models take frequentist approach to weight training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In theory, network weights are drawn from a latent, intractable probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We introduce BayesSpeech for end-to-end Automatic Speech Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' BayesSpeech is a Bayesian Transformer Network where these intractable posteriors are learned through variational inference and the local reparameterization trick without recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We show how the introduction of variance in the weights leads to faster training time and near state-of-the-art perfor- mance on LibriSpeech-960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Introduction In the majority of neural networks, randomness is usually introduced through perturbation of the input or randomly removing nodes from the network (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' There has been great success using these methods across a variety of domains including Automatic Speech Recognition (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Models continue to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' However and data augmentation methods rarely take large leaps in terms of the features they can help express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Newer models are generally larger and larger and require incredible amounts of compute to properly train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We especially see this in the field of Automatic Speech Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Newer models such as Jasper (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019), the Conformer (Gulati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020), LAS (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2016), and the Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017) all require training for multiple days across multiple GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Creating deeper models can certainly help attain better performance on the domain task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' But what if we approach the model differently and try to leverage their probabilistic nature?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='11276v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='AS] 16 Jan 2023 Most Neural Network models take a frequentist approach to model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' As you in- troduce non-linearities and apply gradient methods to solving these optimization problems, we become less and less likely to know we have reached a true minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In theory, our true weights are drawn from an intractable prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' If we approach the problem through a Bayesian lens, we can better contextualize our model’s output and weights on the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Using variational inference techniques, we can design a network whose weights are drawn from a learnable, tractable posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We present BayesSpeech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' a Bayesian Trans- former Network for End-to-End Automatic Speech recognition where feed forward layers are contextualized with probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1 Automatic Speech Recognition Automatic Speech Recognition models have been evolving rapidly in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Models can either be sub-domain specific and focus on speech representation (Mohamed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2012) (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2009) (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020) (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2018) (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019) (Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019) (Maas, 2013) (Baevski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020) attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017) (Chorowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015) (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020) or be end-to-end and incorporate the aforementioned components into one model and jointly train them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1 Connectionist Transporal Classification In order to jointly train end-to-end model including alignment, and encoding/decoding the input/output sequence, Connectionist Transporal (CTC) Loss can be used to better manage the alignments (Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Alignment of the input and output sequence becomes especially challenging in speech recognition tasks as the input sequence is generally longer than the output sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' CTC Loss aids this process by penalizing models based on the joint probability of the current token in the sequence and all other tokens predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' For decoders that only output set-length sequences, we can further augment the CTC loss by utilizing traditional Cross Entropy loss on the predictions (Hori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' By enabling a joint CTC and Cross Entropy (CE) loss function we not only penalize characters based on their sequence but also on their absolute positioning in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' This is covered further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2 Models for End-to-End Speech Recognition End-to-End Speech recognition models combine all of the individual aspects of Automatic Speech Recognition into one model that is trained jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Traditional models rely on RNNs, 2 LSTMs, and generally recurrences for defining the output sequence (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2022)(Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' These models are cumbersome to train and parallelize and create additional operational hurdles in properly tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Recently, new models involving convolutions and linear outputs have been used within Encoder-Decoder frameworks for end-to-end speech recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' These models such as Jasper (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019), Conformer (Gulati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020), and Transformer (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2018) are huge models with one billion or more parameters relying on feed-forward architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The three models were all trained for multiple days on multiple GPUs and required incred- ible compute power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The Speech-Transformer model (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2018) tried to address these issues by creating a thinner model to yield similar performance on the WSJ dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' However, compared to its larger counterparts, it did not have the same performance charac- teristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Although it did lend hope that smaller models could be trained to compete with their larger counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2 Bayesian Methods Bayesian models have begun to show further promise in multiple fields such as Image Recog- nition (Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015), attention mechanisms (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2021) (Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020), and auto-encoders (Kingma & Welling, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In a Bayesian approach, networks weights are samples from an intractable distribution which we can estimate over training iterations through Variational Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1 Variational Inference Variational Inference (VI) is the estimation of an intractable distribution through minimiz- ing the Kullback-Lieblier divergence (DKL) between a sample and some true distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Different works have shown that these estimation methods have value when applied to a Bayesian Neural Network (Graves, 2011) (Kingma & Welling, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' There are two different approaches to VI that largely depend on the what the true distribution is believed to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' If the true prior can be any distribution Monte-Carlo sampling is the only option for estimating the gradient from DKL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' When using MC sampling, a network is sampled multiple times for the same input and the gradients are averaged together across the number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' If the prior is generally assumed to be Gaussian, the KL Divergence can be explicitly calcu- lated and the Local Reparameterization Trick (Kingma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015) can be used for finding the gradient with just one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' This is discussed further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2 Bayes by Backprop Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' introduced the Bayes by Backprop algorithm for jointly learning the in- tractable distribution as well as a domain problem in a Bayesian Neural Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' They introduce the joint loss function in two parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Weighting the KL Divergence of the model against the epoch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Creating an Evidence Lower Bound (ELBO) on the loss function for the purpose of training We discuss the weighting of the KL Divergence loss term over time in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The introduction of ELBO loss serves as the method for tuning an efficient approximator for the Maximum Likelihood given an a-posteriori inference of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Because we are always sampling from an intractable distribution, the KL divergence term can be thought of as a regularization constant on the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Over time, the KL divergence impact on loss will encourage the approximate posterior to be close to the true prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' While not readily apparent, ELBO loss is implicitly involved in the loss function described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Research & Methods As introduced above, sampling network weights (Bayesian approach) rather than explicitly defining them (frequentist approach) has been shown to have increased performance and faster convergence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Our goal is to produce a network, leveraging Bayesian layers, to compete with state-of-the-art models and require less training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' BayesSpeech, is largely based on the no-recurrence model, the Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' While we leverage the model’s general architecture, we introduce modified Encoder and Decoder layers with Bayesian, Pointwise Feed Forward sub-layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In this section, we explore the core components of the model, the model’s architecture, and a new training methodology for an ensemble loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1 Core Components 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1 Attention Mechanisms Part of Vaswani et al.’s Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017) network was the introduction of Scaled Dot-Product attention and, further, Multi-Headed Attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The goal of these mechanisms is to generate a temporal-rich representation of the inputs by attending to different positions within the input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' An attention function maps a query (or set of queries) in a matrix, Q, and a set of key-value pairs in matrices, K, V , to the input 4 sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Scaled Dot-Product Attention first takes the softmax of the matrix multiplication of Q, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Then it matrix multiplies that value with V and normalizes by √dk (the dimension of the keys, K) (Equation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The normalization by the key size is used to prevent the softmax function from suffering the vanishing gradient problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Attention(Q, K, V ) = Softmax(QKT √dk )V (1) Scaled Dot-Product attention only performs a single attention function at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Multi-Head Attention addresses this by linearly projecting the queries, keys, and values h times to each of the input dimensions (dq, dk, dv, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Then Scaled Dot-Product attention is applied to these newly projected inputs in order to attend across the h different ”heads” (Equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Each headi is equal to Attention(QW Q i , KW K i , V W V i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' This way the model jointly attends different input representations across the different subspaces introduced through the projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' MultiHead(Q, K, V ) = Concat(head1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', headh)W O (2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2 Sinusoidal Positional Encoding Because the model does not have recurrence or convolutions, in order to make use of the temporal attentions from the Multi-Head modules, position information about the position of the tokens in the sequence must be introduced (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Positional Encodings are used in the both the Encoder and Decoder modules to align each module’s outputs and allow them to be summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The model leverages Sinusoidal embeddings (Equation 3) where the frequency corresponds to the token position (pos) and the dimension (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' hypothesize that using this encoding function will make it easy for the model to learn the attention weights for relative positions and adapt for longer sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' PositionalEmbedding(pos,i) = � � � sin(pos/10000 2i dmodel ) if i < dmodel 2 cos(pos/10000 2i dmodel ) if i ≥ dmodel 2 (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='4 Proposal: Bayesian, Positionwise Feed Forward Layer Our primary proposal, is the Bayesian, Positionwise Feed Forward Layer (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Rather than use two linear layers with dropout, we substitute the input layer with a Bayesian Linear Layer suplemented with the Local Reparameterization Trick (Bayes Linear LRT) (Kingma 5 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015) and remove the dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Other approaches to producing Bayesian components rely on sampling the same input sequence multiple times from a Monte-Carlo process in order to define an average gradient for modeling the intractable posterior (Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In addition to being computationally expensive, this can also lead to high variance in the gradients increasing training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The Local Reparameterization Trick addresses this by assuming that both the prior and posterior are Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Using only a single sample, the KL divergence between the estimators can be explicitly solved for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' This new estimator is efficient (as it has less computational complexity) and reduces the variance in the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' For each Bayesian Layer in Figure 1, let din, dout be the input and output dimensions of the layer, respectiveley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We define matrices Wµ ∈ RdoutXdin and Wρ ∈ RdoutXdin representing the mean and variance scalar for each weight in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Similarly we have bias vectors for the output of Wµ and Wρ defined as bµ ∈ Rdout and bρ ∈ Rdout, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Finally, at each iteration we sample a term from a standard Gaussian, ϵ ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In order to calculate the output of the layer, we define a function for explicitly calculating the KL divergence when the prior and posterior are both Gaussian (Equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The prior is represented by p and posterior represented by q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' A sum is taken over all elements in the input matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' KLD(µp, σp, µq, σq) = 1 2 � (2 log(σp σq ) − (1 + (σp σq )2) + (µp − µq σp )2) (4) In the forward pass of the algorithm, we first must use our variance parameter ρ for estimating our standard deviation of weight and biases (Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The same applies for the bias Figure 1: Bayesian, Positionwise Feed Forward Layer Diagram 6 Activation (GELU Bayes Linear LRT Linearvector b (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' we arrive at a bσ using bρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Wσ = log(1 + eWρ) (5) Next we sample from our Gaussian and introduce variance in the weights and biases for the input sequence X (Equation 6, Equation 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Wout = XW T µ + � (W 2 σ)TX ∗ ϵ (6) bout = XbT µ + bσ ∗ ϵ (7) Then we calculate the KL divergence between our estimated posteriors and true priors for the weights and biases: WKL = KLD(0, 1, Wµ, Wσ) and bKL = KLD(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1, bµ, bσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Finally, the forward computation sets a global KL divergence term (KL = WKL + bKL) and returns Wout+bout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The KL divergence term is used in the join loss function for tuning our variational posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='4 Encoder Feature Extraction Although the original Transformer architecture does not involve any convolutions, recent work in the Image Recognition domain ((Simonyan & Zisserman, 2014)) has lent itself useful for sequence-to-sequence ASR tasks (Hori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The modified VGG Network from (Hori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017) is used in the Encoder to further enhance in the input feature set drawing ideas from unsupervised speech representation tasks as seen in (Chung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019), (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2009), and (Mohamed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The output from this initial convolutional layer is passed to the encoder layers in the final network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='5 BayesSpeech Model Putting this together, we arrive at our final model architecture (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The model passes the input through an Encoder (Figure 2a) and then passes the encoder output through a Decoder (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In our model, we use 12 encoder block layers (de = 12) and 6 decoder block layers (dd = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' These 18 inner layers contain a Mutli-Head attention block as well as a Bayesian Position-wise Feed Forward block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The model has a dimension of 512 and a feed forward dimension of 2148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2 Model Training In order to train our transformer model, we utilize a variation of the Bayes-By-Backprop algorithm (Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015) with a joint Connectionist Temporal Classification and 7 (a) BayesSpeech Encoder (b) BayesSpeech Decoder Figure 2: BayesSpeech Encoder and Decoder Diagrams Cross Entropy loss function (Joint CTC, CrossEntropy Loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The two-stage training is meant to: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' further tune the sampling mechanics for the variational posterior distribution the weights are drawn from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' and learn the temporal alignments and classification loss of the output tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We have found that trying to optimize each component separately leads to over-fitting in one of the domains of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' If we choose a large step size and seek to minimize the aggregate KL divergence across the Bayesian layers, we cannot further learn the align- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' And if we choose a small step-size and learn the alignments, we introduce too much randomness in the output for our results to be meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Therefore we introduce a scal- ing function, similar to the one in Bayes-by-Backprop for managing the tradeoff over epoch iterations (Minibatch Weighting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Our model was trained on the LibriSpeech-960 dataset (Panayotov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The utterances in the dataset were converted to Mel Spectrogram form with 80 channels a width of 20ms and a stride of 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1 Tuning Variational Posterior The Bayesian part of our model tries to fit a variational posterior distribution (qθ) to a true intractable posterior (p) for each of the weights in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In order to do so, we reduce the problem of fitting qθ to that of a Minimum Description Length (MDL) problem (Hinton & van Camp, 1993a) (Rissanen, 1978) (Hinton & van Camp, 1993b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' While we have introduced the explicit calculation of the Kullback-Leibler divergence between our variational posterior and true prior above (DKL(qθ||p)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' it is important to conceptualize the divergence as the 8 VGG Extractor Linear Norm Sinusoidal Encoding Dropout Layer Norm Multi-Head Attention x de Encoder Layer(s) Bayesian Position-wise Feed Forward Norm LinearEmbedding Sinusoidal Encoding Layer Norm Multi-Head Attention Dropout Layer Norm C xdd Decoder Layer(s) Multi-Head Attention Layer Norm Norm Bayesian Position-wise Linear Feed ForwardMDL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The Minimum Description Length principal is that the best model for a given dataset bal- ances the tradeoff between describing the model and describing the misfit between the model and the data (Hinton & van Camp, 1993b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The KL divergence criteria we use has the goal of keeping weights simple by penalizing the amount of information they contain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Ultimately, this methodology will lead to a better separation between prediction accuracy and model complexity and is explicitly differentiable (Graves, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The variational loss function used has two parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Error Loss - the expected value of negative log probability in samples from qθ(β) (where β are the model’s parameters) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Complexity Loss - the KL divergence between the tractable, variational posterior and the parameterized prior, DKL(qθ(β)||pα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In each batch, we seek to gently tune our model’s variational posterior (qθ) to continue random sampling but isolate different weights that have different levels of kurtosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Due to the minibatch weighting, discussed in a later section, we see a consistent decline in the joint loss value dominated by the KL divergence term (blue, Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' also show that using this relative kurtosis can create thinner models with an explicit scheme for weight pruning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Weights that are more leptokurtic are kept while platykurtic ones are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' While this is beyond the scope of this paper, it would present and interesting future research case for the model presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' (a) CTC Loss Over Training Iterations (b) Joint (Blue) and Scaled (Red) Loss Over Time Figure 3: Loss Functions over Training Iterations 9 CTCLossOverEpochs CTC LoSS CTCLoss(Smooth) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='5 0 1000 2000 3000 4000 Training IterationComparisonofScaledandJointLoss 1e7 1e7 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='57 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='56 Scaled Loss (Red) Joint Loss (Blue) 5 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='55 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='54 2 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='53 0 1000 2000 3000 4000 idx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2 Joint CTC, CrossEntropy Loss In order to penalize the model for alignment of the input sequence to the output tokens, we utilize a joint Connectionist Transporal Classification (Graves et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2006) and Cross Entropy Loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The goal of this two term loss function is to manage a gradient through the alignment of tokens in the feature input (CTC) as well as the actual classification loss of the aligned output and the true tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The loss functions weights the two as so: L(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='3 ∗ CTC(X) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='7 ∗ CE(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We do this in order to help smooth out the gradient while maintaining the proper loss to back-propagate through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Due to the adversarial nature of the Bayesian outputs, we find that this joint loss descends rapidly then continues to descend without adjustment to the original learning rate (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In our training, we held the learning rate fixed at 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='3 Minibatch Weighting Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' found that earlier epochs have a greater importance on tuning of the varia- tional posterior than later ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We adopt a similar methodology where we weight the KL divergence term according to the epoch (e) and number of epochs (ne) (Equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' MinibatchWeight(e, ne) = 2ne−e 2ne − e (8) To better aid training over time, we choose an epoch indexer where the epoch index is integer divided by 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' When the training loop runs for multiple hours, this helps keep the KL divergence more heavily weighted at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We then weight the KL divergence term by the minibatch weight term (Equation 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The KLdiv term is the sum of all KL divergences over the Bayesian layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' L(X, e, ne) = MinibatchWeight(e, ne) ∗ KLdiv + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='3 ∗ CTC(X) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='7 ∗ CE(X) (9) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Results We split our model into two variants: one that outputs a character sequence and one that outputs tokenized word-pieces from a Sentencepiece language model with vocab size of 1000 (Kudo & Richardson, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We trained each model variant on a single A-100 GPU through Google Colab for 8 hours with a batchsize of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' As shown in Table 1, our model performs nearly as well as the state of the art ASR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Our Bayes speech model reaches respectable Word Error Rates with and without a language model on the LibriSpeech dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The Bayes Model as well was trained for just 8 hours on 10 Model WER (w/o LM) WER (w/ LM) test-clean test-other test-clean test-other LAS (Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2016) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='89% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='98% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='33% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='17% Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2017) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='4% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='6% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='6% Conformer (Gulati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2020) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='1% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='3% BayesSpeech 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='5% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='5% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='0% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='7% Table 1: WER Results on LibriSpeech dataset a single GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' For instance, the Conformer model was trained over the course of multiple days on multiple GPUs (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' During evaluation, we use beam search with a beam width of 10 over the set of possible decoded sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' This appears to be the standard decoding methodology giving the probabilistic output of the model’s decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' When the input sequence passes through our Bayesian feed forward layers, we believe this creates an adversarial input stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Rather than artificially augment the input Mel Spec- trogram inputs (Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=', 2019), these layers produce a probabilistic feature encoding of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We believe that this general adversarial training technique allows our model to converge faster with less training time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' The randomness introduced in the model also helps better contextualize outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' As we continue to tune the variational poste- rior over the weights, I imagine we would see a dramatic increase in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Because our model yielded reasonable results after 8 hours, we stopped training but future work may investigate if increased training could further improve our performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' There may also be a benefit to equally weighting the variational component and the CTC loss component of the global loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Similarly, in future work it may be useful to explore systematic model pruning as presented in Blundell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Conclusion Currently, best in class Automatic Speech Recognition solutions require multiple days of training on multiple GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' These models also take a frequentist approach to weight training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In this work, we present BayesSpeech;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' a Bayesian Transformer Network for learning an intractable posterior distribution over which weights are drawn in feed forward layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' We believe this probabilistic encoding of the input feature set creates a better representation of the input Mel Spectrogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' This mechanic in conjunction with a joint loss function yields near state-of-the-art results on the LibriSpeech dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 11 References Baevski, A.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Listen, attend and spell: A neural network for large vocabulary conversational speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' In 2016 ieee interna- tional conference on acoustics, speech and signal processing (icassp) (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' 4960-4964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Bayesian attention belief networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content=' Retrieved from https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='05251 doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='2106 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} +page_content='05251 14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFIT4oBgHgl3EQfeytv/content/2301.11276v1.pdf'} diff --git a/Q9AyT4oBgHgl3EQf7vq-/content/tmp_files/2301.00845v1.pdf.txt b/Q9AyT4oBgHgl3EQf7vq-/content/tmp_files/2301.00845v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..abdd79d144e9be7e8380f63bea85ce075efcc03f --- /dev/null +++ b/Q9AyT4oBgHgl3EQf7vq-/content/tmp_files/2301.00845v1.pdf.txt @@ -0,0 +1,1667 @@ +Weak solutions to the near-field reflector problem with +spatial restrictions approached with generalized +reflectors constructed from ellipsoids +Dylanger S. Pittman +400 Dowman Drive, Atlanta +Abstract +We motivate then formulate a novel variant of the near-field reflector problem +and call it the near-field reflector problem with spatial restrictions. Let O be +an anisotropic point source of light and assume that we are given a bounded +open set U. Suppose that the light emitted from the source at O in directions +defined by the aperture D ⊆ S2, of radiance g(m) for m ∈ D, is reflected off +R ⊂ U, creating the irradiance f(x) for x ∈ T. The inverse problem consists of +constructing the reflector R ⊆ U from the given position of the source O, the +input aperture D, radiance g, ‘target’ set T, and irradiance f. We focus entirely +on the case where the target set T is finite. +Keywords: +partial differential equations, geometric optics, geometry +2020 MSC: 78A05, 35, 51, 53 +1. Introduction +Let O be the origin of R3, and let S2 be the unit sphere centered at O. We +treat points on S2 as unit vectors with initial points at O. Let an aperture be +a subset of S2; in our work, the aperture will be an open set. Physically, it +makes sense to consider O as the location of an anisotropic point source of light +such that rays of light are emitted in a set of directions defined by an aperture +D ⊆ S2. +Email address: dpittm2@emory.edu (Dylanger S. Pittman) +Preprint submitted to Constructive Approximation +January 4, 2023 +arXiv:2301.00845v1 [math.AP] 2 Jan 2023 + +Definition 1.1. Assume that we are given an aperture that is a connected open +set D ⊆ S2, and a function ρ : D → (0, ∞) that is continuous and almost +everywhere differentiable. Then a reflector is the set R = {mρ(m)|m ∈ D} ⊂ +R3. +If ρ is a smooth function, we can call R a smooth reflector. +Given an aperture, D, that is a connected open set, assume that we have a +continuous, almost everywhere differentiable, positive function ρ : D → (0, ∞), +and a corresponding reflector R = {mρ(m)|m ∈ D}. Suppose that a ray origi- +nating from O in the direction m ∈ D is incident on the reflector R at the point +mρ(m). If ρ is differentiable at m, there is a unit vector, n(m), normal to the +reflector R at mρ(m). Therefore, by the reflection law of geometric optics, a +ray from O of direction m reflects off the point mρ(m) in the direction +y(m) = m − 2⟨m, n(m)⟩n(m) +(1) +where ⟨m, n(m)⟩ is the standard Euclidean inner product in R3 and n(m) is +oriented such that ⟨m, n(m)⟩ > 0 [1]. +The reflector R is designed such that the ray described by the point mρ(m) ∈ +R and the direction y(m) corresponds to some element in a prespecified target +set T. What one means by a ‘target set’ changes depending on the context, and +the correspondence between y(m); also, an element of the target set can also +vary depending on one’s needs. Hence a target set can represent many things. +For example, if the target set T is a subset of S2, then a possible correspondence +can be +y(m) +|y(m)| ∈ T; see [2]. Physically, in this case, T can be considered as a set +of directions for rays of light. If T is a subset R3 \ {O}, then for an example of +another possible correspondence, we can say that for every m ∈ D, there exists +an a(m) > 0 such that a(m)y(m) + mρ(m) ∈ T; see [3] and [4]. Physically, in +this case, T can be considered as a region that one wants to illuminate. +Assume that g is an integrable and nonnegative function over an aperture D, +and f is an integrable and nonnegative function over a target set T. Physically +speaking, we say g(m) for m ∈ D is the radiance of the source at O in the +directions m ∈ D, or that g is a radiance distribution over D. We also say f(x) +2 + +for x ∈ T is the irradiance of the target set at x ∈ T, or that f is an irradiance +distribution over T. +A reflector system comprises of an aperture D, O, a reflector R, an integrable +and nonnegative function g over D, and a target set T with an integrable and +nonnegative function f over T. From a physical perspective: light emitted from +the source at O in directions defined by the aperture D, of radiance g(m) for +m ∈ D, is reflected off R, creating the irradiance f(x) for x ∈ T. An example +that can serve as an illustration is shown in Figure 1. +A reflector problem is, in short, an inverse problem that seeks to complete +a reflector system by creating a reflector that fits the other information given. +Specifically, suppose we are given O, an aperture D, an integrable and non- +negative function g over D, and a target set T with an integrable and nonneg- +ative function f over T. The aim of a reflector problem is to find a continu- +ous, almost everywhere differentiable, positive ρ over D such that the reflector +R = {mρ(m)|m ∈ D} produces the specified in advance irradiance distribution +f on T. +Reflector problems have been well studied due to their utility in physics and +engineering. Such problems have found numerous applications in the construc- +tion of reflector antennas (see [5], [6]), mirror design [7], heat transfer [8], and +beam shaping [9]. +We only consider in the high-frequency approximation of +light, where the laws of geometric optics apply. We now proceed with a general +description and motivation for the near-field reflector problem. +2. The Near-Field Reflector Problem +We discuss a reflector problem that we call the ‘near-field reflector prob- +lem.’ In short, the near-field reflector problem aims to design a reflector that +redistributes the light from the origin onto a set a finite distance away from the +origin. +In this part, when we say surface, we mean it in the differential geometric +sense; see Definition 12.4 in [10]. Suppose that we are given a reflector system +3 + +O +The plane reflector R +The surface normal to R +Light rays going to some target set T +Figure 1: Here is the most basic example of a reflector system with a smooth reflector. Here R +is a plane. Every point on R has a normal. Light originates from the point O with directions +represented by points on the unit sphere S2 and travels according to some target set that is +neither shown nor specified. +4 + +consisting of +1. O, +2. an aperture D ⊂ S2, +3. a nonnegative g ∈ L1(D), +4. a bounded Borel set T ⊂ R3 \ {O} (typically either a subset of a surface +or a finite set), +5. a nonnegative and integrable function f : T → [0, ∞), +6. and a smooth function ρ : D → (0, ∞) with a smooth reflector R = +{mρ(m)|m ∈ D}. +From a physical perspective, this setup can be described as follows. The light +is emitted from the source at O in directions defined by the aperture D. Each +ray of direction m ∈ D has radiance g(m) and is reflected off R at the point +mρ(m) in the direction y(m) as described by (1). For every m ∈ D, there exists +an a(m) > 0 such that a(m)y(m) + mρ(m) ∈ T creating the irradiance f(x) for +x ∈ T. A basic illustration of this situation is depicted in Figure 2. With this +setup in mind, we proceed with a formulation of the near-field reflector problem +Let u = (u1, u2) be smooth local coordinates on S2 such that D lies in one +coordinate patch. The position vector of a point m ∈ D is m = m(u). We +choose the coordinates u1, u2 so that ⟨m, m1 × m2⟩ = 1 in D; here, ⟨, ⟩ denotes +the scalar product in R3 and mi = ∂m +∂ui , i = 1, 2. Observe that this implies that +⟨m, mi⟩ = 0, i = 1, 2. The first fundamental form of S2 is given by e = eijduiduj +where eij = ⟨mi, mj⟩. +Set r(m) = mρ(m), then r(m) defines a smooth surface R = {r(m)|m ∈ D}. +Let g = gijduiduj be the first fundamental form of R where gij = ⟨ri, rj⟩ = +ρiρj + ρ2eij, ri = +∂r +∂ui , and ρi = +∂ρ +∂ui . +Let n(m) is the normal vector field on R such that ⟨n(m), m⟩ > 0 everywhere +on R. Then +n(m) = (ρ2 + | ˜∇ρ|2)−1/2(r − ˜∇ρ) +(2) +5 + +O +Reflector R +Target set T with irradiance distribution f +Surface normals to R +Light Rays +Figure 2: Here is an illustration of the near-field reflector problem in R3. +The radiation +intensity at the origin O is given by a nonnegative function g ∈ L1(D). We want to find a +reflector R such that the reflected rays produce the prescribed irradiance distribution f on T. +6 + +where | ˜∇p|2 = ρiρjeij. This combined with equation (1) determines the direction +a ray will go after reflecting off R [11]. +We can now track the path of each ray described by the direction m ∈ D +to a point x(m) ∈ T. A ray, originating at O in direction m, hits the surface +R at a point r(m). Then, said ray reflects off R at r(m) in the direction y(m) +as defined by (1) and reaches T at some point x(m). Thus, from a physical +perspective, an irradiance f(x(m)) is created by the rays reflected at x(m). +This defines a mapping m → x that we call a reflector map; for convenience, +we denote x(m) as the image of m under the reflector map. The reflector map +x : D → T combined with equations (1) and (2) describes the ray tracing from +D to T. +If the reflector map is a diffeomorphism from D to T where T is a subset +of a smooth surface, then one can introduce the first fundamental form of T as +w = wijduiduj, where wij = ⟨xi, xj⟩, xi = +∂x +∂ui . +According to the differential form of the energy conservation law [1], +f(x(m))|J(x(m))| = g(m) +(3) +where J is the Jacobian determinant of the map x. Note that +J(x(m)) = ±dν(x(m)) +dσ(m) += ± +� +det(wij) +� +det(eij) +(4) +where dσ is the surface area element on S2, and dν is the surface area element +on T. We assign a ± sign to the Jacobian according to whether x preserves +the orientation or reverses it. Therefore, by integration of (3), for all Borel sets +ω ⊆ T, +� +x−1[ω] +gdσ = +� +ω +fdν +(5) +where x−1[ω] = {m ∈ D|x(m) ∈ ω} and +� +D gdσ = +� +T fdν. +With this motivation, we can now state the near-field reflector problem. +Assume that we are given O, an aperture D ⊂ S2 with a nonnegative function +g ∈ L1(D), and a bounded Borel set T ⊂ R3\{O} with a nonnegative, integrable +function f : T → [0, ∞). The goal is to find a smooth function ρ over D such +that: +7 + +1. The ray originating from O in the direction m ∈ D reflects off the reflector +R = {mρ(m)|m ∈ D} in accordance with equation (1) and reaches the +target set T. +2. g(m) on D is transformed by the reflector map into f on T; i.e. for all +Borel subsets ω ⊆ T, +� +x−1[ω] +gdσ = +� +ω +fdν +(6) +where x : D → T the reflector map corresponding to the reflector R = +{mρ(m)|m ∈ D}, x−1[ω] = {m ∈ D|x(m) ∈ ω}, dσ is the surface area +element on S2, and dν is the area element on T (ν is typically some discrete +or Lebesgue measure). +3. The law of total energy conservation is obeyed: +� +D gdσ = +� +T fdν. +The case where the reflector map is a diffeomorphism from D to T can be +alternatively formulated as a PDE of Monge-Amp`ere type; specifically equation +(4) from [12]. +There has been a lot of work done on the near-field reflector problem. In +1972, Schruben [3] found that if the target set was a subset of a plane in R3, one +can then derive an implicit integro-differential equation describing the reflector; +the existence of a solution was not proved. Then in [13], Schruben considered +the case where the target set was a small rotationally symmetric patch on the +plane. +In this case, when the radiance and the irradiance distributions are +rotationally symmetric, the equation derived in [3] can be solved as an ODE. +In 1989, Oliker [12] found a formulation of the near-field reflector problem in +the form of a strongly non-linear PDE of Monge-Amp`ere type. The exploration +of the said equation is difficult and in [12] was solved only for the rotationally +symmetric case. +In 1998, Kochengin and Oliker [4] introduced an alternative formulation to +the near-field reflector problem, which was a geometric approach involving the +analysis of the boundaries of convex sets generated by families of supporting +ellipsoids. This approach can also be considered a weak solution to the PDE +8 + +introduced in [12]. The strategy was to assume that the target set was a finite +set on a plane and constructively prove the existence of solutions for that case. +Since the reflectors that were constructed were convex, one can use the Blaschke +selection theorem (for more details, see [14]) to prove the existence of a solution +with a continuous target set on the plane. This method was largely motivated +by previous work done by Caffarelli and Oliker [2] which involved the analysis +of the boundaries of convex sets generated by families of supporting paraboloids +to solve a related problem. +In [15] a provably convergent numerical algorithm was introduced that ex- +plicitly finds the ellipsoids required to construct the reflectors described in [4]. +It was shown that this construction leads to infinitely many solutions; however, +the algorithm has the benefit of converging to a unique solution if we fix an ini- +tial point on the reflector. This algorithm and its variations have been explored +extensively in various scenarios. For example, Fournier, Cassarly, and Rolland +in [16] adapted the algorithm in [15], to situations where the light source is +not a single point; specifically, a flat rotationally symmetric emitter. In [17] a +method was proposed for smoothing out a reflector with a discrete irradiance +distribution to a reflector with a continuous irradiance distribution. Optimal +transport methods have also been studied [18]. +2.1. The Near-Field Reflector Problem with Spatial Restrictions +In this paper, we study a novel variant of the near-field reflector problem +where we have extreme limitations on where we can place and construct the +reflectors. Specifically, we are given an open set U ⊂ R3 \{O}, and our reflector +R must now be a subset of U. +Definition 2.1. Given an x ∈ R3 \ {O} and a subset S ⊆ R3 \ {O}, then we +define Proj(x) = +x +|x| as the projection of x onto S2 and Proj[S] = {Proj(x) ∈ +S2|x ∈ S} as the projection of S onto S2. +Assume that we are given a positive, continuous, almost everywhere differen- +tiable function ρ over Proj[U]. We have a reflector R = {mρ(m)|m ∈ Proj[U]} +9 + +which determines our reflector map x : Proj[U] → T which is determined by +tracking the path of each ray described by the direction m ∈ Proj[U] to a point +x(m) ∈ T. A ray, originating at O in direction m, hits the reflector R at a +point mρ(m). Then, assuming ρ is differentiable at m, said ray reflects off R +at mρ(m) in the direction y(m) as defined by (1) and reaches T at some point +x(m). Thus, from a physical perspective, an irradiance f(x(m)) is created by +the rays reflected at x(m). This defines a mapping m → x(m) that we call the +reflector map; for convenience, we denote x(m) as the image of m under the +reflector map. +We can now formulate the near-field reflector problem with spatial restric- +tions. +Assume that we are given an open set U ⊂ R3 \ {O}, O, an aper- +ture Proj[U] ⊂ S2, a nonnegative g ∈ L1(Proj[U]), and a bounded Borel set +T ⊂ R3 \ {O} with an integrable function f : T → [0, ∞). +The goal is to find a positive, continuous, almost everywhere differentiable +function ρ over Proj[U] such that: +1. R = {mρ(m)|m ∈ Proj[U]} ⊂ U. +2. The ray originating from O in the direction m ∈ Proj[U] reflects off of R +in accordance with equation (1) and reaches the target set T. +3. g(m) on Proj[U] is transformed by the reflector map into f on T, i.e. for +all Borel subsets ω ⊆ T, +� +x−1[ω] +gdσ = +� +ω +fdν +(7) +where x : Proj[U] → T is the reflector map, x−1[ω] = {m ∈ Proj[U]|x(m) ∈ +ω}, dσ is the surface area element on S2, and dν is the area element on T +(ν is typically some discrete or Lebesgue measure). +4. The law of total energy conservation is obeyed: +� +Proj[U] gdσ = +� +T fdν. +This variation of the near-field reflector problem has clear applications to +engineering; as often one has to grapple with restrictions of space in real-world +designs. For example, in the construction of automotive headlights, there are +10 + +strict restrictions, guided purely by aesthetics, as to where a reflector can be +placed and how a reflector must be shaped [19]. +However, to the author’s +knowledge, no mathematical research has been done in this direction. We focus +exclusively on the case where the target set is finite. +3. Ellipsoids of Revolution +We do all our work in R3. We denote S2 to be the unit sphere with the +center at O and kx = x/|x| for all x ∈ R3 \ {O}. We borrow much of this +geometric setup from [4] and [15]. Ellipsoids of revolution are of paramount +importance when solving the near-field reflector problem due to their unique +optical properties. +Let x ∈ R3 \ {O} and d ∈ (0, ∞). +We denote by Ed(x) an ellipsoid of +revolution about the axis Ox and with foci at points O and x. The polar radius +relative to O can be represented as: +ψx,d(m) = +d +1 − ϵ⟨m, kx⟩, m ∈ S2 +(8) +where ϵ is the eccentricity and +ϵ = +� +1 + d2 +x2 − d +|x|. +(9) +So in other words +Ed(x) = {mψx,d(m)|m ∈ S2}. +(10) +From this point on, whenever we use the term ellipsoid we specifically refer to +an ellipsoid of this kind with one of the foci always at O. Note that each Ed(x) +is uniquely defined by the x ∈ R3 \ {O} and the d ∈ (0, ∞). In this paper, we +define Ψx,d(m) = mψx,d(m). +Note that for all possible values of d, we have that ϵ ∈ (0, 1). Also for a +fixed x, as d → 0 the ellipsoid will degenerate into a line segment, i.e. Ed(x) → +{tx + (1 − t)O|t ∈ [0, 1]}. Such an ellipsoid is called degenerate. Observe that as +d → ∞, |ψx,d(m)| → ∞ for all m ∈ S2. +An important property of ellipsoids can be described by the following propo- +sition. +11 + +Proposition 3.1. Let c, d > 0. Then the ellipsoids Ecd(x) and Ed(x) have the +same foci: O and x. +From a physical perspective, the aforementioned property is important be- +cause a reflector that is shaped like an ellipsoid Ed(x) will illuminate the focus +x with the light emitted from O such that the total energy emitted from O is +equal to the total energy reflected onto x. This property is still true no matter +how large or small the ellipsoid is; all that matters is the location of the foci. +4. Generalized Reflectors +Before we proceed, we reiterate that the near-field reflector problem can +be expressed analytically as a PDE of Monge Amp´ere Type. Specifically, the +equation (4) from [12]. Therefore we will consider the following formulation of +the near-field reflector problem with spatial restrictions as a weak formulation +and its solutions, weak solutions. The following formulation only concerns the +case where the target set is finite. +4.1. Weak Solutions Using Generalized Reflectors +Definition 4.1. Assume that we are given an aperture D ⊆ S2 that is an +open set, and a function ρ : D → (0, ∞) that is not necessarily continuous +and almost everywhere differentiable. Then a generalized reflector is the set +R = {mρ(m)|m ∈ D} ⊂ R3. +The upper half-space of R3 be represented as R3+ = {(x, y, z) ∈ R3|z > 0}, +and the lower half-space of R3 be represented as R3− = {(x, y, z) ∈ R3|z < 0}. +Let σ denote the standard measure on S2. Consider an open set U ⊆ R3+, a +corresponding aperture Proj[U], and a finite target set T ⊂ R3−. +Let B be a countable family of open subsets of S2 such that σ(Proj[U] \ +� +B∈B B) = 0, Proj[U] ⊆ � +B∈B B, and σ(B∩B′) = 0 for all distinct B, B′ ∈ B. +Let the set B(U) be the set of all such families. +Since every ellipsoid requires foci and an eccentricity to be well defined, given +a family B ∈ B(U), let UT (B) be the set of all functions B → T and V (B) be +12 + +the set of all functions B → (0, ∞). Thus we define +ET (U) = +� � +B∈B +Ψu(B),v(B)[B] +����� B ∈ B(U), u ∈ UT (B), v ∈ V (B) +� +. +(11) +Assume we are given a Z ∈ ET (U). Let us define +BZ = +� +Int(Proj[Ed(x) ∩ Z]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ Z]) ̸= 0 +� +. +(12) +The geometry of the ellipsoid and the definition of BZ imply that there ex- +ists unique u ∈ UT (B) and v ∈ V (B) such that Z = � +B∈BZ Ψu(B),v(B)[B]. +Define uZ ∈ UT (BZ) and vZ ∈ V (BZ) be the unique functions such that +Z = � +B∈BZ ΨuZ(B),vZ(B)[B]. +Given some Z ∈ ET (U), let y1 +Z(m) = {B ∈ +BZ|m ∈ B} for m ∈ Proj[U]. Given a B ∈ B(U), let N (B) be the set of all +injective functions s : B → N. For Z ∈ ET (U) and s ∈ N (BZ), define +ρs +Z(m) = ψuZ(s−1(min s[y1 +Z(m)])),vZ(s−1(min s[y1 +Z(m)]))(m), m ∈ Proj[U]. +(13) +Observe that the function ρs +Z is positive, not necessarily continuous, and almost +everywhere differentiable. Let W(ρs +Z) = {mρs +Z(m)|m ∈ Proj[U]} and thus we +describe a set of generalized reflectors +RU +1 (T) = +� +W(ρs +Z)| Z ∈ ET (U) where Z ⊂ U, s ∈ N (BZ) +� +. +(14) +Assume we are given a generalized reflector R ∈ RU +1 (T). Let us define +BR = +� +Int(Proj[Ed(x) ∩ R]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ R]) ̸= 0 +� +. +(15) +The geometry of the ellipsoid and the definition of BR imply that there exists +an s ∈ N(BR), unique u ∈ UT (B) and unique v ∈ V (B) such that W(ρs +Z) = R +where Z = � +B∈BR Ψu(B),v(B)[B]. +Therefore, for every generalized reflector +R ∈ RU +1 (T), we may define a unique BR ∈ B(U) such that for each B ∈ BR +there are unique xB ∈ T and dB ∈ (0, ∞) such that, for some s ∈ N(BR), +R = W (ρs +Z) where Z = � +B∈BR ΨxB,dB[B]. +Therefore, given a generalized reflector R ∈ RU +1 (T), we obtain a correspond- +ing BR; for each B ∈ BR we define unique xB and dB. We also obtain an +sR ∈ N(BR) and a unique ZR = � +B∈BR ΨxB,dB[B] such that R = W(ρsR +ZR). +13 + +Given a generalized reflector R ∈ RU +1 (T), for all m ∈ Proj[U] we define +M(m) = xB ∈ T +(16) +where mρsR +ZR(m) = ΨxB,dB(m). Let y2 +R(m) be the points of intersection between +R \ {mρsR +ZR(m)} and the line segment connecting mρsR +ZR(m) to M(m). +Given a generalized reflector R ∈ RU +1 (T), the map α1 : Proj[U] → T ∪ R, +α1(m) = +� +� +� +� +� +M(m) +if y2 +R(m) = ∅ +y2 +R(m) +if y2 +R(m) ̸= ∅ +(17) +is called the generalized reflector map. Physically speaking, a ray of light of +direction m originating from O can only reach the target set if y2 +R(m) is empty. +Assume we are given a nonnegative g ∈ L1(S2). Let us define for all Borel +X ⊆ S2 +µg(X) = +� +X +g(m)dσ(m) +(18) +where σ denotes the standard measure on S2. Assume that g ≡ 0 outside of +Proj[U]. Physically speaking, g is the radiance distribution of the source at O. +In order to formulate and solve the generalized reflector problem (in the +framework of weak solutions to be defined below), we need to define a mea- +sure representing the energy generated by g and redistributed by a generalized +reflector R ∈ RU +1 (T). +Given a generalized reflector R ∈ RU +1 (T) and a set ω ⊆ T we define the +visibility set of ω as +V U +1 (ω) = +� +A∈A +A \ {m ∈ Proj[U]|α1(m) = y2 +R(m)} +(19) +where A = {B ∈ BR|xB ∈ ω}. We now need to show that V U +1 (ω) is measurable. +Note the following definition. +Definition 4.2. For an element x ∈ R3 and a set A ⊂ R3, let the set Cx,A = +{at + x(1 − t)|t ∈ [0, 1], a ∈ A} be the union of all line segments from x to A +and Cx,A,∞ = {at + x(1 − t)|t ∈ [0, ∞), a ∈ A} be the union of all rays from x +that intersect A. +14 + +We proceed with the following lemmas. +Lemma 4.1. Let w : S2 → (0, ∞) be continuous and W(m) = mw(m) for all +m ∈ S2. If B is a Borel set of S2, then CO,W [B] and CO,W [B],∞ are Borel sets +of R3. +Proof. Recall that all Borel sets can be formed from open sets through the +operations of countable union, countable intersection, and relative complement. +Let {Ei} be a countable collection of open sets of S2 such that through said +operations, we obtain B. Then given the countable collection of open sets of R3, +{Int(CO,W [Ei])}, through the same sequence of operations we used to obtain B +from {Ei}, we obtain Int(CO,W [B]). Thus CO,W [B] is Borel. Therefore, assuming +Wi ≡ (im)w(m) for all m ∈ S2 and i ∈ (0, ∞), �∞ +n=1 CO,Wn[B] = CO,W [B],∞ is +Borel. +Lemma 4.2. If B is a Borel set of S2, x ∈ R3 \ {O} and d ∈ (0, ∞), then +Cx,Ψx,d[B] and Cx,Ψx,d[B],∞ are Borel sets of R3. +Proof. Let S2 +x = {m+x|m ∈ S2} be the set of all unit vectors originating from x, +i.e the unit sphere centered at x. Since x is another focus of the ellipsoid, there +exists a continuous function w : S2 → (0, ∞) such that Ed(x) = {mw(m)+x|m ∈ +S2}. Let Wx(m) = mw(m) + x and let W(m) = mw(m). Note that since B is +Borel in S2, Ψx,d[B] is Borel in Ed(x). Thus W −1 +x [Ψx,d[B]] is Borel in S2. By +Lemma 4.1, CO,W [W −1 +x +[Ψx,d[B]]] and CO,W [W −1 +x +[Ψx,d[B]]],∞ are Borel sets of R3. +Thus, by translation, Cx,Ψx,d[B] and Cx,Ψx,d[B],∞ are Borel sets of R3. +We can now prove the following proposition. +Proposition 4.1. Let R be a generalized reflector in RU +1 (T). For any set ω ⊆ T +the visibility set V U +1 (ω) is Borel. +Proof. We make use of the fact that sets formed from Borel sets through the +operations of countable union, countable intersection, and relative complement +are Borel. Recall that we obtain a sR ∈ N(BR). Note that by the definition +of a generalized reflector in RU +1 (T), R = � +B∈BR ΨxB,dB [B′] where B′ = {m ∈ +15 + +B|mρsR +ZR = ΨxB,dB(m)} = B\� +K∈K K where K = {A ∈ BR|sR(A) < sR(B)}; +note that B′ is clearly Borel and B ⊆ B′ ⊆ B. +For B ∈ BR, we have that CxB,ΨxB,dB [B′] and CO,ΨxB,dB [B′] are Borel sets +by Lemmas 4.2 and 4.1 respectively. Since BR is countable and functions of +the form ΨxB,dB are continuous and bijective, R is Borel. Thus for all B ∈ BR, +the set QB = CxB,ΨxB,dB [B′] ∩ (R \ ΨxB,dB[B′]) is Borel and therefore the set +LB = CxB,QB,∞ ∩ ΨxB,dB[B′] is Borel. Thus Proj[LB] is Borel, as it is the +preimage of LB under ΨxB,dB. Since +{m ∈ Proj[U]|α1(m) = y2 +R(m)} = +� +B∈BR +Proj[LB], +(20) +we have that {m ∈ Proj[U]|α1(m) = y2 +R(m)} is Borel and thus V U +1 (ω) is Borel. +Define for any generalized reflector R ∈ RU +1 (T), +G1(ω) = µg(V U +1 (ω)) +(21) +which we will deem the energy function of the generalized reflector problem. +Let F be a nonnegative, finite measure on the finite set T. We say that a +generalized reflector R ∈ RU +1 (T) is a weak solution to the generalized reflector +problem if the generalized reflector map α1 determined by R is such that +F(ω) = G1(ω) for any Borel set ω ⊆ T. +(22) +It would be useful to point out the similarity of condition (22) and condition +(6). +4.2. Geometric Lemmas +One thing that should be noted is that the definition of the generalized +reflector map takes into account that there could potentially be a part of the +generalized reflector that intercepts an already reflected ray before it can reach +the target set. That fact inspires some key geometric lemmas. +16 + +Lemma 4.3. Let R ∈ RU +1 (T) for some finite set T ⊂ R3− and open set U ⊆ +R3+. For all B ∈ BR, if m ∈ B, then α1(m) = xB if and only if y2 +R(m) = ∅. +Proof. This follows directly from the definition of the reflector map of the gen- +eralized reflector problem. +Note the following definition. +Definition 4.3. When we say that r is a ray in Cx,B,∞, then r = {at + x(1 − +t)|t ∈ [0, ∞)} for some a ∈ B, similarly if we say r is a line segment in Cx,B, +then r = {at + x(1 − t)|t ∈ [0, 1]} for some a ∈ B. +Lemma 4.4. Let A, B ⊂ S2 be disjoint sets. Then for any x ∈ R3 \ {O} and +a, b ∈ (0, ∞), Cx,Ψx,a[A] ∩ Ψx,b[B] = ∅ and Cx,Ψx,b[B] ∩ Ψx,a[A] = ∅ if and only +if Cx,Ψx,a[A],∞ ∩ Ψx,b[B] = ∅. +Proof. If Cx,Ψx,a[A],∞∩Ψx,b[B] ̸= ∅, then there exists a ray r in Cx,Ψx,a[A],∞ such +that r intersects Ψx,b[B]. Thus, either there exists a line segment in Cx,Ψx,a[A] +that intersects Ψx,b[B] and thus Cx,Ψx,a[A] ∩ Ψx,b[B] ̸= ∅, or there exists a line +segment in Cx,Ψx,b[B] that intersects Ψx,a[A] and thus Cx,Ψx,b[B] ∩ Ψx,a[A] ̸= ∅. +Conversely, if Cx,Ψx,a[A] ∩ Ψx,b[B] ̸= ∅, then there exists a line segment +in Cx,Ψx,a[A] that intersects Ψx,b[B], said line segment coincides with a ray in +Cx,Ψx,a[A],∞; thus Cx,Ψx,a[A],∞ ∩ Ψx,b[B] ̸= ∅. If Cx,Ψx,b[B] ∩ Ψx,a[A] ̸= ∅, then +there exists a line segment in Cx,Ψx,b[B] that intersects Ψx,a[A], said line segment +coincides with a ray in Cx,Ψx,a[A],∞; thus Cx,Ψx,a[A],∞ ∩ Ψx,b[B] ̸= ∅. +These two lemmas give us the following result. +Lemma 4.5. Assume that U is an open set in R3+, and T is a finite target +set in R3−. Let R ∈ RU +1 (T) be a generalized reflector and A, B ∈ BR such that +A ̸= B and x = xA = xB. Then the following conditions are equivalent: +1. for all m ∈ A and m′ ∈ B, α1(m) = α1(m′) = x, +2. Cx,Ψx,dA[A] ∩ Ψx,dB[B] = ∅ and Cx,Ψx,dB [B] ∩ Ψx,dA[A] = ∅, +3. Cx,Ψx,dA[A],∞ ∩ Ψx,dB[B] = ∅. +17 + +Proof. (2) and (3) are equivalent by Lemma 4.4. By Lemma 4.3, for all m ∈ A +α1(m) = x if and only if y2 +R(m) = ∅. By definition, y2 +R(m) = ∅ if and only if +the line segment between Ψx,dA(m) and x does not intersect R \ {Ψx,dA(m)}. +Similarly, By Lemma 4.3, for all m′ ∈ B, α1(m′) = x if and only if y2 +R(m′) = ∅. +By definition, y2 +R(m′) = ∅ if and only if the line segment between Ψx,dB(m′) +and x does not intersect R\{Ψx,dB(m′)}. Therefore, statements (1) and (2) are +equivalent. +4.3. Generalized Reflectors Constructed in an Open Conical Cylinder of Arbi- +trary Thickness +Let S2 ++ = {m ∈ S2|⟨m, (0, 0, 1)⟩ > 0} be the open hemisphere of the S2 +oriented towards the positive z−axis. Similarly, S2 +− = {m ∈ S2|⟨m, (0, 0, 1)⟩ < +0} be the open hemisphere of the S2 oriented towards the negative z−axis. +Given an open U ⊆ S2 ++, and δ, z′ > 0, we then define an open conical cylinder +of thickness δ as C δ +U(z′) = CO,U,∞ ∩ {(x, y, z) ∈ R3|z′ + δ > z > z′}. +In this paper, given a finite target set T ⊂ R3−, we aim to construct a +generalized reflector R ∈ RC δ +U(z′) +1 +(T) that is a weak solution of the generalized +reflector problem. This condition is very strict and the following strategies can +potentially be applied to other kinds of open subsets in R3+. +We first consider the case where the target set is a single point. We proceed +with the following lemmas. +Lemma 4.6. Let U be an open set in S2 ++ and z′, δ > 0. +Let {Si}i∈N be a +countable collection of open subsets in U, {di}i∈N is a countable collection of +distinct positive numbers, and x ∈ R3−. Assume that each Ψx,di[Si] ⊂ C δ +U(z′) +and denote Ψi = Ψx,di[Si]. Then we have that +Proj +� +C δ +U(z′) \ +� +i∈N +(CO,Ψi,∞ ∪ Cx,Ψi,∞) +� += Proj +� +C δ +U(z′) \ +� +i∈N +CO,Ψi,∞ +� +. (23) +Proof. Assume to the contrary that +Proj +� +C δ +U(z′) \ +� +i∈N +(CO,Ψi,∞ ∪ Cx,Ψi,∞) +� +̸= Proj +� +C δ +U(z′) \ +� +i∈N +CO,Ψi,∞ +� +. (24) +18 + +Then there exists a ray r in CO,U\� +i∈N Si,∞ = CO,U,∞ \ � +i∈N CO,Si,∞ such that +r ∩ C δ +U(z′) ⊂ � +i∈N Cxi,Si,∞. Equivalently, one can say that there must be a ray +of direction m ∈ U \ � +i∈N Si originating from O that we denote as r such that +r ∩ (C δ +U(z′) \ � +i∈N(CO,Ψi,∞ ∪ Cx,Ψi,∞)) = ∅. +Consider the plane P(α) = {(x, y, z) ∈ R3|z = α}. Let m ∈ U \ � +i∈N Si. +Assume that there exists a set P(z′) ∩ � +i∈N Cx,Ψi,∞ such that +�� +P(z′) ∩ +� +i∈N +Cx,Ψi,∞ +� +\ +� +P(z′) ∩ +� +i∈N +CO,Ψi,∞ +�� +∩ CO,U,∞ ̸= ∅. +(25) +Otherwise there does not exist a ray r of direction m ∈ U \ � +i∈N Si originating +from O such that r ∩ (C δ +U(z′) \ � +i∈N(CO,Ψi,∞ ∪ Cx,Ψi,∞)) = ∅; a contradiction. +Thus we assume such a ray exists r exists. Then m must be in a direction such +that there exists a dmin > 0 where +Ψx,dmin(m) ∈ +�� +P(z′) ∩ +� +i∈N +Cx,Ψi,∞ +� +\ +� +P(z′) ∩ +� +i∈N +CO,Ψi,∞ +�� +∩ CO,U,∞. +(26) +Since C δ +U(z′) is bounded, there must also exist a dmax > 0 such that +Ψx,dmax(m) ∈ P(z′ + δ) ∩ CO,U,∞. +(27) +Note that by our assumptions, for all d ∈ (dmin, dmax), there exists an α ∈ N +such that the line segment between Ψx,d(m) and x is a subset of a line segment +in Cx,Ψα. However, since all Ψi are closed, then for all d ∈ [dmin, dmax], there +exists an α such that the line segment between Ψx,d(m) and x is a subset of a +line segment in Cx,Ψα. +Case 1. dmax > di for all i ∈ N. +Recall that by our assumptions, Ψx,d(m) ∈ � +i∈N Cx,Ψi,∞ for all d ∈ [dmin, dmax]. +However, since ψx,dmax(m) > ψx,di(m) for all i ∈ N, Ψx,dmax(m) cannot reside +on the interior of any ellipsoid Ed′(x) where d′ ∈ {di}i∈N, thus Ψx,d(m) ̸∈ +� +i∈N Cx,Ψi,∞. A contradiction. +Case 2. There exists some α ∈ N such that dmax = dα. +If there exists some α such that dmax = dα, then, since tΨx,dmax(m) + (1 − +19 + +t)x ̸∈ C δ +U(z′) for all t > 1, Ψx,dmax(m) resides on the ellipsoid Edα(x). Therefore +Ψx,dmax(m) ∈ Ψα ∩ P(z′ + δ) and thus m ∈ Sα. A contradiction. +Case 3. There exists some α ∈ N such that dα > dmax. +Assume that {di}i∈N is arranged such that di+1 ≥ di If there exists some α +such that dα > dmax, then there exists a ray originating from x that intersects +the point Ψx,dmax(m) that also intersects a point (xβ, yβ, zβ) ∈ Ψβ where dβ ≥ +dmax. The case where dβ = dmax has already been covered. When dβ > dmax: +since x ∈ R3− and Ψx,dmax(m) ∈ P(z′ + δ), this implies that zβ > z′ + δ. A +contradiction. +Lemma 4.7. Recall that σ is the standard measure on S2. Let U be a Borel +set in R3 \ {O} such that Int(U) ̸= ∅. Let x ∈ R3 \ {O}. Consider the set +K(d) = Proj[Ed(x) ∩ U] and the corresponding function D(d) = σ(K(d)) for +d ∈ (0∞). Then D(d) cannot be identically zero. +Furthermore, if U is open, K(d) is open in S2 for all d ∈ (0, ∞). +Proof. Clearly there exists a d′ ∈ (0, ∞) such that K(d′) ∩ Int(U) ̸= ∅. Then +Ed′(x) ∩ Int(U) is open in Ed′(x) and thus Proj[Ed′(x) ∩ Int(U)] is open in S2. +Therefore, D(d′) ≥ σ(Proj[K(d′) ∩ Int(U)]) > 0. +Theorem 4.1. Let U be an open set in S2 ++, δ, z′ > 0, and T = {x} ∈ R3−. +Assume that we are given a nonnegative g ∈ L1(S2) where g ≡ 0 outside U. +Then there exists a generalized reflector R ∈ RC δ +U(z′) +1 +(T) such that G1({x}) = +µg(U). +Proof. For convenience, label C∗ = C δ +U(z′). Recall that σ is the standard mea- +sure on S2. +Consider the set K1(d) = Proj[Ed(x) ∩ C∗] and its corresponding function +D1(d) = σ(K1(d)) where d ∈ (0, ∞). Note that since C∗ is bounded, D1(d) → 0 +as d → ∞ and D1(d) → 0 as d → 0. By construction, it is clear that D1 is +bounded by 0 and σ(U). Therefore, Dmax +1 += sup{D1(d)|d ∈ (0, ∞)} exists and +is finite, and by Lemma 4.7, Dmax +1 +> 0. +20 + +Let ϵ1 ∈ [0, Dmax +1 +). We define dmax1 to be a value such that D1(dmax1) = +Dmax +1 +− ϵ1 where ϵ1 = 0 if Dmax +1 +∈ {D1(d)|d ∈ (0, ∞)}. We now eliminate the +parts of U that had already been accounted for and the parts of C∗ that can no +longer be used: let E1 = Proj[Edmax1 (x) ∩ C∗], Ψ1 = Ψx,dmax1 [E1], +Q2 = C∗ \ (Cx,Ψ1,∞ ∪ CO,Ψ1,∞), +(28) +and U2 = U \ E1. Note by Lemma 4.6, U2 = Proj[Q2]. Let us define K2(d) = +Proj[Ψx,d[U2] ∩ Q2] and D2(d) = σ(K2(d)). +Note that since Q2 is bounded, D2(d) → 0 as d → ∞ and D2(d) → 0 +as d → 0. By construction, it is clear that D2 is bounded by 0 and σ(U2). +Therefore, Dmax +2 += sup{D2(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma +4.7, Dmax +2 +> 0. Let ϵ2 ∈ [0, Dmax +2 +). We define dmax2 to be a value such that +D1(dmax1) ≥ D2(dmax2) = Dmax +2 +− ϵ2. +Given that U1 = U and Q1 = C∗, we can now recursively define a sequence +of functions and sets for k ≥ 2: +Ek−1 = Proj[Ψx,dmaxk−1 [Uk−1] ∩ Qk−1], +(29) +Ψk−1 = Ψx,dmaxk−1 [Ek−1], +(30) +Qk = Qk−1 \ (Cx,Ψk−1,∞ ∪ CO,Ψk−1,∞), +(31) +Uk = U \ +� +� +k−1 +� +j=1 +Ej +� +� = Proj[Qk], +(32) +Kk(d) = Proj[Ed(x) ∩ Qk], +(33) +Dk(d) = σ(Kk(d)). +(34) +Also, note that since Qk is bounded, Dk(d) → 0 as d → ∞ and Dk(d) → 0 +as d → 0. By construction, it is clear that Dk is bounded by 0 and σ(Uk). +Therefore, Dmax +k += sup{Dk(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma +4.7, Dmax +k +> 0. Let ϵk ∈ [0, Dmax +k +). We define dmaxk to be a value such that +Dk−1(dmaxk−1) ≥ Dk(dmaxk) = Dmax +k +− ϵk. +Observe that the set Kk(d) is open for all d > 0. We can therefore construct +21 + +a sequence +� +� +�σ +� +� +k� +j=1 +Ej +� +� +� +� +� +∞ +k=1 +. +(35) +Claim 4.1. There exists {ϵi}i∈N such that +� +σ +��k +j=1 Ej +��∞ +k=1 converges to +σ(U). +Proof. By construction, the sequence increases monotonically and is bounded +between 0 and σ(U); thus it converges. Assume to the contrary that for every +possible {ϵi}i∈N, +� +σ +��k +j=1 Ej +��∞ +k=1 that converges to an L ∈ (0, σ(U)). Then +σ +� +U \ �∞ +j=1 Ej +� += σ(U) − L > 0. +Consider the function +D∗(d) = σ +� +Proj +� +Ed(x) ∩ lim +j→∞ Qj +�� +. +(36) +Observe that limj→∞ Qj = C∗\�∞ +i=1(Cx,Ψi,∞∪CO,Ψi,∞). Note that �∞ +i=1(Cx,Ψi,∞∪ +CO,Ψi,∞) ⊆ �∞ +i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞). Observe that for all i ∈ N, Int(Cx,Ψi,∞ ∪ CO,Ψi,∞) = +Cx,Ψi,∞∪CO,Ψi,∞; thus �∞ +i=1 Int(Cx,Ψi,∞ ∪ CO,Ψi,∞) = �∞ +i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞). +Thus limj→∞ Qj = C∗\�∞ +i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞) is open and thus Int(limj→∞ Qj) ̸= +∅. +Thus, by Lemma 4.7, there exists a d′ such that D∗(d′) > 0. By the def- +inition of convergence, there exists an M such that for all m ≥ M, D∗(d′) > +σ +��∞ +j=m Ej +� +. Note that σ +��∞ +j=m Ej +� += �∞ +j=m σ(Ej) = �∞ +j=m Di(dmaxi) ≥ +Dm(dmaxm). For all k ∈ N, Dmax +k +is a limit point of {Dk(d)|d ∈ (0, ∞)}, there- +fore as ϵk → 0, Dk(dmaxk) → Dmax +k +. Observe that Dm(d′) ≥ D∗(d′) because +limj→∞ Qj ⊆ Qm and U \ �∞ +j=1 Ej ⊆ Um. Therefore there exists a sequence +{ϵi}i∈N such that Dm(dmaxm) ≥ D∗(d′). For this sequence σ +��∞ +j=m Ej +� +≥ +D∗(d′); a contradiction. Thus, there exists {ϵi}i∈N such that +� +σ +��k +j=1 Ej +��∞ +k=1 +converges to σ(U). +Let +Z = +∞ +� +j=1 +Ψj. +(37) +22 + +For some s ∈ N(BZ), consider the generalized reflector R = W(ρs +Z) ∈ RC∗ +1 (T). +By construction, +Cx,Ψx,dmaxj [Int(Ej)],∞ ∩ Ψx,dmaxj′ [Int(Ej′)] = ∅ +(38) +when j ̸= j′. Observe that if j′, j ∈ N where j′ > j, then dmaxj ̸= dmaxj′ +because otherwise Ej′ ⊆ Ej; thus BR = {Int(Ej) ⊂ S2|j ∈ N}. +Thus, by +Lemma 4.5, for all m ∈ B where B ∈ BR, we have α1(m) = x. Then, for any +s ∈ N(BZ), the generalized reflector R = W(ρs +Z) ∈ RC∗ +1 (T) is a weak solution +to the generalized reflector problem such that G1({x}) = µg(U). +We can now prove a result where our target set is made up of finitely many +points. First, we prove the following lemma. +Lemma 4.8. Assume that U is an open set in R3+, and T is a finite target +set in R3−. Let R ∈ RU +1 (T) be a generalized reflector and A, B ∈ BR such that +A ̸= B. Then the following conditions are equivalent: +1. for all m ∈ A and m′ ∈ B, α1(m) = xA and α1(m′) = xB, +2. CxA,ΨxA,dA[A] ∩ ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B] ∩ ΨxA,dA[A] = ∅. +Proof. For the case where xA = xB, we have Lemma 4.5. We now consider +the case where xB ̸= xB. By Lemma 4.3, for all m ∈ A α1(m) = xA if and +only if y2 +R(m) = ∅. By definition, y2 +R(m) = ∅ if and only if the line segment +between ΨxA,dA(m) and xA does not intersect R \ {ΨxA,dA(m)}. +Similarly, +By Lemma 4.3, for all m′ ∈ B α1(m′) = xB if and only if y2 +R(m′) = ∅. By +definition, y2 +R(m′) = ∅ if and only if the line segment between ΨxB,dB(m′) and +x does not intersect R \ {ΨxB,dB(m′)}. Therefore, statements (1) and (2) are +equivalent. +Theorem 4.2. Let U be an open set in S2 ++, δ, z′ > 0, and {x1, . . . , xk} ∈ R3− +where k ≥ 2. Assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0 +23 + +outside U. Let f1, f2, . . . , fk be nonnegative real numbers such that +k +� +i=1 +fi = µg(U). +(39) +Assume that there exists n ≥ k disjoint open sets Bi in U where � +i∈[n] Bi = U. +Also assume that there exists a collection of k subsets of [n], {Ai}i∈[k], such that: +At ∩At′ = ∅ where t ̸= t′, � +i∈[k] Ai = [n], and µg(� +i∈At Bi) = ft. Suppose that +for all i ∈ [n] there exists ai, bi > 0 where z′ ≤ ai < ai + bi ≤ z′ + δ such that +C bi +Bi(ai) ∩ Cxj,C +bj +Bj (aj) = ∅ for all j ∈ [n] \ {i}. +Then there exists a generalized reflector in R ∈ RC δ +U(z′) +1 +(T) such that G1({xi}) = +fi for all i ∈ [k]. +Proof. For each i ∈ [n], C bi +Bi(ai) is open and a generalized reflector Ri ∈ +R +C +bi +Bi(ai) +1 +({xi}) is constructed in the exact same way as Theorem 4.1. By our +assumptions, since Ri ⊆ C bi +Bi(ai), then +(Ri ∩ C bi +Bi(ai)) ∩ Cxj,Rj∩C +bj +Bj (aj) = ∅ +(40) +for all j ∈ [n] \ {i}. +Let F = � +i∈[k] Ri and consider the generalized reflector R = W(ρF ) ∈ +RC δ +U(z′) +1 +(T). By construction, for an A, B ∈ BR such that A ̸= B, we have +that CxA,ΨxA,dA[A] ∩ ΨxB,dB[B] = ∅ and CxB,Ψx,dB [B] ∩ ΨxA,dA[A] = ∅. Thus +by Lemma 4.8, for any B ∈ BR, for all m ∈ B we have α1(m) = xB. Then, +for any s ∈ N(BF ), the generalized reflector R = W(ρs +F ) ∈ RC δ +U (z′) +1 +(T) is a +weak solution to the generalized reflector problem such that G1({xi}) = fi for +all i ∈ [k]. +We now will use Theorem 4.2 to construct a specific type of generalized +reflector. Note the following definition. +Definition 4.4. Let k ≥ 2, d > 0, ξ ∈ (−1, 0), and t ∈ R. Recall that, given a +24 + +point (x, y, z) ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], θ ∈ [0, 2π), such that +x = r cos θ sin φ +(41) +y = r sin θ sin φ +(42) +z = r cos φ. +(43) +Define the set of points T ξ +k,d(t) as +�� +d cos +�2πj +k ++ t +� +sin (arccos(ξ)) , d sin +�2πj +k ++ t +� +sin (arccos(ξ)) , dξ +� +|j ∈ I +� +(44) +where I = {0, 1, . . . , k − 1}. +If we are additionally given an i ∈ {0, 1, . . . , k − 1}, we may define the set +Pk,i(t) ⊂ S2, as +�� +cos +� +θ + π(2i − 1) +k ++ t +� +sin φ, sin +� +θ + π(2i − 1) +k ++ t +� +sin φ, cos φ +� +����φ ∈ [0, π], θ ∈ +� +0, 2π +k +�� +. +(45) +If k = 1, define the set of points T ξ +1,d(t) = {(0, 0, −d)} and P1,0(t) = S2. +It is good to observe that T ξ +k,d(t) defines the points of a regular k-gon centered +at the z-axis and that Pk,i(t) defines a spherical wedge. +Theorem 4.3. Let δ, z′ > 0. Consider the open disk U = {m ∈ S2 ++|⟨(0, 0, 1), m⟩ > +c} where 0 < c < 1. Let d1, . . . , dn be a collection of not necessarily distinct +positive numbers. Let k1, . . . , kn be a collection of not necessarily distinct pos- +itive integers. Let ξ1, . . . , ξn be a collection of not necessarily distinct numbers +such that ξi ∈ (−1, 0). Let t′ +1, . . . , t′ +n be a collection of not necessarily distinct +elements of R. Let us denote Ti = T ξi +ki,di(t′ +i) and let T = �n +i=1 Ti. +Assume that we a given a nonnegative g ∈ L1(S2 ++) that is rotationally sym- +metric about the z-axis such that g ≡ 0 outside U. Let f1, . . . , fn be a collection +of positive numbers such that +µg(U) = +n +� +i=1 +fi. +(46) +25 + +Then there exists a generalized reflector R ∈ RC δ +U(z′) +1 +(T) such that +G1({x}) = +� +{j∈[n]|x∈Tj} +fj +kj +(47) +for all x ∈ T. +Proof. By the intermediate value theorem, there exists a collection of numbers +ζ1, . . . , ζn ∈ [c, 1) where ζn = c and µg +� +{m ∈ S2 ++|⟨(0, 0, 1), m⟩ > ζi} +� += �i +j=1 fj. +Define +Bi = {m ∈ S2 ++|⟨(0, 0, 1), m⟩ > ζi} \ {m ∈ S2 ++|⟨(0, 0, 1), m⟩ ≥ ζi−1} +(48) +for all i ∈ {2, . . . , n} and B1 = {m ∈ S2 ++|⟨(0, 0, 1), m⟩ > ζ1}. Thus µg(Bi) = fi. +Consider the set Ti, let +Ti(j) = +� +di cos +�2πj +ki ++ t′ +i +� +sin (arccos(ξi)) , di sin +�2πj +ki ++ t′ +i +� +sin (arccos(ξi)) , diξi +� +(49) +where j ∈ {0, . . . , ki − 1} if k ≥ 2 and Ti(0) = (0, 0, −di). +Let Pi(j) = Pki,j(t′ +i) where j ∈ {0, . . . , ki −1}, then by construction µg(Bi ∩ +Pi(j)) = fi +ki . Let Ui(j) = C +δ +n +Bi∩Int(Pi(j)) +� +z′ + (i − 1) δ +n +� +where j ∈ {0, . . . , ki −1}. +Since Ti(j) ∈ R3−, any given line segment between a point in Ui(j) and the point +Ti(j) will not intersect any set Ui′(j′) where i′ > i and j′ ∈ {0, . . . , ki′ − 1}. +Therefore CTi(j),Ui(j) ∩ Ui′(j′) = ∅ where i′ > i and j′ ∈ {0, . . . , ki′ − 1}. Also, +since CO,Pi(j),∞ is a convex set and Ui(j), {Ti(j)} are both subsets of CO,Pi(j),∞, +we then have CTi(j),Ui(j) ⊂ CO,Pi(j),∞. Therefore, CTi(j),Ui(j) ∩ Ui(j′) = ∅ +where j′ ̸= j. Finally, if there exists a line segment between a point in Ui(j) +and Ti(j) that intersects a Ui′(j′) where i > i′ and j′ ∈ {0, . . . , ki′ − 1}, then it +must intersect CO,{m∈S2 ++|⟨(0,0,1),m⟩>ζi−1},∞. However, CTi(j),Ui(j) is disjoint from +CO,{m∈S2 ++|⟨(0,0,1),m⟩>ζi−1},∞ and thus CTi(j),Ui(j) ∩ Ui′(j′) = ∅ where i > i′ and +j′ ∈ {0, . . . , ki′ − 1}. Therefore, CTi(j),Ui(j) ∩ Ui′(j′) = ∅ when (i, j) ̸= (i′, j′). +Therefore, by Theorem 4.2, there exists a generalized reflector R ∈ RCδ +U(z′) +1 +(T) +such that +G1({x}) = +� +{j∈[n]|x∈Tj} +fj +kj +(50) +26 + +for all x ∈ T. +5. Interpolated Reflectors +The generalized reflector presented in the previous section might be impos- +sible or, at best, very difficult to construct in the real world. Thus we introduce +the following notion. +Definition 5.1. Assume that we are given an aperture that is a connected open +set D ⊆ S2 and a not necessarily continuous, almost everywhere differentiable +function ρ : D → (0, ∞). Then an interpolated reflector is the set R = +∂(CO,S) \ ∂(CO,S,∞) ⊂ R3 where S = {mρ(m)|m ∈ D}. +It is interesting to note that, given an aperture that is a connected open set +D ⊆ S2, a set is a reflector if and only if it is both a generalized reflector and +an interpolated reflector. +The type of interpolated reflector we construct below is a topological surface +(see Chapter 4.36 in [20]) and thus consists of one connected component instead +of countably many. In a practical sense, when designing an interpolated reflector +as opposed to a generalized reflector, new challenges are introduced. Thus, we +settle for finding a necessary and sufficient condition for the existence of an +interpolated reflector. +We will consider the following formulation of the near-field reflector problem +as a weak formulation of equation (4) from [12] and its solutions, weak solutions. +The following formulation only concerns the case where the target set is finite. +5.1. Weak Solutions using Interpolated Reflectors +Consider a connected open set U ⊆ R3+, a corresponding aperture Proj[U], +and a finite target set T ⊆ R3−. Also, consider the set RU +1 (T) as defined by +(14). We then describe a set of interpolated reflectors +RU +2 (T) = +� +∂(CO,S) \ ∂(CO,S,∞)| S ∈ RU +1 (T) +� +. +(51) +27 + +It is interesting to note that the interpolated reflectors in RU +2 (T) are all topo- +logical surfaces. +Assume we are given an interpolated reflector R ∈ RU +2 (T). Let us define +BR = +� +Int(Proj[Ed(x) ∩ R]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ R]) ̸= 0 +� +. +(52) +The geometry of the ellipsoid and the definition of BR imply that there +exists an s ∈ N(BR), unique u ∈ UT (B) and unique v ∈ V (B) such that +∂(CO,W (ρs +Z))\∂(CO,W (ρs +Z),∞) = R where Z = � +B∈BR Ψu(B),v(B)[B]. Therefore, +for every interpolated reflector R ∈ RU +2 (T), we may define a unique BR ∈ B(U) +such that for each B ∈ BR there are unique xB ∈ T and dB ∈ (0, ∞) such +that, for some s ∈ N(BR), R = ∂(CO,W (ρs +Z)) \ ∂(CO,W (ρs +Z),∞) where Z = +� +B∈BR ΨxB,dB[B]. +Therefore, given a interpolated reflector R ∈ RU +2 (T), we obtain a cor- +responding BR; for each B ∈ BR we define unique xB and dB. +We also +obtain an sR ∈ N(BR) and a unique ZR = � +B∈BR ΨxB,dB[B] such that +R = ∂(CO,S) \ ∂(CO,S,∞) where S = W(ρsR +ZR). +Given an interpolated reflector R ∈ RU +2 (T), let +M(m) = xB ∈ T +(53) +where mρsR +ZR(m) = ΨxB,dB(m). Let y2 +R(m) be the points of intersection between +R \ {mρsR +ZR(m)} and the line segment connecting mρsR +ZR(m) to M(m). +Given an interpolated reflector R ∈ RU +2 (T), the map α2 : Proj[U] → T ∪ R, +α2(m) = +� +� +� +� +� +M(m) +if y2 +R(m) = ∅ +y2 +R(m) +if y2 +R(m) ̸= ∅ +(54) +is called the interpolated reflector map. Physically speaking, a ray of light of +direction m originating from O can only reach the target set if y2 +R(m) is empty. +As before, we denote by g ∈ L1(S2) the energy density of the source O. Let +us define for all Borel X ⊆ S2 +µg(X) = +� +X +g(m)dσ(m) +(55) +28 + +where σ denotes the standard measure on S2. Assume that g is a nonnegative +function where g ≡ 0 outside of Proj[U]. Physically speaking, g is the radiance +distribution of the source at O +In order to formulate and solve the interpolated reflector problem (in the +framework of weak solutions to be defined below), we need to define a measure +representing the energy generated by g and redistributed by an interpolated +reflector R ∈ RU +2 (T). +Given a interpolated reflector R ∈ RU +2 (T) and a set ω ⊆ T we define the +visibility set of ω as +V U +2 (ω) = +� +A∈A +A \ {m ∈ Proj[U]|α2(m) = y2 +R(m)} +(56) +where A = {B ∈ BR|xB ∈ ω}. We now need to show that V U +2 (ω) is measurable. +Proposition 5.1. Let R be a interpolated reflector in RU +2 (T). +For any set +ω ⊆ T, the visibility set V U +2 (ω) is Borel. +Proof. We make use of the fact that sets formed from Borel sets through the +operations of countable union, countable intersection, and relative complement +are Borel. Recall that we obtain a sR ∈ N(BR). Note that by the definition of +an interpolated reflector in RU +1 (T), R = � +B∈BR ΨxB,dB [B′] ∪ SR where B′ = +{m ∈ B|mρsR +ZR = ΨxB,dB(m)} = B \ � +K∈K K where K = {A ∈ BR|sR(A) < +sR(B)} and SR = R \ � +B∈BR ΨxB,dB[B′]. Note that B′ is clearly Borel and +B ⊆ B′ ⊆ B. +For B ∈ BR, we have that CxB,ΨxB,dB [B′] and CO,ΨxB,dB [B′] are Borel sets +by Lemmas 4.2 and 4.1 respectively. Note that R is Borel as it is a boundary of +an open set minus the boundary of another open set. Since BR is countable and +functions of the form ΨxB,dB are continuous and bijective, � +B∈BR ΨxB,dB [B′] +is Borel. +Therefore SR is also Borel. +Thus for all B ∈ BR, the set QB = +CxB,ΨxB,dB [B′]∩(R\ΨxB,dB[B′]) is Borel and therefore the set LB = CxB,QB,∞∩ +ΨxB,dB[B′] is Borel. Thus Proj[LB] is Borel, as it is the preimage of LB under +29 + +ΨxB,dB. Since +{m ∈ Proj[U]|α2(m) = y2 +R(m)} = +� +B∈BR +Proj[LB], +(57) +we have that {m ∈ Proj[U]|α2(m) = y2 +R(m)} is Borel and thus V U +2 (ω) is Borel. +Define for any interpolated reflector R ∈ RU +2 (T), +G2(ω) = µg(V U +2 (ω)) +(58) +which we will deem the energy function of interpolated reflector problem. +Let F be a nonnegative, finite measure on the finite set T. We say that an +interpolated reflector R ∈ RU +2 (T) is a weak solution to the interpolated reflector +problem if the interpolated reflector map α2 determined by R is such that +F(ω) = G2(ω) for any Borel set ω ⊆ T. +(59) +5.2. Main Results +Here we prove a necessary and sufficient condition for the existence of weak +solutions to the interpolated reflector problem. We proceed with the following +lemma. +Lemma 5.1. Assume that U is an open set in R3+, and T is a finite target +set in R3−. Assume we are given a nonnegative g ∈ L1(S2) such that g ≡ 0 +outside Proj[U] and g > 0 inside Proj[U]. Let R ∈ RU +1 (T) be a generalized +reflector, then we define the set Bx +R = {B ∈ BR|x = xB} for x ∈ T. Then for +any z, y ∈ T the following conditions are equivalent: +1. for all m ∈ � +A∈Bz +R A and m′ ∈ � +B∈By +R B, α1(m) = z and α1(m′) = y, +2. Cz,Ψz,dA[A] ∩Ψy,dB[B] = ∅ and Cy,Ψy,dB [B] ∩Ψz,dA[A] = ∅ for all A ∈ Bz +R +and B ∈ By +R where A ̸= B, +3. CxA,ΨxA,dA[A] ∩ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B] ∩ΨxA,dA[A] = ∅ for all +A, B ∈ BR where A ̸= B, +30 + +4. G1({x}) = µg(� +A∈Bx +R A) and G1({y}) = µg(� +B∈By +R B). +Proof. (1) ⇔ (2) by Lemma 4.8, clearly (3) ⇔ (2), and (1) trivially implies (4). +We only need to prove that (4) implies (2); we prove the contrapositive. +Assume that there exists an A ∈ Bx +R and a B′ ∈ By +R such that, without loss of +generality, Cz,Ψz,dA[A] ∩ Ψy,dB′ [B′] ̸= ∅. Let Q = Cz,Ψz,dA[A] ∩ Ψy,dB′ [B′]. Note +that Cz,Ψz,dA[A],∞ ∩ Cz,Ψy,dB′ [B′],∞ = Cz,Q,∞. Since Cz,Ψz,dA[A],∞ \ {O} and +Cz,Ψy,dB′ [B′],∞\{O} are open, Cz,Q,∞\{O} is open; thus Cz,Q,∞∩EdB(y) is open +in EdB(y). Thus Proj[Q] ⊆ B is open and, since g > 0 in U, µg(Proj[Q]) > 0, +and thus G1({y}) ≤ µg(� +B∈By +R B) − µg(Proj[Q]). +Note the following definition. +Definition 5.2. Let K be a subset of Rn where n ≥ 2 such that K is compact. +The complement U = Rn \ K is an open set. For sufficiently large R > 0, +the set V = {x|R < |x|} is contained in U. Since V is connected, there exists +a connected component of U that contains V . This is the unique unbounded +connected component of U. +We define the exterior boundary of K as the boundary of the unbounded +connected component of Rn \ K. We denote this as ∂E(K). +The following result gives a condition that is necessary and sufficient for the +existence of weak solutions to the interpolated reflector problem. +Theorem 5.1. Let U ⊂ R3+ be a simply connected open set such that U ⊂ R3+, +and T ⊂ R3− be a finite set. Assume we are given a nonegative g ∈ L1(S2) such +that g ≡ 0 outside Proj[U] and g > 0 inside Proj[U]. Let F be a measure over +T such that +F(T) = µg(Proj[U]). +(60) +Then there exists an interpolated reflector R2 ∈ RU +2 (T) that is a weak solu- +tion to the interpolated reflector problem as defined in (59) if and only if there +exists a generalized reflector R1 ∈ RU +1 (T) that is a weak solution to the gen- +eralized reflector problem as defined in (22) where R1 is a subset of a simply +31 + +connected subset of +U ∩ ∂E +� +�CO,R1 ∪ +� +B∈BR1 +CxB,ΨxB,dB [B] +� +� . +(61) +Proof. It is clear that if R2 ∈ RU +2 (T) that is a weak solution to the interpolated +reflector problem as defined in (59), then, for any s ∈ N(BR2), R1 = W(ρs +F ) +where F = � +B∈BR2 ΨxB,dB[B] is a weak solution to the generalized reflector +problem as defined in (22). Note that this implies that BR1 = BR2. Since +g is positive in Proj[U], for our reflector R1, by Lemma 5.1, for all distinct +A, B ∈ BR1, CxA,ΨxA,dA[A]∩ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B]∩ΨxA,dA[A] = ∅. +Therefore, for all A ∈ BR1, ΨxA,dA[A]∩Int +� +CO,R1 ∪ � +B∈BR1 CxB,ΨxB,dB [B] +� += +∅. +Also observe that for all A ∈ BR1, (CO,ΨxA,dA[A],∞ \ CO,ΨxA,dA[A]) ∩ +Int +� +CO,R1 ∪ � +B∈BR1 CxB,ΨxB,dB [B] +� += ∅. +Therefore, R1 is a subset of a simply connected subset of +∂E +� +�CO,R1 ∪ +� +B∈BR1 +CxB,ΨxB,dB [B] +� +� . +(62) +Since the interpolated reflector must be contained in U, we obtain (61). +Conversely, if there exists a reflector R1 ∈ RU +1 (T) that is a weak solution of +the generalized reflector problem as defined in (22) where R1 ⊂ Q such that Q +is a simply connected subset of (61), then R2 = ∂(CO,R1) \ ∂(CO,R1,∞) is also +a subset of Q. +6. Discussion +In this note, with respect to the near-field reflector problem with spatial +restrictions, we defined two different kinds of weak solutions. For the first weak +solution, we proved, under certain assumptions, the existence of a generalized +reflector where the target set is multiple points. A possible avenue for further +research is to attempt to expand Theorem 4.3 for different target sets, apertures, +and spatial restrictions. Another idea might be to try to come up with designs +such that the generalized reflectors have finitely many connected components +32 + +instead of countably many. The author believes that the following statement is +true. +Conjecture 6.1. Let U be an open set in S2 ++, δ, z′ > 0, and {x1, . . . , xk} ∈ R3− +where k ≥ 2. Assume we are given a positive g ∈ L1(S2) where g ≡ 0 outside +U. Let f1, f2, . . . , fk be nonnegative real numbers such that +k +� +i=1 +fi = µg(U). +(63) +Then there exists a generalized reflector R ∈ RC δ +U(z′) +1 +(T) such that G1({xi}) = fi +for all i ∈ [k]. +For the second weak solution, we proved a theorem that detailed a necessary +and sufficient condition for the existence of an interpolated reflector. The ad- +vantage of our interpolated reflectors, as opposed to our generalized reflectors, +is that our interpolated reflector design is a topological surface; thus it is easier +to construct from an engineering perspective. An obvious avenue for further +work would be to create some practically useful interpolated reflectors; using +Theorem 4.2 might be useful in this regard. In fact, it would be very useful if +the following conjecture is true. +Conjecture 6.2. Let U be a simply connected open set in S2 ++, δ, z′ > 0, and +{x1, . . . , xk} ∈ R3−. Assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0 +outside U. Let f1, f2, . . . , fk be nonnegative real numbers such that +k +� +i=1 +fi = µg(U). +(64) +Then there exists an interpolated reflector in R ∈ RC δ +U(z′) +2 +(T) such that +G2({xi}) = fi for all i ∈ [k]. +Another fruitful avenue of research might be to somehow expand these def- +initions of weak solutions to account for cases where the target set is not finite. +Then, proving the existence of generalized and interpolated reflectors with con- +tinuous irradiance distributions. +33 + +As the reader might have noticed, we make no attempt to address the near- +field reflector problem with spatial conditions with a reflector. +Instead, we +exclusively use generalized or interpolated reflectors. While it may be interesting +to research reflectors in order to get a ‘stronger’ solution, it is the author’s +view that, in general, it is not possible to construct a reflector under spatial +conditions. For example, if the target set is a single point, the solution to the +near-field reflector problem is an ellipsoid. However, if given spatial restrictions, +a single ellipsoid, in general, cannot fit those restrictions; this is demonstrated +in Theorem 4.1. If no reflector exists for a single point, the prospects for more +complicated target sets and irradiance distributions appear limited. +References +[1] M. Born, E. Wolf, Principles of Optics: 60th Anniversary Edition, 7th +Edition, Cambridge University Press, 2019. +[2] L. Caffarelli, V. Oliker, Weak solutions of one inverse problem in geometric +optics, Journal of Mathematical Sciences 154 (2008) 39–49. +[3] J. Schruben, Formulation of relector-design problem for a lighting fixture, +J. Opt. Soc. Am. 62 (12) (1972) 1498–1501. +[4] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field +scattering data, Inverse Problems 13 (2) (1997) 363–373. +[5] X.-J. Wang, On design of reflector antenna, Inverse Problems 12 (1996) +351–375. +[6] B. Kinber, On two reflector antennas, Radio Eng. Electron. Phys. 7 (1962) +973–979. +[7] D. Burkhard, D. Shealy, Design of reflectors which will distribute sunlight +in a specified manner, Solar Energy 17 (1975) 221–307. +34 + +[8] T. E. Horton, J. H. McDermit, Design of a specular aspheric surface to +uniformly radiate a flat surface using a nonuniform collimated radiation +source, J. Heat Transfer (1972) 453–458. +[9] F. Fournier, A review of beam shaping strategies for LED lighting, in: T. E. +Kidger, S. David (Eds.), Illumination Optics II, Vol. 8170, SPIE, 2011, pp. +55 – 65. +[10] A. Gray, Modern Differential Geometry of Curves and Surfaces with Math- +ematica, 2nd Edition, CRC Press, Boca Raton, FL, 1997. +[11] V. Oliker, E. Newman, The energy conservation equation in the reflector +mapping problem, Applied Mathematics Letters 6 (1) (1993) 91–95. +[12] V. Oliker, On reconstructing a reflecting surface from the scattering data in +the geometric optics approximation, Inverse Problems 5 (1) (1989) 51–65. +[13] J. Schruben, Analysis of rotationally symmetric reflectors for illuminating +systems∗, J. Opt. Soc. Am. 64 (1) (1974) 55–58. +[14] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd Edition, +Encyclopedia of Mathematics and its Applications, Cambridge University +Press, 2013. +[15] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field +scattering data ii. numerical solution ., Numer. Math. 79 (1998) 553–568. +[16] F. Fournier, B. Cassarly, J. Rolland, Optimization of single reflectors for +extended sources, Proc SPIE 7103 (09 2008). +[17] F. Fournier, B. Cassarly, J. Rolland, Fast freeform reflector generation using +source-target maps, Opt. Express 18 (5) (2010) 5295–5304. +[18] T. Graf, V. Oliker, An optimal mass transport approach to the near-field +reflector problem in optical design, Inverse Problems 28 (2012) 025001. +35 + +[19] F. R. Fournier, Freeform reflector design with extended sources, Ph.D. +thesis, CREOL, the College of Optics and Photonics at the University of +Central Florida (2010). +[20] J. Munkres, Topology, 2nd Edition, Featured Titles for Topology, Prentice +Hall, Incorporated, 2000. +36 + diff --git a/Q9AyT4oBgHgl3EQf7vq-/content/tmp_files/load_file.txt b/Q9AyT4oBgHgl3EQf7vq-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa0815d2e6ce913cacf306964e6b506ab07e486d --- /dev/null +++ b/Q9AyT4oBgHgl3EQf7vq-/content/tmp_files/load_file.txt @@ -0,0 +1,818 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf,len=817 +page_content='Weak solutions to the near-field reflector problem with spatial restrictions approached with generalized reflectors constructed from ellipsoids Dylanger S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Pittman 400 Dowman Drive, Atlanta Abstract We motivate then formulate a novel variant of the near-field reflector problem and call it the near-field reflector problem with spatial restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let O be an anisotropic point source of light and assume that we are given a bounded open set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Suppose that the light emitted from the source at O in directions defined by the aperture D ⊆ S2, of radiance g(m) for m ∈ D, is reflected off R ⊂ U, creating the irradiance f(x) for x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The inverse problem consists of constructing the reflector R ⊆ U from the given position of the source O, the input aperture D, radiance g, ‘target’ set T, and irradiance f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We focus entirely on the case where the target set T is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Keywords: partial differential equations, geometric optics, geometry 2020 MSC: 78A05, 35, 51, 53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Introduction Let O be the origin of R3, and let S2 be the unit sphere centered at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We treat points on S2 as unit vectors with initial points at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let an aperture be a subset of S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' in our work, the aperture will be an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically, it makes sense to consider O as the location of an anisotropic point source of light such that rays of light are emitted in a set of directions defined by an aperture D ⊆ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Email address: dpittm2@emory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='edu (Dylanger S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Pittman) Preprint submitted to Constructive Approximation January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='00845v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='AP] 2 Jan 2023 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given an aperture that is a connected open set D ⊆ S2, and a function ρ : D → (0, ∞) that is continuous and almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then a reflector is the set R = {mρ(m)|m ∈ D} ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If ρ is a smooth function, we can call R a smooth reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given an aperture, D, that is a connected open set, assume that we have a continuous, almost everywhere differentiable, positive function ρ : D → (0, ∞), and a corresponding reflector R = {mρ(m)|m ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Suppose that a ray origi- nating from O in the direction m ∈ D is incident on the reflector R at the point mρ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If ρ is differentiable at m, there is a unit vector, n(m), normal to the reflector R at mρ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, by the reflection law of geometric optics, a ray from O of direction m reflects off the point mρ(m) in the direction y(m) = m − 2⟨m, n(m)⟩n(m) (1) where ⟨m, n(m)⟩ is the standard Euclidean inner product in R3 and n(m) is oriented such that ⟨m, n(m)⟩ > 0 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The reflector R is designed such that the ray described by the point mρ(m) ∈ R and the direction y(m) corresponds to some element in a prespecified target set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' What one means by a ‘target set’ changes depending on the context, and the correspondence between y(m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' also, an element of the target set can also vary depending on one’s needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Hence a target set can represent many things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For example, if the target set T is a subset of S2, then a possible correspondence can be y(m) |y(m)| ∈ T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' see [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically, in this case, T can be considered as a set of directions for rays of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If T is a subset R3 \\ {O}, then for an example of another possible correspondence, we can say that for every m ∈ D, there exists an a(m) > 0 such that a(m)y(m) + mρ(m) ∈ T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' see [3] and [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically, in this case, T can be considered as a region that one wants to illuminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that g is an integrable and nonnegative function over an aperture D, and f is an integrable and nonnegative function over a target set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically speaking, we say g(m) for m ∈ D is the radiance of the source at O in the directions m ∈ D, or that g is a radiance distribution over D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We also say f(x) 2 for x ∈ T is the irradiance of the target set at x ∈ T, or that f is an irradiance distribution over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A reflector system comprises of an aperture D, O, a reflector R, an integrable and nonnegative function g over D, and a target set T with an integrable and nonnegative function f over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' From a physical perspective: light emitted from the source at O in directions defined by the aperture D, of radiance g(m) for m ∈ D, is reflected off R, creating the irradiance f(x) for x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' An example that can serve as an illustration is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A reflector problem is, in short, an inverse problem that seeks to complete a reflector system by creating a reflector that fits the other information given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Specifically, suppose we are given O, an aperture D, an integrable and non- negative function g over D, and a target set T with an integrable and nonneg- ative function f over T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The aim of a reflector problem is to find a continu- ous, almost everywhere differentiable, positive ρ over D such that the reflector R = {mρ(m)|m ∈ D} produces the specified in advance irradiance distribution f on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Reflector problems have been well studied due to their utility in physics and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Such problems have found numerous applications in the construc- tion of reflector antennas (see [5], [6]), mirror design [7], heat transfer [8], and beam shaping [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We only consider in the high-frequency approximation of light, where the laws of geometric optics apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We now proceed with a general description and motivation for the near-field reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The Near-Field Reflector Problem We discuss a reflector problem that we call the ‘near-field reflector prob- lem.’ In short, the near-field reflector problem aims to design a reflector that redistributes the light from the origin onto a set a finite distance away from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In this part, when we say surface, we mean it in the differential geometric sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' see Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='4 in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Suppose that we are given a reflector system 3 O The plane reflector R The surface normal to R Light rays going to some target set T Figure 1: Here is the most basic example of a reflector system with a smooth reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Here R is a plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Every point on R has a normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Light originates from the point O with directions represented by points on the unit sphere S2 and travels according to some target set that is neither shown nor specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 4 consisting of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' O, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' an aperture D ⊂ S2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' a nonnegative g ∈ L1(D), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' a bounded Borel set T ⊂ R3 \\ {O} (typically either a subset of a surface or a finite set), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' a nonnegative and integrable function f : T → [0, ∞), 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' and a smooth function ρ : D → (0, ∞) with a smooth reflector R = {mρ(m)|m ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' From a physical perspective, this setup can be described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The light is emitted from the source at O in directions defined by the aperture D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Each ray of direction m ∈ D has radiance g(m) and is reflected off R at the point mρ(m) in the direction y(m) as described by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For every m ∈ D, there exists an a(m) > 0 such that a(m)y(m) + mρ(m) ∈ T creating the irradiance f(x) for x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A basic illustration of this situation is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' With this setup in mind, we proceed with a formulation of the near-field reflector problem Let u = (u1, u2) be smooth local coordinates on S2 such that D lies in one coordinate patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The position vector of a point m ∈ D is m = m(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We choose the coordinates u1, u2 so that ⟨m, m1 × m2⟩ = 1 in D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' here, ⟨, ⟩ denotes the scalar product in R3 and mi = ∂m ∂ui , i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Observe that this implies that ⟨m, mi⟩ = 0, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The first fundamental form of S2 is given by e = eijduiduj where eij = ⟨mi, mj⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Set r(m) = mρ(m), then r(m) defines a smooth surface R = {r(m)|m ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let g = gijduiduj be the first fundamental form of R where gij = ⟨ri, rj⟩ = ρiρj + ρ2eij, ri = ∂r ∂ui , and ρi = ∂ρ ∂ui .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let n(m) is the normal vector field on R such that ⟨n(m), m⟩ > 0 everywhere on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then n(m) = (ρ2 + | ˜∇ρ|2)−1/2(r − ˜∇ρ) (2) 5 O Reflector R Target set T with irradiance distribution f Surface normals to R Light Rays Figure 2: Here is an illustration of the near-field reflector problem in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The radiation intensity at the origin O is given by a nonnegative function g ∈ L1(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We want to find a reflector R such that the reflected rays produce the prescribed irradiance distribution f on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 6 where | ˜∇p|2 = ρiρjeij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This combined with equation (1) determines the direction a ray will go after reflecting off R [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We can now track the path of each ray described by the direction m ∈ D to a point x(m) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A ray, originating at O in direction m, hits the surface R at a point r(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then, said ray reflects off R at r(m) in the direction y(m) as defined by (1) and reaches T at some point x(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, from a physical perspective, an irradiance f(x(m)) is created by the rays reflected at x(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This defines a mapping m → x that we call a reflector map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for convenience, we denote x(m) as the image of m under the reflector map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The reflector map x : D → T combined with equations (1) and (2) describes the ray tracing from D to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If the reflector map is a diffeomorphism from D to T where T is a subset of a smooth surface, then one can introduce the first fundamental form of T as w = wijduiduj, where wij = ⟨xi, xj⟩, xi = ∂x ∂ui .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' According to the differential form of the energy conservation law [1], f(x(m))|J(x(m))| = g(m) (3) where J is the Jacobian determinant of the map x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that J(x(m)) = ±dν(x(m)) dσ(m) = ± � det(wij) � det(eij) (4) where dσ is the surface area element on S2, and dν is the surface area element on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We assign a ± sign to the Jacobian according to whether x preserves the orientation or reverses it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, by integration of (3), for all Borel sets ω ⊆ T, � x−1[ω] gdσ = � ω fdν (5) where x−1[ω] = {m ∈ D|x(m) ∈ ω} and � D gdσ = � T fdν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' With this motivation, we can now state the near-field reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given O, an aperture D ⊂ S2 with a nonnegative function g ∈ L1(D), and a bounded Borel set T ⊂ R3\\{O} with a nonnegative, integrable function f : T → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The goal is to find a smooth function ρ over D such that: 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The ray originating from O in the direction m ∈ D reflects off the reflector R = {mρ(m)|m ∈ D} in accordance with equation (1) and reaches the target set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' g(m) on D is transformed by the reflector map into f on T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for all Borel subsets ω ⊆ T, � x−1[ω] gdσ = � ω fdν (6) where x : D → T the reflector map corresponding to the reflector R = {mρ(m)|m ∈ D}, x−1[ω] = {m ∈ D|x(m) ∈ ω}, dσ is the surface area element on S2, and dν is the area element on T (ν is typically some discrete or Lebesgue measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The law of total energy conservation is obeyed: � D gdσ = � T fdν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The case where the reflector map is a diffeomorphism from D to T can be alternatively formulated as a PDE of Monge-Amp`ere type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' specifically equation (4) from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' There has been a lot of work done on the near-field reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In 1972, Schruben [3] found that if the target set was a subset of a plane in R3, one can then derive an implicit integro-differential equation describing the reflector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' the existence of a solution was not proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then in [13], Schruben considered the case where the target set was a small rotationally symmetric patch on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In this case, when the radiance and the irradiance distributions are rotationally symmetric, the equation derived in [3] can be solved as an ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In 1989, Oliker [12] found a formulation of the near-field reflector problem in the form of a strongly non-linear PDE of Monge-Amp`ere type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The exploration of the said equation is difficult and in [12] was solved only for the rotationally symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In 1998, Kochengin and Oliker [4] introduced an alternative formulation to the near-field reflector problem, which was a geometric approach involving the analysis of the boundaries of convex sets generated by families of supporting ellipsoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This approach can also be considered a weak solution to the PDE 8 introduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The strategy was to assume that the target set was a finite set on a plane and constructively prove the existence of solutions for that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since the reflectors that were constructed were convex, one can use the Blaschke selection theorem (for more details, see [14]) to prove the existence of a solution with a continuous target set on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This method was largely motivated by previous work done by Caffarelli and Oliker [2] which involved the analysis of the boundaries of convex sets generated by families of supporting paraboloids to solve a related problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In [15] a provably convergent numerical algorithm was introduced that ex- plicitly finds the ellipsoids required to construct the reflectors described in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' It was shown that this construction leads to infinitely many solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' however, the algorithm has the benefit of converging to a unique solution if we fix an ini- tial point on the reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This algorithm and its variations have been explored extensively in various scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For example, Fournier, Cassarly, and Rolland in [16] adapted the algorithm in [15], to situations where the light source is not a single point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' specifically, a flat rotationally symmetric emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In [17] a method was proposed for smoothing out a reflector with a discrete irradiance distribution to a reflector with a continuous irradiance distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Optimal transport methods have also been studied [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The Near-Field Reflector Problem with Spatial Restrictions In this paper, we study a novel variant of the near-field reflector problem where we have extreme limitations on where we can place and construct the reflectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Specifically, we are given an open set U ⊂ R3 \\{O}, and our reflector R must now be a subset of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given an x ∈ R3 \\ {O} and a subset S ⊆ R3 \\ {O}, then we define Proj(x) = x |x| as the projection of x onto S2 and Proj[S] = {Proj(x) ∈ S2|x ∈ S} as the projection of S onto S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given a positive, continuous, almost everywhere differen- tiable function ρ over Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We have a reflector R = {mρ(m)|m ∈ Proj[U]} 9 which determines our reflector map x : Proj[U] → T which is determined by tracking the path of each ray described by the direction m ∈ Proj[U] to a point x(m) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A ray, originating at O in direction m, hits the reflector R at a point mρ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then, assuming ρ is differentiable at m, said ray reflects off R at mρ(m) in the direction y(m) as defined by (1) and reaches T at some point x(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, from a physical perspective, an irradiance f(x(m)) is created by the rays reflected at x(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This defines a mapping m → x(m) that we call the reflector map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for convenience, we denote x(m) as the image of m under the reflector map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We can now formulate the near-field reflector problem with spatial restric- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given an open set U ⊂ R3 \\ {O}, O, an aper- ture Proj[U] ⊂ S2, a nonnegative g ∈ L1(Proj[U]), and a bounded Borel set T ⊂ R3 \\ {O} with an integrable function f : T → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The goal is to find a positive, continuous, almost everywhere differentiable function ρ over Proj[U] such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' R = {mρ(m)|m ∈ Proj[U]} ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The ray originating from O in the direction m ∈ Proj[U] reflects off of R in accordance with equation (1) and reaches the target set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' g(m) on Proj[U] is transformed by the reflector map into f on T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for all Borel subsets ω ⊆ T, � x−1[ω] gdσ = � ω fdν (7) where x : Proj[U] → T is the reflector map, x−1[ω] = {m ∈ Proj[U]|x(m) ∈ ω}, dσ is the surface area element on S2, and dν is the area element on T (ν is typically some discrete or Lebesgue measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The law of total energy conservation is obeyed: � Proj[U] gdσ = � T fdν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This variation of the near-field reflector problem has clear applications to engineering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' as often one has to grapple with restrictions of space in real-world designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For example, in the construction of automotive headlights, there are 10 strict restrictions, guided purely by aesthetics, as to where a reflector can be placed and how a reflector must be shaped [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' However, to the author’s knowledge, no mathematical research has been done in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We focus exclusively on the case where the target set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Ellipsoids of Revolution We do all our work in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We denote S2 to be the unit sphere with the center at O and kx = x/|x| for all x ∈ R3 \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We borrow much of this geometric setup from [4] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Ellipsoids of revolution are of paramount importance when solving the near-field reflector problem due to their unique optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let x ∈ R3 \\ {O} and d ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We denote by Ed(x) an ellipsoid of revolution about the axis Ox and with foci at points O and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The polar radius relative to O can be represented as: ψx,d(m) = d 1 − ϵ⟨m, kx⟩, m ∈ S2 (8) where ϵ is the eccentricity and ϵ = � 1 + d2 x2 − d |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (9) So in other words Ed(x) = {mψx,d(m)|m ∈ S2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (10) From this point on, whenever we use the term ellipsoid we specifically refer to an ellipsoid of this kind with one of the foci always at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that each Ed(x) is uniquely defined by the x ∈ R3 \\ {O} and the d ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In this paper, we define Ψx,d(m) = mψx,d(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that for all possible values of d, we have that ϵ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Also for a fixed x, as d → 0 the ellipsoid will degenerate into a line segment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Ed(x) → {tx + (1 − t)O|t ∈ [0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Such an ellipsoid is called degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Observe that as d → ∞, |ψx,d(m)| → ∞ for all m ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' An important property of ellipsoids can be described by the following propo- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 11 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let c, d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then the ellipsoids Ecd(x) and Ed(x) have the same foci: O and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' From a physical perspective, the aforementioned property is important be- cause a reflector that is shaped like an ellipsoid Ed(x) will illuminate the focus x with the light emitted from O such that the total energy emitted from O is equal to the total energy reflected onto x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This property is still true no matter how large or small the ellipsoid is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' all that matters is the location of the foci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Generalized Reflectors Before we proceed, we reiterate that the near-field reflector problem can be expressed analytically as a PDE of Monge Amp´ere Type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Specifically, the equation (4) from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore we will consider the following formulation of the near-field reflector problem with spatial restrictions as a weak formulation and its solutions, weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The following formulation only concerns the case where the target set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Weak Solutions Using Generalized Reflectors Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given an aperture D ⊆ S2 that is an open set, and a function ρ : D → (0, ∞) that is not necessarily continuous and almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then a generalized reflector is the set R = {mρ(m)|m ∈ D} ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The upper half-space of R3 be represented as R3+ = {(x, y, z) ∈ R3|z > 0}, and the lower half-space of R3 be represented as R3− = {(x, y, z) ∈ R3|z < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let σ denote the standard measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider an open set U ⊆ R3+, a corresponding aperture Proj[U], and a finite target set T ⊂ R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let B be a countable family of open subsets of S2 such that σ(Proj[U] \\ � B∈B B) = 0, Proj[U] ⊆ � B∈B B, and σ(B∩B′) = 0 for all distinct B, B′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let the set B(U) be the set of all such families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since every ellipsoid requires foci and an eccentricity to be well defined, given a family B ∈ B(U), let UT (B) be the set of all functions B → T and V (B) be 12 the set of all functions B → (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus we define ET (U) = � � B∈B Ψu(B),v(B)[B] ����� B ∈ B(U), u ∈ UT (B), v ∈ V (B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (11) Assume we are given a Z ∈ ET (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us define BZ = � Int(Proj[Ed(x) ∩ Z]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ Z]) ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (12) The geometry of the ellipsoid and the definition of BZ imply that there ex- ists unique u ∈ UT (B) and v ∈ V (B) such that Z = � B∈BZ Ψu(B),v(B)[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Define uZ ∈ UT (BZ) and vZ ∈ V (BZ) be the unique functions such that Z = � B∈BZ ΨuZ(B),vZ(B)[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given some Z ∈ ET (U), let y1 Z(m) = {B ∈ BZ|m ∈ B} for m ∈ Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given a B ∈ B(U), let N (B) be the set of all injective functions s : B → N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For Z ∈ ET (U) and s ∈ N (BZ), define ρs Z(m) = ψuZ(s−1(min s[y1 Z(m)])),vZ(s−1(min s[y1 Z(m)]))(m), m ∈ Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (13) Observe that the function ρs Z is positive, not necessarily continuous, and almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let W(ρs Z) = {mρs Z(m)|m ∈ Proj[U]} and thus we describe a set of generalized reflectors RU 1 (T) = � W(ρs Z)| Z ∈ ET (U) where Z ⊂ U, s ∈ N (BZ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (14) Assume we are given a generalized reflector R ∈ RU 1 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us define BR = � Int(Proj[Ed(x) ∩ R]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ R]) ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (15) The geometry of the ellipsoid and the definition of BR imply that there exists an s ∈ N(BR), unique u ∈ UT (B) and unique v ∈ V (B) such that W(ρs Z) = R where Z = � B∈BR Ψu(B),v(B)[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, for every generalized reflector R ∈ RU 1 (T), we may define a unique BR ∈ B(U) such that for each B ∈ BR there are unique xB ∈ T and dB ∈ (0, ∞) such that, for some s ∈ N(BR), R = W (ρs Z) where Z = � B∈BR ΨxB,dB[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, given a generalized reflector R ∈ RU 1 (T), we obtain a correspond- ing BR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for each B ∈ BR we define unique xB and dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We also obtain an sR ∈ N(BR) and a unique ZR = � B∈BR ΨxB,dB[B] such that R = W(ρsR ZR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 13 Given a generalized reflector R ∈ RU 1 (T), for all m ∈ Proj[U] we define M(m) = xB ∈ T (16) where mρsR ZR(m) = ΨxB,dB(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let y2 R(m) be the points of intersection between R \\ {mρsR ZR(m)} and the line segment connecting mρsR ZR(m) to M(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given a generalized reflector R ∈ RU 1 (T), the map α1 : Proj[U] → T ∪ R, α1(m) = � � � � � M(m) if y2 R(m) = ∅ y2 R(m) if y2 R(m) ̸= ∅ (17) is called the generalized reflector map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically speaking, a ray of light of direction m originating from O can only reach the target set if y2 R(m) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us define for all Borel X ⊆ S2 µg(X) = � X g(m)dσ(m) (18) where σ denotes the standard measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that g ≡ 0 outside of Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically speaking, g is the radiance distribution of the source at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In order to formulate and solve the generalized reflector problem (in the framework of weak solutions to be defined below), we need to define a mea- sure representing the energy generated by g and redistributed by a generalized reflector R ∈ RU 1 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given a generalized reflector R ∈ RU 1 (T) and a set ω ⊆ T we define the visibility set of ω as V U 1 (ω) = � A∈A A \\ {m ∈ Proj[U]|α1(m) = y2 R(m)} (19) where A = {B ∈ BR|xB ∈ ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We now need to show that V U 1 (ω) is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For an element x ∈ R3 and a set A ⊂ R3, let the set Cx,A = {at + x(1 − t)|t ∈ [0, 1], a ∈ A} be the union of all line segments from x to A and Cx,A,∞ = {at + x(1 − t)|t ∈ [0, ∞), a ∈ A} be the union of all rays from x that intersect A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 14 We proceed with the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let w : S2 → (0, ∞) be continuous and W(m) = mw(m) for all m ∈ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If B is a Borel set of S2, then CO,W [B] and CO,W [B],∞ are Borel sets of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that all Borel sets can be formed from open sets through the operations of countable union, countable intersection, and relative complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let {Ei} be a countable collection of open sets of S2 such that through said operations, we obtain B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then given the countable collection of open sets of R3, {Int(CO,W [Ei])}, through the same sequence of operations we used to obtain B from {Ei}, we obtain Int(CO,W [B]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus CO,W [B] is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, assuming Wi ≡ (im)w(m) for all m ∈ S2 and i ∈ (0, ∞), �∞ n=1 CO,Wn[B] = CO,W [B],∞ is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If B is a Borel set of S2, x ∈ R3 \\ {O} and d ∈ (0, ∞), then Cx,Ψx,d[B] and Cx,Ψx,d[B],∞ are Borel sets of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let S2 x = {m+x|m ∈ S2} be the set of all unit vectors originating from x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='e the unit sphere centered at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since x is another focus of the ellipsoid, there exists a continuous function w : S2 → (0, ∞) such that Ed(x) = {mw(m)+x|m ∈ S2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let Wx(m) = mw(m) + x and let W(m) = mw(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that since B is Borel in S2, Ψx,d[B] is Borel in Ed(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus W −1 x [Ψx,d[B]] is Borel in S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1, CO,W [W −1 x [Ψx,d[B]]] and CO,W [W −1 x [Ψx,d[B]]],∞ are Borel sets of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, by translation, Cx,Ψx,d[B] and Cx,Ψx,d[B],∞ are Borel sets of R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We can now prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let R be a generalized reflector in RU 1 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For any set ω ⊆ T the visibility set V U 1 (ω) is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We make use of the fact that sets formed from Borel sets through the operations of countable union, countable intersection, and relative complement are Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that we obtain a sR ∈ N(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that by the definition of a generalized reflector in RU 1 (T), R = � B∈BR ΨxB,dB [B′] where B′ = {m ∈ 15 B|mρsR ZR = ΨxB,dB(m)} = B\\� K∈K K where K = {A ∈ BR|sR(A) < sR(B)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' note that B′ is clearly Borel and B ⊆ B′ ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For B ∈ BR, we have that CxB,ΨxB,dB [B′] and CO,ΨxB,dB [B′] are Borel sets by Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since BR is countable and functions of the form ΨxB,dB are continuous and bijective, R is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus for all B ∈ BR, the set QB = CxB,ΨxB,dB [B′] ∩ (R \\ ΨxB,dB[B′]) is Borel and therefore the set LB = CxB,QB,∞ ∩ ΨxB,dB[B′] is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus Proj[LB] is Borel, as it is the preimage of LB under ΨxB,dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since {m ∈ Proj[U]|α1(m) = y2 R(m)} = � B∈BR Proj[LB], (20) we have that {m ∈ Proj[U]|α1(m) = y2 R(m)} is Borel and thus V U 1 (ω) is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Define for any generalized reflector R ∈ RU 1 (T), G1(ω) = µg(V U 1 (ω)) (21) which we will deem the energy function of the generalized reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let F be a nonnegative, finite measure on the finite set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We say that a generalized reflector R ∈ RU 1 (T) is a weak solution to the generalized reflector problem if the generalized reflector map α1 determined by R is such that F(ω) = G1(ω) for any Borel set ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (22) It would be useful to point out the similarity of condition (22) and condition (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Geometric Lemmas One thing that should be noted is that the definition of the generalized reflector map takes into account that there could potentially be a part of the generalized reflector that intercepts an already reflected ray before it can reach the target set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' That fact inspires some key geometric lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 16 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let R ∈ RU 1 (T) for some finite set T ⊂ R3− and open set U ⊆ R3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For all B ∈ BR, if m ∈ B, then α1(m) = xB if and only if y2 R(m) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This follows directly from the definition of the reflector map of the gen- eralized reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' When we say that r is a ray in Cx,B,∞, then r = {at + x(1 − t)|t ∈ [0, ∞)} for some a ∈ B, similarly if we say r is a line segment in Cx,B, then r = {at + x(1 − t)|t ∈ [0, 1]} for some a ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let A, B ⊂ S2 be disjoint sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then for any x ∈ R3 \\ {O} and a, b ∈ (0, ∞), Cx,Ψx,a[A] ∩ Ψx,b[B] = ∅ and Cx,Ψx,b[B] ∩ Ψx,a[A] = ∅ if and only if Cx,Ψx,a[A],∞ ∩ Ψx,b[B] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If Cx,Ψx,a[A],∞∩Ψx,b[B] ̸= ∅, then there exists a ray r in Cx,Ψx,a[A],∞ such that r intersects Ψx,b[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, either there exists a line segment in Cx,Ψx,a[A] that intersects Ψx,b[B] and thus Cx,Ψx,a[A] ∩ Ψx,b[B] ̸= ∅, or there exists a line segment in Cx,Ψx,b[B] that intersects Ψx,a[A] and thus Cx,Ψx,b[B] ∩ Ψx,a[A] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Conversely, if Cx,Ψx,a[A] ∩ Ψx,b[B] ̸= ∅, then there exists a line segment in Cx,Ψx,a[A] that intersects Ψx,b[B], said line segment coincides with a ray in Cx,Ψx,a[A],∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus Cx,Ψx,a[A],∞ ∩ Ψx,b[B] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If Cx,Ψx,b[B] ∩ Ψx,a[A] ̸= ∅, then there exists a line segment in Cx,Ψx,b[B] that intersects Ψx,a[A], said line segment coincides with a ray in Cx,Ψx,a[A],∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus Cx,Ψx,a[A],∞ ∩ Ψx,b[B] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' These two lemmas give us the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that U is an open set in R3+, and T is a finite target set in R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let R ∈ RU 1 (T) be a generalized reflector and A, B ∈ BR such that A ̸= B and x = xA = xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for all m ∈ A and m′ ∈ B, α1(m) = α1(m′) = x, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Cx,Ψx,dA[A] ∩ Ψx,dB[B] = ∅ and Cx,Ψx,dB [B] ∩ Ψx,dA[A] = ∅, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Cx,Ψx,dA[A],∞ ∩ Ψx,dB[B] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (2) and (3) are equivalent by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3, for all m ∈ A α1(m) = x if and only if y2 R(m) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By definition, y2 R(m) = ∅ if and only if the line segment between Ψx,dA(m) and x does not intersect R \\ {Ψx,dA(m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Similarly, By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3, for all m′ ∈ B, α1(m′) = x if and only if y2 R(m′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By definition, y2 R(m′) = ∅ if and only if the line segment between Ψx,dB(m′) and x does not intersect R\\{Ψx,dB(m′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, statements (1) and (2) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Generalized Reflectors Constructed in an Open Conical Cylinder of Arbi- trary Thickness Let S2 + = {m ∈ S2|⟨m, (0, 0, 1)⟩ > 0} be the open hemisphere of the S2 oriented towards the positive z−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Similarly, S2 − = {m ∈ S2|⟨m, (0, 0, 1)⟩ < 0} be the open hemisphere of the S2 oriented towards the negative z−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given an open U ⊆ S2 +, and δ, z′ > 0, we then define an open conical cylinder of thickness δ as C δ U(z′) = CO,U,∞ ∩ {(x, y, z) ∈ R3|z′ + δ > z > z′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In this paper, given a finite target set T ⊂ R3−, we aim to construct a generalized reflector R ∈ RC δ U(z′) 1 (T) that is a weak solution of the generalized reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This condition is very strict and the following strategies can potentially be applied to other kinds of open subsets in R3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We first consider the case where the target set is a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We proceed with the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U be an open set in S2 + and z′, δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let {Si}i∈N be a countable collection of open subsets in U, {di}i∈N is a countable collection of distinct positive numbers, and x ∈ R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that each Ψx,di[Si] ⊂ C δ U(z′) and denote Ψi = Ψx,di[Si].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then we have that Proj � C δ U(z′) \\ � i∈N (CO,Ψi,∞ ∪ Cx,Ψi,∞) � = Proj � C δ U(z′) \\ � i∈N CO,Ψi,∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (23) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume to the contrary that Proj � C δ U(z′) \\ � i∈N (CO,Ψi,∞ ∪ Cx,Ψi,∞) � ̸= Proj � C δ U(z′) \\ � i∈N CO,Ψi,∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (24) 18 Then there exists a ray r in CO,U\\� i∈N Si,∞ = CO,U,∞ \\ � i∈N CO,Si,∞ such that r ∩ C δ U(z′) ⊂ � i∈N Cxi,Si,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Equivalently, one can say that there must be a ray of direction m ∈ U \\ � i∈N Si originating from O that we denote as r such that r ∩ (C δ U(z′) \\ � i∈N(CO,Ψi,∞ ∪ Cx,Ψi,∞)) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider the plane P(α) = {(x, y, z) ∈ R3|z = α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let m ∈ U \\ � i∈N Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that there exists a set P(z′) ∩ � i∈N Cx,Ψi,∞ such that �� P(z′) ∩ � i∈N Cx,Ψi,∞ � \\ � P(z′) ∩ � i∈N CO,Ψi,∞ �� ∩ CO,U,∞ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (25) Otherwise there does not exist a ray r of direction m ∈ U \\ � i∈N Si originating from O such that r ∩ (C δ U(z′) \\ � i∈N(CO,Ψi,∞ ∪ Cx,Ψi,∞)) = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus we assume such a ray exists r exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then m must be in a direction such that there exists a dmin > 0 where Ψx,dmin(m) ∈ �� P(z′) ∩ � i∈N Cx,Ψi,∞ � \\ � P(z′) ∩ � i∈N CO,Ψi,∞ �� ∩ CO,U,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (26) Since C δ U(z′) is bounded, there must also exist a dmax > 0 such that Ψx,dmax(m) ∈ P(z′ + δ) ∩ CO,U,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (27) Note that by our assumptions, for all d ∈ (dmin, dmax), there exists an α ∈ N such that the line segment between Ψx,d(m) and x is a subset of a line segment in Cx,Ψα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' However, since all Ψi are closed, then for all d ∈ [dmin, dmax], there exists an α such that the line segment between Ψx,d(m) and x is a subset of a line segment in Cx,Ψα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' dmax > di for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that by our assumptions, Ψx,d(m) ∈ � i∈N Cx,Ψi,∞ for all d ∈ [dmin, dmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' However, since ψx,dmax(m) > ψx,di(m) for all i ∈ N, Ψx,dmax(m) cannot reside on the interior of any ellipsoid Ed′(x) where d′ ∈ {di}i∈N, thus Ψx,d(m) ̸∈ � i∈N Cx,Ψi,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' There exists some α ∈ N such that dmax = dα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If there exists some α such that dmax = dα, then, since tΨx,dmax(m) + (1 − 19 t)x ̸∈ C δ U(z′) for all t > 1, Ψx,dmax(m) resides on the ellipsoid Edα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore Ψx,dmax(m) ∈ Ψα ∩ P(z′ + δ) and thus m ∈ Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' There exists some α ∈ N such that dα > dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that {di}i∈N is arranged such that di+1 ≥ di If there exists some α such that dα > dmax, then there exists a ray originating from x that intersects the point Ψx,dmax(m) that also intersects a point (xβ, yβ, zβ) ∈ Ψβ where dβ ≥ dmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The case where dβ = dmax has already been covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' When dβ > dmax: since x ∈ R3− and Ψx,dmax(m) ∈ P(z′ + δ), this implies that zβ > z′ + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that σ is the standard measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U be a Borel set in R3 \\ {O} such that Int(U) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let x ∈ R3 \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider the set K(d) = Proj[Ed(x) ∩ U] and the corresponding function D(d) = σ(K(d)) for d ∈ (0∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then D(d) cannot be identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Furthermore, if U is open, K(d) is open in S2 for all d ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Clearly there exists a d′ ∈ (0, ∞) such that K(d′) ∩ Int(U) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then Ed′(x) ∩ Int(U) is open in Ed′(x) and thus Proj[Ed′(x) ∩ Int(U)] is open in S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, D(d′) ≥ σ(Proj[K(d′) ∩ Int(U)]) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U be an open set in S2 +, δ, z′ > 0, and T = {x} ∈ R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given a nonnegative g ∈ L1(S2) where g ≡ 0 outside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then there exists a generalized reflector R ∈ RC δ U(z′) 1 (T) such that G1({x}) = µg(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For convenience, label C∗ = C δ U(z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that σ is the standard mea- sure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider the set K1(d) = Proj[Ed(x) ∩ C∗] and its corresponding function D1(d) = σ(K1(d)) where d ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that since C∗ is bounded, D1(d) → 0 as d → ∞ and D1(d) → 0 as d → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By construction, it is clear that D1 is bounded by 0 and σ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, Dmax 1 = sup{D1(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='7, Dmax 1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 20 Let ϵ1 ∈ [0, Dmax 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We define dmax1 to be a value such that D1(dmax1) = Dmax 1 − ϵ1 where ϵ1 = 0 if Dmax 1 ∈ {D1(d)|d ∈ (0, ∞)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We now eliminate the parts of U that had already been accounted for and the parts of C∗ that can no longer be used: let E1 = Proj[Edmax1 (x) ∩ C∗], Ψ1 = Ψx,dmax1 [E1], Q2 = C∗ \\ (Cx,Ψ1,∞ ∪ CO,Ψ1,∞), (28) and U2 = U \\ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='6, U2 = Proj[Q2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us define K2(d) = Proj[Ψx,d[U2] ∩ Q2] and D2(d) = σ(K2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that since Q2 is bounded, D2(d) → 0 as d → ∞ and D2(d) → 0 as d → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By construction, it is clear that D2 is bounded by 0 and σ(U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, Dmax 2 = sup{D2(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='7, Dmax 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let ϵ2 ∈ [0, Dmax 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We define dmax2 to be a value such that D1(dmax1) ≥ D2(dmax2) = Dmax 2 − ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given that U1 = U and Q1 = C∗, we can now recursively define a sequence of functions and sets for k ≥ 2: Ek−1 = Proj[Ψx,dmaxk−1 [Uk−1] ∩ Qk−1], (29) Ψk−1 = Ψx,dmaxk−1 [Ek−1], (30) Qk = Qk−1 \\ (Cx,Ψk−1,∞ ∪ CO,Ψk−1,∞), (31) Uk = U \\ � � k−1 � j=1 Ej � � = Proj[Qk], (32) Kk(d) = Proj[Ed(x) ∩ Qk], (33) Dk(d) = σ(Kk(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (34) Also, note that since Qk is bounded, Dk(d) → 0 as d → ∞ and Dk(d) → 0 as d → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By construction, it is clear that Dk is bounded by 0 and σ(Uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, Dmax k = sup{Dk(d)|d ∈ (0, ∞)} exists and is finite, and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='7, Dmax k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let ϵk ∈ [0, Dmax k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We define dmaxk to be a value such that Dk−1(dmaxk−1) ≥ Dk(dmaxk) = Dmax k − ϵk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Observe that the set Kk(d) is open for all d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We can therefore construct 21 a sequence � � �σ � � k� j=1 Ej � � � � � ∞ k=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (35) Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' There exists {ϵi}i∈N such that � σ ��k j=1 Ej ��∞ k=1 converges to σ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By construction, the sequence increases monotonically and is bounded between 0 and σ(U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus it converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume to the contrary that for every possible {ϵi}i∈N, � σ ��k j=1 Ej ��∞ k=1 that converges to an L ∈ (0, σ(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then σ � U \\ �∞ j=1 Ej � = σ(U) − L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider the function D∗(d) = σ � Proj � Ed(x) ∩ lim j→∞ Qj �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (36) Observe that limj→∞ Qj = C∗\\�∞ i=1(Cx,Ψi,∞∪CO,Ψi,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that �∞ i=1(Cx,Ψi,∞∪ CO,Ψi,∞) ⊆ �∞ i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Observe that for all i ∈ N, Int(Cx,Ψi,∞ ∪ CO,Ψi,∞) = Cx,Ψi,∞∪CO,Ψi,∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus �∞ i=1 Int(Cx,Ψi,∞ ∪ CO,Ψi,∞) = �∞ i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus limj→∞ Qj = C∗\\�∞ i=1(Cx,Ψi,∞ ∪ CO,Ψi,∞) is open and thus Int(limj→∞ Qj) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='7, there exists a d′ such that D∗(d′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By the def- inition of convergence, there exists an M such that for all m ≥ M, D∗(d′) > σ ��∞ j=m Ej � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that σ ��∞ j=m Ej � = �∞ j=m σ(Ej) = �∞ j=m Di(dmaxi) ≥ Dm(dmaxm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For all k ∈ N, Dmax k is a limit point of {Dk(d)|d ∈ (0, ∞)}, there- fore as ϵk → 0, Dk(dmaxk) → Dmax k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Observe that Dm(d′) ≥ D∗(d′) because limj→∞ Qj ⊆ Qm and U \\ �∞ j=1 Ej ⊆ Um.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore there exists a sequence {ϵi}i∈N such that Dm(dmaxm) ≥ D∗(d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For this sequence σ ��∞ j=m Ej � ≥ D∗(d′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, there exists {ϵi}i∈N such that � σ ��k j=1 Ej ��∞ k=1 converges to σ(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let Z = ∞ � j=1 Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (37) 22 For some s ∈ N(BZ), consider the generalized reflector R = W(ρs Z) ∈ RC∗ 1 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By construction, Cx,Ψx,dmaxj [Int(Ej)],∞ ∩ Ψx,dmaxj′ [Int(Ej′)] = ∅ (38) when j ̸= j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Observe that if j′, j ∈ N where j′ > j, then dmaxj ̸= dmaxj′ because otherwise Ej′ ⊆ Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus BR = {Int(Ej) ⊂ S2|j ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='5, for all m ∈ B where B ∈ BR, we have α1(m) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then, for any s ∈ N(BZ), the generalized reflector R = W(ρs Z) ∈ RC∗ 1 (T) is a weak solution to the generalized reflector problem such that G1({x}) = µg(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We can now prove a result where our target set is made up of finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' First, we prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that U is an open set in R3+, and T is a finite target set in R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let R ∈ RU 1 (T) be a generalized reflector and A, B ∈ BR such that A ̸= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for all m ∈ A and m′ ∈ B, α1(m) = xA and α1(m′) = xB, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' CxA,ΨxA,dA[A] ∩ ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B] ∩ ΨxA,dA[A] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For the case where xA = xB, we have Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We now consider the case where xB ̸= xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3, for all m ∈ A α1(m) = xA if and only if y2 R(m) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By definition, y2 R(m) = ∅ if and only if the line segment between ΨxA,dA(m) and xA does not intersect R \\ {ΨxA,dA(m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Similarly, By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3, for all m′ ∈ B α1(m′) = xB if and only if y2 R(m′) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By definition, y2 R(m′) = ∅ if and only if the line segment between ΨxB,dB(m′) and x does not intersect R \\ {ΨxB,dB(m′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, statements (1) and (2) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U be an open set in S2 +, δ, z′ > 0, and {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , xk} ∈ R3− where k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0 23 outside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , fk be nonnegative real numbers such that k � i=1 fi = µg(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (39) Assume that there exists n ≥ k disjoint open sets Bi in U where � i∈[n] Bi = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Also assume that there exists a collection of k subsets of [n], {Ai}i∈[k], such that: At ∩At′ = ∅ where t ̸= t′, � i∈[k] Ai = [n], and µg(� i∈At Bi) = ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Suppose that for all i ∈ [n] there exists ai, bi > 0 where z′ ≤ ai < ai + bi ≤ z′ + δ such that C bi Bi(ai) ∩ Cxj,C bj Bj (aj) = ∅ for all j ∈ [n] \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then there exists a generalized reflector in R ∈ RC δ U(z′) 1 (T) such that G1({xi}) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For each i ∈ [n], C bi Bi(ai) is open and a generalized reflector Ri ∈ R C bi Bi(ai) 1 ({xi}) is constructed in the exact same way as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By our assumptions, since Ri ⊆ C bi Bi(ai), then (Ri ∩ C bi Bi(ai)) ∩ Cxj,Rj∩C bj Bj (aj) = ∅ (40) for all j ∈ [n] \\ {i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let F = � i∈[k] Ri and consider the generalized reflector R = W(ρF ) ∈ RC δ U(z′) 1 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By construction, for an A, B ∈ BR such that A ̸= B, we have that CxA,ΨxA,dA[A] ∩ ΨxB,dB[B] = ∅ and CxB,Ψx,dB [B] ∩ ΨxA,dA[A] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='8, for any B ∈ BR, for all m ∈ B we have α1(m) = xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then, for any s ∈ N(BF ), the generalized reflector R = W(ρs F ) ∈ RC δ U (z′) 1 (T) is a weak solution to the generalized reflector problem such that G1({xi}) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We now will use Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2 to construct a specific type of generalized reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let k ≥ 2, d > 0, ξ ∈ (−1, 0), and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that, given a 24 point (x, y, z) ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], θ ∈ [0, 2π), such that x = r cos θ sin φ (41) y = r sin θ sin φ (42) z = r cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (43) Define the set of points T ξ k,d(t) as �� d cos �2πj k + t � sin (arccos(ξ)) , d sin �2πj k + t � sin (arccos(ξ)) , dξ � |j ∈ I � (44) where I = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' If we are additionally given an i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , k − 1}, we may define the set Pk,i(t) ⊂ S2, as �� cos � θ + π(2i − 1) k + t � sin φ, sin � θ + π(2i − 1) k + t � sin φ, cos φ � ����φ ∈ [0, π], θ ∈ � 0, 2π k �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (45) If k = 1, define the set of points T ξ 1,d(t) = {(0, 0, −d)} and P1,0(t) = S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' It is good to observe that T ξ k,d(t) defines the points of a regular k-gon centered at the z-axis and that Pk,i(t) defines a spherical wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let δ, z′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider the open disk U = {m ∈ S2 +|⟨(0, 0, 1), m⟩ > c} where 0 < c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , dn be a collection of not necessarily distinct positive numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , kn be a collection of not necessarily distinct pos- itive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ξn be a collection of not necessarily distinct numbers such that ξi ∈ (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let t′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , t′ n be a collection of not necessarily distinct elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us denote Ti = T ξi ki,di(t′ i) and let T = �n i=1 Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we a given a nonnegative g ∈ L1(S2 +) that is rotationally sym- metric about the z-axis such that g ≡ 0 outside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , fn be a collection of positive numbers such that µg(U) = n � i=1 fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (46) 25 Then there exists a generalized reflector R ∈ RC δ U(z′) 1 (T) such that G1({x}) = � {j∈[n]|x∈Tj} fj kj (47) for all x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' By the intermediate value theorem, there exists a collection of numbers ζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ζn ∈ [c, 1) where ζn = c and µg � {m ∈ S2 +|⟨(0, 0, 1), m⟩ > ζi} � = �i j=1 fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Define Bi = {m ∈ S2 +|⟨(0, 0, 1), m⟩ > ζi} \\ {m ∈ S2 +|⟨(0, 0, 1), m⟩ ≥ ζi−1} (48) for all i ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , n} and B1 = {m ∈ S2 +|⟨(0, 0, 1), m⟩ > ζ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus µg(Bi) = fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Consider the set Ti, let Ti(j) = � di cos �2πj ki + t′ i � sin (arccos(ξi)) , di sin �2πj ki + t′ i � sin (arccos(ξi)) , diξi � (49) where j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki − 1} if k ≥ 2 and Ti(0) = (0, 0, −di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let Pi(j) = Pki,j(t′ i) where j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki −1}, then by construction µg(Bi ∩ Pi(j)) = fi ki .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let Ui(j) = C δ n Bi∩Int(Pi(j)) � z′ + (i − 1) δ n � where j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since Ti(j) ∈ R3−, any given line segment between a point in Ui(j) and the point Ti(j) will not intersect any set Ui′(j′) where i′ > i and j′ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki′ − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore CTi(j),Ui(j) ∩ Ui′(j′) = ∅ where i′ > i and j′ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki′ − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Also, since CO,Pi(j),∞ is a convex set and Ui(j), {Ti(j)} are both subsets of CO,Pi(j),∞, we then have CTi(j),Ui(j) ⊂ CO,Pi(j),∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, CTi(j),Ui(j) ∩ Ui(j′) = ∅ where j′ ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Finally, if there exists a line segment between a point in Ui(j) and Ti(j) that intersects a Ui′(j′) where i > i′ and j′ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki′ − 1}, then it must intersect CO,{m∈S2 +|⟨(0,0,1),m⟩>ζi−1},∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' However, CTi(j),Ui(j) is disjoint from CO,{m∈S2 +|⟨(0,0,1),m⟩>ζi−1},∞ and thus CTi(j),Ui(j) ∩ Ui′(j′) = ∅ where i > i′ and j′ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , ki′ − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, CTi(j),Ui(j) ∩ Ui′(j′) = ∅ when (i, j) ̸= (i′, j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2, there exists a generalized reflector R ∈ RCδ U(z′) 1 (T) such that G1({x}) = � {j∈[n]|x∈Tj} fj kj (50) 26 for all x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Interpolated Reflectors The generalized reflector presented in the previous section might be impos- sible or, at best, very difficult to construct in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus we introduce the following notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that we are given an aperture that is a connected open set D ⊆ S2 and a not necessarily continuous, almost everywhere differentiable function ρ : D → (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then an interpolated reflector is the set R = ∂(CO,S) \\ ∂(CO,S,∞) ⊂ R3 where S = {mρ(m)|m ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' It is interesting to note that, given an aperture that is a connected open set D ⊆ S2, a set is a reflector if and only if it is both a generalized reflector and an interpolated reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The type of interpolated reflector we construct below is a topological surface (see Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='36 in [20]) and thus consists of one connected component instead of countably many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In a practical sense, when designing an interpolated reflector as opposed to a generalized reflector, new challenges are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus, we settle for finding a necessary and sufficient condition for the existence of an interpolated reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We will consider the following formulation of the near-field reflector problem as a weak formulation of equation (4) from [12] and its solutions, weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The following formulation only concerns the case where the target set is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Weak Solutions using Interpolated Reflectors Consider a connected open set U ⊆ R3+, a corresponding aperture Proj[U], and a finite target set T ⊆ R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Also, consider the set RU 1 (T) as defined by (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We then describe a set of interpolated reflectors RU 2 (T) = � ∂(CO,S) \\ ∂(CO,S,∞)| S ∈ RU 1 (T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (51) 27 It is interesting to note that the interpolated reflectors in RU 2 (T) are all topo- logical surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given an interpolated reflector R ∈ RU 2 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us define BR = � Int(Proj[Ed(x) ∩ R]) ⊆ S2 |d ∈ (0, ∞), x ∈ T, σ(Proj[Ed(x) ∩ R]) ̸= 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (52) The geometry of the ellipsoid and the definition of BR imply that there exists an s ∈ N(BR), unique u ∈ UT (B) and unique v ∈ V (B) such that ∂(CO,W (ρs Z))\\∂(CO,W (ρs Z),∞) = R where Z = � B∈BR Ψu(B),v(B)[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, for every interpolated reflector R ∈ RU 2 (T), we may define a unique BR ∈ B(U) such that for each B ∈ BR there are unique xB ∈ T and dB ∈ (0, ∞) such that, for some s ∈ N(BR), R = ∂(CO,W (ρs Z)) \\ ∂(CO,W (ρs Z),∞) where Z = � B∈BR ΨxB,dB[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, given a interpolated reflector R ∈ RU 2 (T), we obtain a cor- responding BR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for each B ∈ BR we define unique xB and dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We also obtain an sR ∈ N(BR) and a unique ZR = � B∈BR ΨxB,dB[B] such that R = ∂(CO,S) \\ ∂(CO,S,∞) where S = W(ρsR ZR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given an interpolated reflector R ∈ RU 2 (T), let M(m) = xB ∈ T (53) where mρsR ZR(m) = ΨxB,dB(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let y2 R(m) be the points of intersection between R \\ {mρsR ZR(m)} and the line segment connecting mρsR ZR(m) to M(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given an interpolated reflector R ∈ RU 2 (T), the map α2 : Proj[U] → T ∪ R, α2(m) = � � � � � M(m) if y2 R(m) = ∅ y2 R(m) if y2 R(m) ̸= ∅ (54) is called the interpolated reflector map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically speaking, a ray of light of direction m originating from O can only reach the target set if y2 R(m) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' As before, we denote by g ∈ L1(S2) the energy density of the source O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let us define for all Borel X ⊆ S2 µg(X) = � X g(m)dσ(m) (55) 28 where σ denotes the standard measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that g is a nonnegative function where g ≡ 0 outside of Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Physically speaking, g is the radiance distribution of the source at O In order to formulate and solve the interpolated reflector problem (in the framework of weak solutions to be defined below), we need to define a measure representing the energy generated by g and redistributed by an interpolated reflector R ∈ RU 2 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Given a interpolated reflector R ∈ RU 2 (T) and a set ω ⊆ T we define the visibility set of ω as V U 2 (ω) = � A∈A A \\ {m ∈ Proj[U]|α2(m) = y2 R(m)} (56) where A = {B ∈ BR|xB ∈ ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We now need to show that V U 2 (ω) is measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let R be a interpolated reflector in RU 2 (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For any set ω ⊆ T, the visibility set V U 2 (ω) is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We make use of the fact that sets formed from Borel sets through the operations of countable union, countable intersection, and relative complement are Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Recall that we obtain a sR ∈ N(BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that by the definition of an interpolated reflector in RU 1 (T), R = � B∈BR ΨxB,dB [B′] ∪ SR where B′ = {m ∈ B|mρsR ZR = ΨxB,dB(m)} = B \\ � K∈K K where K = {A ∈ BR|sR(A) < sR(B)} and SR = R \\ � B∈BR ΨxB,dB[B′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that B′ is clearly Borel and B ⊆ B′ ⊆ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For B ∈ BR, we have that CxB,ΨxB,dB [B′] and CO,ΨxB,dB [B′] are Borel sets by Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that R is Borel as it is a boundary of an open set minus the boundary of another open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since BR is countable and functions of the form ΨxB,dB are continuous and bijective, � B∈BR ΨxB,dB [B′] is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore SR is also Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus for all B ∈ BR, the set QB = CxB,ΨxB,dB [B′]∩(R\\ΨxB,dB[B′]) is Borel and therefore the set LB = CxB,QB,∞∩ ΨxB,dB[B′] is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus Proj[LB] is Borel, as it is the preimage of LB under 29 ΨxB,dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since {m ∈ Proj[U]|α2(m) = y2 R(m)} = � B∈BR Proj[LB], (57) we have that {m ∈ Proj[U]|α2(m) = y2 R(m)} is Borel and thus V U 2 (ω) is Borel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Define for any interpolated reflector R ∈ RU 2 (T), G2(ω) = µg(V U 2 (ω)) (58) which we will deem the energy function of interpolated reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let F be a nonnegative, finite measure on the finite set T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We say that an interpolated reflector R ∈ RU 2 (T) is a weak solution to the interpolated reflector problem if the interpolated reflector map α2 determined by R is such that F(ω) = G2(ω) for any Borel set ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (59) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Main Results Here we prove a necessary and sufficient condition for the existence of weak solutions to the interpolated reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We proceed with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that U is an open set in R3+, and T is a finite target set in R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) such that g ≡ 0 outside Proj[U] and g > 0 inside Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let R ∈ RU 1 (T) be a generalized reflector, then we define the set Bx R = {B ∈ BR|x = xB} for x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then for any z, y ∈ T the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' for all m ∈ � A∈Bz R A and m′ ∈ � B∈By R B, α1(m) = z and α1(m′) = y, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Cz,Ψz,dA[A] ∩Ψy,dB[B] = ∅ and Cy,Ψy,dB [B] ∩Ψz,dA[A] = ∅ for all A ∈ Bz R and B ∈ By R where A ̸= B, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' CxA,ΨxA,dA[A] ∩ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B] ∩ΨxA,dA[A] = ∅ for all A, B ∈ BR where A ̸= B, 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' G1({x}) = µg(� A∈Bx R A) and G1({y}) = µg(� B∈By R B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (1) ⇔ (2) by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='8, clearly (3) ⇔ (2), and (1) trivially implies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We only need to prove that (4) implies (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' we prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume that there exists an A ∈ Bx R and a B′ ∈ By R such that, without loss of generality, Cz,Ψz,dA[A] ∩ Ψy,dB′ [B′] ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let Q = Cz,Ψz,dA[A] ∩ Ψy,dB′ [B′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that Cz,Ψz,dA[A],∞ ∩ Cz,Ψy,dB′ [B′],∞ = Cz,Q,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since Cz,Ψz,dA[A],∞ \\ {O} and Cz,Ψy,dB′ [B′],∞\\{O} are open, Cz,Q,∞\\{O} is open;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus Cz,Q,∞∩EdB(y) is open in EdB(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Thus Proj[Q] ⊆ B is open and, since g > 0 in U, µg(Proj[Q]) > 0, and thus G1({y}) ≤ µg(� B∈By R B) − µg(Proj[Q]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let K be a subset of Rn where n ≥ 2 such that K is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The complement U = Rn \\ K is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For sufficiently large R > 0, the set V = {x|R < |x|} is contained in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since V is connected, there exists a connected component of U that contains V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' This is the unique unbounded connected component of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We define the exterior boundary of K as the boundary of the unbounded connected component of Rn \\ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' We denote this as ∂E(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The following result gives a condition that is necessary and sufficient for the existence of weak solutions to the interpolated reflector problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U ⊂ R3+ be a simply connected open set such that U ⊂ R3+, and T ⊂ R3− be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given a nonegative g ∈ L1(S2) such that g ≡ 0 outside Proj[U] and g > 0 inside Proj[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let F be a measure over T such that F(T) = µg(Proj[U]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (60) Then there exists an interpolated reflector R2 ∈ RU 2 (T) that is a weak solu- tion to the interpolated reflector problem as defined in (59) if and only if there exists a generalized reflector R1 ∈ RU 1 (T) that is a weak solution to the gen- eralized reflector problem as defined in (22) where R1 is a subset of a simply 31 connected subset of U ∩ ∂E � �CO,R1 ∪ � B∈BR1 CxB,ΨxB,dB [B] � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (61) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' It is clear that if R2 ∈ RU 2 (T) that is a weak solution to the interpolated reflector problem as defined in (59), then, for any s ∈ N(BR2), R1 = W(ρs F ) where F = � B∈BR2 ΨxB,dB[B] is a weak solution to the generalized reflector problem as defined in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Note that this implies that BR1 = BR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Since g is positive in Proj[U], for our reflector R1, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1, for all distinct A, B ∈ BR1, CxA,ΨxA,dA[A]∩ΨxB,dB[B] = ∅ and CxB,ΨxB,dB [B]∩ΨxA,dA[A] = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, for all A ∈ BR1, ΨxA,dA[A]∩Int � CO,R1 ∪ � B∈BR1 CxB,ΨxB,dB [B] � = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Also observe that for all A ∈ BR1, (CO,ΨxA,dA[A],∞ \\ CO,ΨxA,dA[A]) ∩ Int � CO,R1 ∪ � B∈BR1 CxB,ΨxB,dB [B] � = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Therefore, R1 is a subset of a simply connected subset of ∂E � �CO,R1 ∪ � B∈BR1 CxB,ΨxB,dB [B] � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (62) Since the interpolated reflector must be contained in U, we obtain (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Conversely, if there exists a reflector R1 ∈ RU 1 (T) that is a weak solution of the generalized reflector problem as defined in (22) where R1 ⊂ Q such that Q is a simply connected subset of (61), then R2 = ∂(CO,R1) \\ ∂(CO,R1,∞) is also a subset of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Discussion In this note, with respect to the near-field reflector problem with spatial restrictions, we defined two different kinds of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For the first weak solution, we proved, under certain assumptions, the existence of a generalized reflector where the target set is multiple points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' A possible avenue for further research is to attempt to expand Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='3 for different target sets, apertures, and spatial restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Another idea might be to try to come up with designs such that the generalized reflectors have finitely many connected components 32 instead of countably many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The author believes that the following statement is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U be an open set in S2 +, δ, z′ > 0, and {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , xk} ∈ R3− where k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given a positive g ∈ L1(S2) where g ≡ 0 outside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , fk be nonnegative real numbers such that k � i=1 fi = µg(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (63) Then there exists a generalized reflector R ∈ RC δ U(z′) 1 (T) such that G1({xi}) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For the second weak solution, we proved a theorem that detailed a necessary and sufficient condition for the existence of an interpolated reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' The ad- vantage of our interpolated reflectors, as opposed to our generalized reflectors, is that our interpolated reflector design is a topological surface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' thus it is easier to construct from an engineering perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' An obvious avenue for further work would be to create some practically useful interpolated reflectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2 might be useful in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' In fact, it would be very useful if the following conjecture is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Conjecture 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let U be a simply connected open set in S2 +, δ, z′ > 0, and {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , xk} ∈ R3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0 outside U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Let f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' , fk be nonnegative real numbers such that k � i=1 fi = µg(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' (64) Then there exists an interpolated reflector in R ∈ RC δ U(z′) 2 (T) such that G2({xi}) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Another fruitful avenue of research might be to somehow expand these def- initions of weak solutions to account for cases where the target set is not finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Then, proving the existence of generalized and interpolated reflectors with con- tinuous irradiance distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 33 As the reader might have noticed, we make no attempt to address the near- field reflector problem with spatial conditions with a reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Instead, we exclusively use generalized or interpolated reflectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' While it may be interesting to research reflectors in order to get a ‘stronger’ solution, it is the author’s view that, in general, it is not possible to construct a reflector under spatial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' For example, if the target set is a single point, the solution to the near-field reflector problem is an ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' However, if given spatial restrictions, a single ellipsoid, in general, cannot fit those restrictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' this is demonstrated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' Munkres, Topology, 2nd Edition, Featured Titles for Topology, Prentice Hall, Incorporated, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} +page_content=' 36' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9AyT4oBgHgl3EQf7vq-/content/2301.00845v1.pdf'} diff --git a/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/2301.13679v1.pdf.txt b/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/2301.13679v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..45447a0edd595982d90fec2bfd41b2d17de9caec --- /dev/null +++ b/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/2301.13679v1.pdf.txt @@ -0,0 +1,1313 @@ +Density of states and spectral function of a superconductor out of a quantum-critical +metal +Shang-Shun Zhang1 and Andrey V. Chubukov1 +1School of Physics and Astronomy and William I. Fine Theoretical Physics Institute, +University of Minnesota, Minneapolis, MN 55455, USA +(Dated: February 1, 2023) +We analyze the validity of a quasiparticle description of a superconducting state at a metallic +quantum-critical point (QCP). A normal state at a QCP is a non-Fermi liquid with no coherent +quasiparticles. +A superconducting order gaps out low-energy excitations, except for a sliver of +states for non-s-wave gap symmetry, and at a first glance, should restore a coherent quasiparticle +behavior. We argue that this does not necessarily hold as in some cases the fermionic self-energy +remains singular slightly above the gap edge. +This singularity gives rise to markedly non-BCS +behavior of the density of states and to broadening and eventual vanishing of the quasiparticle peak +in the spectral function. We analyze the set of quantum-critical models with an effective dynamical +4-fermion interaction, mediated by a gapless boson at a QCP, V (Ω) ∝ 1/Ωγ. We show that coherent +quasiparticle behavior in a superconducting state holds for γ < 1/2, but breaks down for larger γ. +We discuss signatures of quasiparticle breakdown and compare our results with the data. +Introduction. +Metals near a quantum critical +point (QCP) display a number of non-Fermi liquid +properties like linear-in-T resistivity, a broad peak in +the spectral function near kF with linear-in-ω width, +singular behavior of optical conductivity, etc [1–18]. +These properties are often thought to be caused by +the coupling of fermions to near-gapless fluctuations of +an order parameter, which condenses at a QCP [19– +29]. +The same fermion-boson interaction gives rise to +superconductivity near a QCP [30–42]. +A +superconducting +order +gaps +out +low-energy +excitations, leaving at most a tiny subset of gapless +states for a non-s−wave order parameter. +A general +belief has been that this restores fermionic coherence. A +frequently cited experimental evidence is the observed +re-emergence +of +a +quasiparticle +peak +below +Tc +in +near-optimally doped cuprates (see e.g., +Ref. [43]). +From theory side, the argument is that the fermionic +self-energy in a superconductor has a conventional Fermi- +liquid form Σ(ω) ∼ ω at the lowest ω, in distinction from +a non-Fermi-liquid Σ(ω) ∝ ωa with a < 1 in the normal +state +[44–53]. +In this paper, we analyze theoretically +whether fermions in a superconducting state at a QCP +can be viewed as well-defined coherent quasiparticles. +We argue that this is not necessarily the case as fermionic +self-energy can still be singular on a real frequency axis +immediately above the gap edge. This singularity gives +rise to markedly non-BCS behavior of the density of +states (DoS) and to broadening and eventual vanishing +of the quasiparticle peak. +For superconductivity away from a QCP, mediated by +a massive boson, numerous earlier studies have found +that the spectral function A(k, ω) at T += 0 has a +δ-functional peak at ω = (∆2 + (ξk/Z)2)1/2, where +ξk = vF (k − kF ) is a fermionic dispersion (vF is a +Fermi velocity), ∆ is a superconducting gap, and Z is +an inverse quasiparticle residue. +A δ-functional peak +FIG. 1. +Three possible forms of the electronic spectral +function +A(k, ω) +at +T += +0 +in +a +quantum +critical +superconductor at a small but finite k − kF and in the +absence of impurity broadening. +(a): +A(k, ω) vanishes at +|ω| = ∆ and has a well-defined peak at ω > ∆, (b): A(k, ω) +diverges at |ω| = ∆, but it non-monotonic at larger ω. The +peak in A(k, ω) at |ω| > ∆ broadens, but still exists. (c): +A(k, ω) diverges at |ω| = ∆, and monotonically decreases at +larger ω. In case (a) fermions can be viewed as well-defined +quasiparticles, in case (c) the quasiparticle picture completely +breaks down. The case (b) is the intermediate one between +(a) and (c). +holds for momenta near the Fermi surface, as long as +ω < ∆ + ω0, where ω0 is a mass of a pairing boson +in energy units. At larger ω, fermionic damping kicks +in, and the peak broadens. +The same physics leads +to peak-dip-hump behavior of A(k, ω) as a function of +ω, observed most spectacularly in near-optimally doped +cuprate Bi2Sr2CaCu2O8+δ (see, e.g, Refs. [54, 55]). At +a QCP, the pairing boson becomes massless and ω0 +vanishes. This creates a singular behavior near the gap +edge at ω = ∆, which holds even when ξk is finite. +A simple experimentation shows that there are three +possible forms of A(k, ω), which we present in Fig. 1: +it (i) either vanishes at ω = ∆ and has a well-defined +peak at ω > ∆ whose width at small ξk is parametrically +smaller than its energy; or (ii) diverges at ω = ∆, +arXiv:2301.13679v1 [cond-mat.supr-con] 31 Jan 2023 + +Three possible forms of A( k,w) in quantum-critical superconductor +(a) Quasiparticle picture +(b) Partial breakdown +(c) Complete breakdown +of quasiparticle +of quasiparticle +(k,w) +A +-△ +0 +-△ +0 +-△ +0 +3 +3 +32 +0 +0.5 +1 +2 +. +0 +0.5 +1 +1.5 +2 +Leading exponent +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +Subleading exponent +8 +c +0 +0.2 +0.4 +0.6 +0.8 +!=7g +0 +2 +4 +6 +8 +10 +D(!) +. = 0:8; 8 ' 1:18 +gap edge +0 +0.1 +0.2 +(" ! !)8 +0 +0.1 +0.2 +D(!) ! 1 +-10 +-5 +0 +5 +10 +!=7g +0 +1 +2 +3 +4 +5 +DoS +10!2 +10!1 +100 +(! ! ")=7g +100 +101 +9 1=x0:5 +9 1=x0:59 +. = 0:35 +. = 0:8 +(a) +(b) +(c) +FIG. 2. +(a) Exponents ν and c for the leading and the subleading terms in the expansion D(ω) ≃ 1+α(∆−ω)ν +β(∆−ω)ν+c, +where D(ω) = ∆(ω)/ω and the gap edge ∆ is the solution of D(ω = ∆) = 1. (b) Numerical result for D(ω) for γ = 0.8. Inset +shows the power-law behavior near the gap edge with ν = 1.18, consistent with (a). (c) Fermionic DoS at T = 0 for γ = 0.35 +(thick green line) and γ = 0.8 (thin pink line). In both cases, the DoS vanishes below the gap edge ∆ and has a power-law +singularity above it N(ω) ∝ 1/(ω − ∆)ν/2, but the exponent ν is different in the two cases, as we show in the right panel. +but is non-monotonic at larger ω and displays a broad +maximum at some ω > ∆, or (iii) diverges at ω = ∆ and +monotonically decreases at larger ω. +In the first case, +fermions in a quantum-critical superconductor can be +viewed as well-defined quasiparticles; in the last case the +quasiparticle picture completely breaks down; the second +case is the intermediate one between the other two. Our +goal is to understand under what circumstances A(k, ω) +of a quantum-critical superconductor has one of these +forms. +Model. +For our study, we consider dispersion- +full fermions, Yukawa-coupled to a massless boson. We +assume, like in earlier works (see, e.g., Refs. [56]), that +a boson is Landau overdamped, and its effective velocity +is far smaller than vF . In this situation, the interaction +that gives rise to non-Fermi liquid in the normal state +and to superconductivity, is a purely dynamical V (Ω). +The fermionic self-energy and the pairing gap, tuned into +a proper spatial pairing channel, are then determined +by two coupled equations in the frequency domain. At +a QCP, V (Ω) is singular at vanishing Ω in spatial +dimension D ≤ 3, and behaves as V (Ω) ∝ (¯g/Ω)γ, +where ¯g is the effective fermion-boson coupling, and the +exponent γ is determined by the underlying microscopic +model. +The most studied models of this kind are of +fermions near an Ising-nematic or Ising/ferromagnetic +QCP (γ = 1/3) and near an antiferromagnetic or charge +density wave QCP (γ += 1/2). +The same effective +interaction emerges for dispersion-less fermions in a +quantum dot coupled to Einstein bosons (the Yuakawa- +SYK model) [57–60]. For this last case, the exponent γ is +a continuous variable γ ∈ (0, 1), depending on the ratio of +fermion and boson flavors. An extension of the Yukawa- +SYK model to γ ∈ (1, 2) has recently been proposed [61]. +We follow these works and consider γ as a continuous +variable. We note that the value of γ is generally larger +deep in a superconducting state because of feedback +from superconductivity on the bosonic polarization. For +simplicity, we neglect potential in-gap states associated +with non-s-wave pairing symmetry and focus on the +spectral function of fermions away from the nodal points +and on features in the density of states (DoS) above the +gap edge. An extension to models with in-gap states is +straightforward. +In previous studies of the γ-model, we focused on +the novel superconducting behavior at γ > 1, when the +pairing interaction is attractive on the Matsubara axis, +while on the real axis ReV (Ω) is repulsive [62, 63]. We +argued that this dichotomy gives rise to phase slips of +the gap function on the real axis. +Here, we restrict +ourselves to γ ≤ 1, when this physics is not present and, +hence, does not interfere with the analysis of the validity +of a quasiparticle description in a superconducting state. +Pairing gap and quasiparticle residue. +For +superconductivity mediated by a dynamical interaction, +the paring gap ∆(ω) and the inverse quasiparticle +residue Z(ω) are functions of the running real fermionic +frequency ω. We define the gap edge ∆ (often called the +gap) from the condition ∆(ω) = ω at ω = ∆. +For our purposes, it is convenient to introduce D(ω) = +∆(ω)/ω. The gap edge is at |D| = 1. The equation for +D(ω) that we need to solve is +ωB(ω)D(ω) = A(ω) + C(ω), +(1) +where B(ω) and A(ω) are regular functions of ω (see +[64, 65]). The C(ω) term depends on the running D(ω), +C(ω) = ¯gγ sin πγ +2 +� ω +0 +dΩ +Ωγ +D(ω − Ω) − D(ω) +� +D2(ω − Ω) − 1 +. +(2) +Its presence makes Eq. (S3) an integral equation. The +inverse residue Z(ω) is expressed via D(ω′) as +Z(ω) = B(ω) + ¯gγ sin πγ +2 +ω +� ω +0 +dΩ +Ωγ +1 +� +D2(ω − Ω) − 1 +(3) + +3 +FIG. 3. +Spectral function A(k, ω) at T = 0 for four representative γ. The broadening in the plots is intrinsic. (a-d): color- +coded plot at negative ω, as measured by the ARPES intensity at T = 0. (e-f): constant-k cuts of A(k, ω) at ξk = 0 and at +ξk = ±4¯g. For γ < 1/2, the spectral function has a sharp quasiparticle peak at ω + ∆ ∝ ξ2 +k. For γ > 1/2, the peak moves to +ω + ∆ ∝ |ξk|1/(1−γ) and broadens up, which eventually disappears (see text). +and is readily obtained once D(ω) is known. +At γ += 0, which models a BCS superconductor, +C(ω) = 0 and D(ω) = A(ω)/(ωB(ω)) is a regular +function of frequency. +Near the gap edge at ω > 0, +D(ω)−1 ∼ ω−∆ and Z(ω) ≈ Z(∆) ≡ Z. We assume and +then verify that D(ω) remains regular in some range of +γ > 0. Substituting D(ω)−1 ∼ ω−∆ into (S7) for γ > 0, +we obtain C(ω)−C(∆) ∼ (ω−∆)3/2−γ. We see that C(ω) +is non-analytic near the gap edge, but for γ < 1/2, the +exponent 3/2−γ is larger than one. In this situation, the +non-analytic term in C(ω) generates a non-analytic term +in D(ω) of order (ω − ∆)3/2−γ, which is smaller than the +regular ω−∆ term. Evaluating the prefactors, we obtain +slightly above the gap edge, at ω = ∆ + δ +D′(∆ + δ) = 1 + αδ + A cos[π(3/2 − γ)]δ3/2−γ, +D′′(∆ + δ) = −A sin[π(3/2 − γ)]δ3/2−γ, +(4) +where α ∼ 1/¯g, A = � α +2 +¯gγ sin(πγ/2) +∆B(∆) +J(γ, 1) and J(γ, ν) is +expressed via Beta functions: +J(γ, ν) = B(1 − γ, γ − 1 − ν +2) − B(1 − γ, γ − 1 + ν +2). (5) +For γ > 1/2, 3/2 − γ > 1, and the calculation of D(ω) +has to be done differently. We find after straightforward +analysis that the leading δ-dependent term in D(∆ + δ) +is non-analytic and of order δν, where ν is the solution +of J(γ, ν) = 0. The exponent ν ≈ 1 + 0.67(γ − 1/2) for +γ ≈ 1/2 and ν ≈ 1.3 for γ = 1. The subleading term in +D(∆ + δ) scales as δν+c, where c > 0 is approximately +linear in γ − 1/2. In Fig. 2, we plot ν(γ) and c(γ) along +with the numerical results of D(ω) for a representative +γ = 0.8. The exponent ν extracted from this numerical +D(ω) is 1.18, which matches perfectly with the analytical +result. The behavior at γ = 1/2 is special, and we discuss +it in Ref. [64]. +Substituting D(∆ + δ) into the formula for Z(ω), Eq. +(S8), we obtain +Z′(∆ + δ)=Z(∆)+B cos(π(γ + ν/2 − 1))δ1−γ−ν/2,(6) +Z′′(∆ + δ)=B sin(π(γ + ν/2 − 1))δ1−γ−ν/2. +(7) +where B = +¯gγ sin πγ +2 +∆ +√ +2α B(1 − γ, ν +2 + γ − 1). For γ < 1/2, +Z(ω) = Z(∆) + O(δ1/2−γ) is approximately a constant +near the gap edge. +For γ > 1/2, the inverse residue +diverges at the gap edge, indicating a qualitative change +in the system behavior. +Spectral function and DoS. +The spectral function +and the DoS per unit volume are given by +A(k, ω) = − 1 +π ImGR(k, ω), +N(ω) = 1 +V +� +k +A(k, ω) = NF ωIm +� +1 +∆2(ω) − ω2 , (8) +where +the +retarded +Green’s +function +GR(k, ω) += +−(ωZ(ω) + ξk)/(ξ2 +k + (∆2(ω) − ω2)Z2(ω)). +ARPES +intensity is proportional to A(k, ω)nF (ω), which at T = 0 +selects negative ω. +At γ = 0 (BCS limit), N(ω) ∼ +1/(ω−∆)1/2, and the spectral function has a δ-functional +peak at ω = (∆2 + (ξk/Z)2)1/2. In Fig. 2 (c,d), we show +the DoS N(ω), obtained from the numerical solution of +the full gap equation (S3) for representative γ = 0.35 +and 0.8. We see that in both cases the DoS describes a +gapped continuum, but there is a qualitative difference in +the behavior near the gap edge: for γ = 0.35, N(ω) has +the same 1/δ1/2 singularity as for γ = 0, and for γ = 0.8 + +(e) = 0.35 +(f) = 0.45 +1 +1 +A(k,w) +Sk = 0 +0> +...... +0< +0.5 +0.5 +0 +0 +-10 +6- +-8 +-7 +-6 +-5 +-4.5 +-4 +-3.5 +-3 +-2.5 +-2 +w/g +w/g(g) = 0.65 +(h) = 0.8 +0.4 +0.4 +0.3 +0.3 +0.2 +0.2 +0.1 +0.1 +0 +0 +-2 +-1.5 +-1 +-0.5 +-1.5 +-1 +-0.5 +w/g +w/g(a) = 0.35 +(b) = 0.45 +-2 +-1 +-4 +-2 +19 +-6 +3 +-8 +-3 +-10 +-4 +-12 +-10 +-5 +0 +5 +10 +-10 +-5 +0 +5 +10 +Sk/g +Sk/g +1.5. +1.5.(c) = 0.65 +(d) = 0.8 +-0.5 +-0.5 +-1 +-1 +-1.5 +-1.5 +-2 +-10 +-5 +0 +5 +10 +-10 +-5 +0 +5 +10 +Sk/g +Sk/g +0.5 +0.5Max +00. +0.0. +0.4 +the DOS behaves as 1/δ0.59, which perfectly matches the +analytical form δ−ν/2, given that ν = 1.18 for γ = 0.8. +The spectral function A(k, ω) is shown in Fig. +(3). +For comparison with ARPES, we set ω to be negative: +ω = −(∆ + δ). +For any γ, there is no frequency +range, where A(k, ω) is a δ-function, simply because +the bosonic mass vanishes at a QCP. Still, for γ < 1/2, +D(−(∆+δ))−1 ∝ δ and Z(−(∆+δ)) ≈ Z(−∆) = Z(∆). +In this situation, the spectral weight on the Fermi +surface, integrated over an infinitesimally small range +around ω += +−∆ immediately above the real axis, +is finite, like in BCS case. +Away from the Fermi +surface, the spectral function vanishes as |ω + ∆|1/2−γ +at the gap edge and displays a quasiparticle peak at +ω ≈ −(∆2 + (ξk/Z(∆))2)1/2. The peak is well defined +at small δ as its width O(δ1/2−γ) is parametrically +smaller than its frequency. +This is the same behavior +as in Fig. 1 (a). +For γ +> +1/2, the situation is +qualitatively different. +Now Z(−∆ − δ) diverges at +δ → 0 and D(−∆ − δ) − 1 ∼ |δ|ν ≪ |δ|. +In this +case, the integral of A(kF , ω) over an infinitesimally +small range around ω = −∆ vanishes, which can be +interpreted as a vanishing of a quasiparticle peak. +At +finite ξk, the spectral function diverges at the gap edge +as 1/|ω + ∆|γ/2+γ−1. For γ slightly above 1/2, A(k, ω) +is non-monotonic and possess a broad maximum at +|ω + ∆| ∼ (ξk/¯gγ) +1 +1−γ . +This is the same behavior as +in Fig. 1 (b). +For larger γ, the maximum disappears, +and A(k, ω) monotonically decreases at |ω| > ∆. This +is the same behavior as in Fig. 1 (c). +For small ξk, +the maximum disappears at γ ∼ 0.9. +For larger ξk, +it disappears at smaller γ, first for positive ξk (see Fig. 4). +Comparison with ARPES +The behavior shown +in Fig. 4 is our result in some range of γ > 1/2. For +positive ξk (i.e., outside the Fermi surface), the spectral +function has a single non-dispersing maximum at the +gap edge, except for the smallest ξk, while for negative +ξk, A(k, ω) has a kink at the gap edge ω = −∆ and +a dispersing maximum at ω = −∆ − O +� +|ξk|1/(1−γ)� +. +This behavior is consistent with the ARPES data for +Bi2201, Ref. [66]. +The data shows that the spectral +function near the antinode, where our analysis is valid, +displays an almost non-dispersing maximum at positive +ξk, while for negative ξk it displays a non-dispersing +kink at the same energy and a dispersing maximum +at larger |ω|. +We associate the non-dispersing feature +at both positive and negative ξk with the gap edge +∆, and associate the dispersing maximum, observed +in [66] at ξk < 0, with the dispersing maximum in Fig. 4. +Discussion and summary. +In this work, we +analyzed the applicability of quasiparticle description of +a superconducting state which emerges out of a non- +Fermi liquid at a metallic QCP. We considered the +model with an effective dynamical 4-fermion interaction +FIG. 4. (a) Spectral function A(k, ω) at positive and negative +ξk = ±4¯g at γ = 0.6. To account for impurity scattering, +we convoluted the spectral function with a Lorentzian of +width ∼ 0.03¯g. +(b) Spectral function at a set of discrete +momenta. It displays a non-dispersing gap edge singularity +(green dots) and a dispersing maximum (blue circles). This +theoretical A(k, ω) is consistent with the ARPES data for +Bi2201, Ref. [66] (see text). +V (Ω) ∝ 1/Ωγ, mediated by a gapless boson at a QCP +and analyzed the spectral function and the DoS for +γ ∈ (0, 1). Interaction V (Ω) gives rise to a non-Fermi +liquid in the normal state with self-energy Σ(ω) ∝ ω1−γ +and to pairing below some finite Tc. A superconducting +order gaps out low-energy excitations and, at a first +glance, should restore fermionic coherence. +We found, +however, that this holds only for γ < 1/2. For larger γ +the spectral function and the DoS exhibit qualitatively +different behavior than that in a superconductor with +coherent quasiparticles. +(different power-laws). +We +argued that the quasiparticle peak broadens up and +completely disappears for γ close to one. +Away from a QCP, a pairing boson is massive and +at the lowest energies a Fermi-liquid description holds +already in the normal state and continue to hold in +a superconductor. +In particular, in the immediate +vicinity of the gap edge, the system displays a BCS- +like behavior for all γ. Still, the system behavior over +a broad frequency range is governed by the physics at +a QCP, as numerous experiments on the cuprates and +other correlated systems indicate. We argued that our +results are quite consistent with the ARPES data for +Bi2201 [11, 66]. +We acknowledge with thanks useful conversations with +a number of our colleagues. 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Tanaka, et al., Science 331, 1579 (2011). + +7 +Supplementary information for “Density of states and spectral function of a +superconductor out of a quantum-critical metal” +by Shang-Shun Zhang and Andrey V Chubukov +GAP EQUATION ALONG THE REAL-FREQUENCY AXIS AND ITS SOLUTION +We will use the approach pioneered by Marsiglio, Shossmann, and Carbotte [1]. In this approach, one first solves +non-linear gap equation along the Matsubara axis, which can be done rather straightforwardly as the gap function +∆(ωm) can be chosen to be real for all frequencies and is a regular function of ωm even when the pairing boson is +massless. One then uses this ∆(ωm) as an input for the equation for complex ∆(ω) along the real frequency axis. +The non-linear integral equation for D(ωm) = ωm∆(ωm) on the Matsubara axis, and the equation for the inverse +quasiparticle residue Z(ωm) = 1 + Σ(ωm)/ωm (Σ(ωm) is the fermionic self-energy), have the form +ωmD(ωm) = πT +� +ω′m +(D(ω′ +m) − D(ωm)) sgn(ω′ +m) +� +1 + D2(ω′m) +V (ω − ω′ +m), +(S1) +Z(ωm) = 1 + 1 +ωm +πT +� +ω′m +sgn(ω′ +m) +� +1 + D2(ω′m) +V (ω − ω′ +m), +(S2) +where V (Ωm) = (¯g/|Ωm|)γ is the same as in the main text (¯g is an effective fermion-boson coupling, and γ depends +on the underlying microscopic model).This set of equations has a non-zero solution D(ωm) below a finite pairing +temperature Tp ∼ ¯g. Fig. S1 shows the numerical solution for ∆(ωm) at T = 10−6¯g ≪ Tp, for different γ. We see +from the figure that ∆(ωm) approaches a constant value at small frequencies and decays as ω−γ +m +at high frequencies. +This behavior holds for all γ and can be easily verified analytically. +The gap equation along the real frequency axis is +ωB(ω)D(ω) = A(ω) + C(ω), +(S3) +(Eqn. (1) in the main text). This equation is obtained by using the spectral representation of an analytic function +on the upper frequency half-plane +f(iωm) = 1 +π +� +dx Imf(x) +x − iωm +(S4) +and, where possible, keeping D(ωm) as an input function. This approach was pioneered for electron-phonon interaction +in Refs. [3–6] for the electron-phonon problem. +FIG. S1. Numerical resulls for the gap function ∆(ωm) along the Matsubara axis. The calculation is performed at temperature +T = 10−6¯g using the hybrid-frequency method [2]. + +10 +10° +6/(um)V +1.0 +0. +10 +102 +100 +104 +wm +/g8 +20 +40 +60 +80 +100 120 +0 +0.5 +1 +1.5 +D(!) +(a) +. = 0:25 +#5 +D0 +D00 +20 +40 +60 +80 +100 120 +!=7g +0 +0.5 +1 +1.5 +2 +Z(!) +(b) +. = 0:25 +Z0 +Z00 +1 +2 +3 +0 +0.5 +1 +1.5 +(c) +. = 0:5 +1 +2 +3 +!=7g +0 +2 +4 +6 +(d) +. = 0:5 +0 +1 +2 +3 +4 +5 +0 +0.5 +1 +1.5 +(e) +. = 0:8 +0 +1 +2 +3 +4 +5 +!=7g +0 +2 +4 +6 +8 +10 +(f) +. = 0:8 +FIG. S2. Numerical results for D(ω) and Z(ω) for γ = 0.25, γ = 0.5 and γ = 0.8. +For our case, the functions A(ω) and B(ω) are directly expressed via D(ωm) along the Matsubara axis as +A(ω) = 1 +2 +� ∞ +0 +dωm +D(ωm) +� +1 + D2(ωm) +× +� +¯gγ +(ωm + iω)γ + +¯gγ +(ωm − iω)γ +� +, +(S5) +B(ω) = 1 + i +2ω +� ∞ +0 +dωm +1 +� +1 + D2(ωm) +× +� +¯gγ +(ωm + iω)γ − +¯gγ +(ωm − iω)γ +� +. +(S6) +and C(ω) is given by +C(ω) = ¯gγ sin πγ +2 +� ω +0 +dΩ +Ωγ +D(ω − Ω) − D(ω) +� +D2(ω − Ω) − 1 +, +(S7) +(Eqn (3) in the main text). This function depends on the running D(ω − Ω), which makes Eq. (S3) an integral +equation. The inverse residue Z(ω) is expressed via D(ω′) as +Z(ω) = B(ω) + ¯gγ sin πγ +2 +ω +� ω +0 +dΩ +Ωγ +1 +� +D2(ω − Ω) − 1 +(S8) +(Eqn (4) in the main text) and is readily obtained once D(ω) is known. +The gap equation along the real-frequency axis has an iterative structure in the sense that D(ω) depends on D(ω′) +at ω′ < ω. This allows us to solve this equation iteratively, using the low-frequency form D(ω) ≃ ∆(0)/ω as an input, +with ∆(0) ≡ ∆(ωm = πT). In Fig. S2 we show the results for D(ω) and Z(ω) for three representative values of γ. In +all cases, D(ω) and Z(ω) are real below the gap edge ω = ∆ and are complex above the gap edge, where ∆ is defined +as ∆(ω) = 1 at ω = ∆. We see that for γ > 1/2, Z(ω) diverges at the gap edge. We use this fact in the main text in +the analysis of the spectral function. + +9 +FIG. S3. The real and imaginary parts of the gap function near the gap edge ω = ∆ for γ = 1/2, obtained by solving the +non-linear gap equation numerically. The leading term in D′(∆ − δ) − 1, shown in the inset of (a), is linear in δ∆ − ω, and the +subleading scales as δ/| log |δ||, as is confirmed by the linear relation in panel (a), The imaginary part D′′ appears at negative +δ above the gap edge. The numerical result in panel (b) clearly shows the scaling relation D′′ ∼ δ/ log2 (|δ|/¯g), expected from +the Kramers-Kronig relation with D′. +THE CASE OF γ = 1/2 +In the main text we argued that for γ < 1/2, the function D(∆ − δ) − 1 ∝ δ, where δ = ∆ − ω, and the correction +scales as δ3/2−γ. More specifically, we found iteratively that +D(∆ − δ) = 1 + δ +∞ +� +n=0 +αnδnϵ +(S9) +where ϵ = 1/2−γ and α0 = O(1/¯g). The expression for α1 is presented in the main text, after Eq. (4). It is proportional +to J(γ, 1) = B(1 − γ, γ − 3/2) − B(1 − γ, γ − 1/2), where B(a, b) is a Beta function (B(a, b) = Γ(a)Γ(b)/Γ(a + b)). +For small ϵ (i.e., for γ ≤ 1/2), α1 ∼ J(γ, 1) ∼ 1/ϵ. For the next term in (S9) we find α2 ∼ 1/ϵ2, and so on. +We see that the perturbative expansion in δϵ in (S9) holds for (δ/¯g)ϵ/ϵ ≤ 1. Outside this range, all terms in Eq. +(S9) are relevant. As γ approaches 1/2 from below and ϵ decreases, the perturbative regime shrinks to exponentially +small δ < ¯g exp(−| log ϵ|/ϵ). +To understand the form of D(ω) outside the perturbative regime, we express (δ/¯g)ϵ as eϵ log (δ/¯g) and expand (S10) +in powers of log (δ/¯g). We obtain +D(∆ − δ) = 1 + δ +∞ +� +n=0 +˜αn(log δ +¯g )n +(S10) +where ˜α0 = α0 + α1 + α2 + .., ˜α1 = ϵα1 + 2ϵα2 + .., ˜α2 = 2ϵ2α2 + .... We see that each ˜αn is a series, in which the +first term is independent on ϵ, and the others diverge as powers of 1/ϵ, because α1 ∼ 1/ϵ, α2 ∼ 1/ϵ2, and so on. +We now argue that singular parts of ˜αn can be neglected. The argument is two-fold. First, in the calculations, the +1/ϵ divergencies originate from the divergence of J(1/2 − ϵ, 1) ≈ 1/(2ϵ). This divergence is regularized by a finite +boson mass, such that strictly at ϵ = 0, one has J(1/2, 1) = 0 instead of infinity. Second, if we assume that ˜α0 +in (S10) remains finite at ϵ = 0 and substitute the trial D(∆ − δ) = 1 + δ˜α0 in the gap equation at γ = 1/2 and +compute iteratively the next term in D(∆−δ), we find it in the form ˜α1δ log (δ/¯g) with a finite ˜α1 = √¯g˜α0/(4∆B(∆)). +Extending the iterative analysis, we find that all ˜αn are finite at γ = 1/2 (i.e., ϵ = 0), as we anticipated. +We didn’t manage to sum up analytically the logarithmic series in (S10). The numerical solution for D(∆ − δ) for +γ = 1/2 shows that D(∆ − δ) − 1 remains linear in δ (see Fig. S2 (c)), and the corrections scale as 1/| log (|δ|/¯g)| +(see Fig. S3 (a)). By Kramers-Kronig relation, this implies that at negative δ, when ω > ∆ is above the gap edge, +the imaginary part of D(ω) scales as D +′′(∆ + |δ|) ∝ δ/ log2 (|δ|/¯g) (the same form is obtained by just noticing that +log(−|δ|) = log(−(ω − ∆ + i0)) = log |δ| − iπ). This form of D +′′(∆ + |δ|) is consistent with our numerical solution +above the gap edge, Fig. S3 (b). The solution clearly shows that the ratio D +′′(∆ + |δ|)/δ decreases at the smallest δ. +We next use the result for D(ω) to obtain the inverse quasi-particle residue near the gap edge. +Substituting +D(∆ − δ) ≈ 1 + ˜α0δ into (S8), we obtain at γ = 1/2 +Z(∆ − δ) = +1 +2∆ +� ¯g +˜α0 +| log δ|. +(S11) + +4 +3.5 +1 +1.2 +1.4 +1.6 +1. +I log(8)|12 +000 +11.5 +GO +8 +5 +10 +1og2 [8]15(a) +5.5 +×10-4 +gap +edge +1 +5 +0 +1.843 +1.8435 +/ +3 +1 +4.5 +G0000(b) 13.5 +13 +12.510 +FIG. S4. (a) The spectral function A(k, ω) for γ = 1/2. (b) Constant ξk cuts along the blue lines in panel (a). +Analytically continuing this function to negative δ, i.e., to ω above the threshold, we obtain +Z(∆ + |δ|) = +1 +2∆ +� ¯g +˜α0 +(| log |δ|| + iπ) , +(S12) +Note that the imaginary part of Z(ω) jumps to a finite value at ω infinitesimally above the threshold. This behavior +is consistent with the numerical solution for Z(ω), see Fig. S2 (d). +Finally, we use the results for D(ω) and Z(ω) and compute the spectral function near the gap edge. On the Fermi +surface, the spectral function at negative ω and |ω| > ∆ takes the form +A(kF , ω) ∝ +1 +|ω + ∆| log(¯g/|ω + ∆|). +(S13) +Slightly away from the Fermi surface, the spectral function has a peak at |ω| = ∆+δk where δk ∼ (ξ2 +k/¯g)/ log2(|¯g/ξk|). +The peak width scales as δk/ log(|¯g/ξk|) and is logarithmically smaller than the energy variation |ω| − ∆. Also, for +any non-zero ξk, the spectral function jumps at the gap edge to a finite value of order 1/ξ2 +k. In Fig. S4 we show the +numerical result for the spectral function. It is consistent with the behavior we just described. +UNIVERSAL FORM OF THE SPECTRAL FUNCTION AT 1/2 < γ < 1 +For frequencies ω near the gap edge and for momenta near the Fermi surface, when ξk is much smaller than |ωZ(ω)|, +a straightforward calculation shows that for γ > 1/2, the spectral function can be expressed as a scaling function of +ξk/|ω + ∆|1−γ¯gγ (we set ω < 0). Namely, +A(k, ω) ∝ +1 +|ω + ∆| +ν +2 +1−γ Φ +� +ξk +|ω + ∆|1−γ¯gγ +� +, +(S14) +where +Φ(x) ≡ +x2 + Q2 +γ sin[π(ν − c)]/ sin(πc) +� +x2 + Q2γ cos(2πγ) +�2 + Q4γ sin2(2πγ) +(S15) +with Qγ = sin(πγ/2)B(1 − γ, ν/2 + γ − 1). In Fig. S5, we plot the dimensionless function Φ(x) for different values of +γ. At x ≫ 1, Φ(x) ∼ 1/x2; at x ≪ 1, Φ(x) ∼ const. For γ < γc ≃ 0.9, function Φ(x) contains a local maximum at +x2 +∗ ∼ +� +(u − v)2 + w2 − u, +(S16) +where u = Q2 +γ sin[π(ν − c)]/ sin(πc), v = Q2 +γ cos(2πγ), and w = Q2 +γ sin(2πγ). This maximum can be interpreted as an +over-damped, but still existing quasi-particle peak. At γ > γc, the function Φ(x) monotonically decreases with x. In +this case, the quasiparticle description breaks down completely. + +-2.5 +-3 +-3.5 +-10 +-5 +0 +5 +10 +Sk/g2 +1 +0 +-2.8 +-2.6 +-2.4 +-2.2 +-2 +-1.8 +-1.6 +w/gMin(a) +0 +-0.5 +-1 +-1.5 +19 +3 +-2(b) +6 +5 +4 +ntensity +3Max11 +100 +101 +102 +x +10!4 +10!3 +10!2 +10!1 +)(x) +0:5 +. +1:0 +. = .c ' 0:9 +FIG. S5. The function Φ(x), Eqn (S15), for different values of γ. +We emphasize that this behavior holds only for small enough ξk. For larger ξk, the spectral function does depend +on the sign of ξk and as γ increases, the quasiparticle behavior gets completely destroyed first for positive ξk and +then, at larger γ, for negative ξk. +[1] F. Marsiglio, M. Schossmann, and J. P. Carbotte, Phys. Rev. B 37, 4965 (1988). +[2] Y.-M. Wu, A. Abanov, Y. Wang, and A. V. Chubukov, Phys. Rev. B 102, 024525 (2020). +[3] F. Marsiglio and J. P. Carbotte, Phys. Rev. B 43, 5355 (1991), for more recent results see F. Marsiglio and J.P. Carbotte, +“Electron-Phonon Superconductivity”, in “The Physics of Conventional and Unconventional Superconductors”, Bennemann +and Ketterson eds., Springer-Verlag, (2006) and references therein; F. Marsiglio, Annals of Physics 417, 168102-1-23 (2020). +[4] A. Karakozov, E. Maksimov, and A. Mikhailovsky, Solid State Communications 79, 329 (1991). +[5] R. Combescot, Phys. Rev. B 51, 11625 (1995). +[6] Y.-M. Wu, A. Abanov, Y. Wang, and A. V. Chubukov, Phys. Rev. B 99, 144512 (2019). + diff --git a/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/load_file.txt b/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..488a6585c43bdf6c0f4e5710c9a43d1dd7a5c287 --- /dev/null +++ b/QtFRT4oBgHgl3EQf7Tgp/content/tmp_files/load_file.txt @@ -0,0 +1,1099 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf,len=1098 +page_content='Density of states and spectral function of a superconductor out of a quantum-critical metal Shang-Shun Zhang1 and Andrey V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov1 1School of Physics and Astronomy and William I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA (Dated: February 1, 2023) We analyze the validity of a quasiparticle description of a superconducting state at a metallic quantum-critical point (QCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A normal state at a QCP is a non-Fermi liquid with no coherent quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A superconducting order gaps out low-energy excitations, except for a sliver of states for non-s-wave gap symmetry, and at a first glance, should restore a coherent quasiparticle behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argue that this does not necessarily hold as in some cases the fermionic self-energy remains singular slightly above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This singularity gives rise to markedly non-BCS behavior of the density of states and to broadening and eventual vanishing of the quasiparticle peak in the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We analyze the set of quantum-critical models with an effective dynamical 4-fermion interaction, mediated by a gapless boson at a QCP, V (Ω) ∝ 1/Ωγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We show that coherent quasiparticle behavior in a superconducting state holds for γ < 1/2, but breaks down for larger γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We discuss signatures of quasiparticle breakdown and compare our results with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Metals near a quantum critical point (QCP) display a number of non-Fermi liquid properties like linear-in-T resistivity, a broad peak in the spectral function near kF with linear-in-ω width, singular behavior of optical conductivity, etc [1–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' These properties are often thought to be caused by the coupling of fermions to near-gapless fluctuations of an order parameter, which condenses at a QCP [19– 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The same fermion-boson interaction gives rise to superconductivity near a QCP [30–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A superconducting order gaps out low-energy excitations, leaving at most a tiny subset of gapless states for a non-s−wave order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A general belief has been that this restores fermionic coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A frequently cited experimental evidence is the observed re-emergence of a quasiparticle peak below Tc in near-optimally doped cuprates (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' From theory side, the argument is that the fermionic self-energy in a superconductor has a conventional Fermi- liquid form Σ(ω) ∼ ω at the lowest ω, in distinction from a non-Fermi-liquid Σ(ω) ∝ ωa with a < 1 in the normal state [44–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this paper, we analyze theoretically whether fermions in a superconducting state at a QCP can be viewed as well-defined coherent quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argue that this is not necessarily the case as fermionic self-energy can still be singular on a real frequency axis immediately above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This singularity gives rise to markedly non-BCS behavior of the density of states (DoS) and to broadening and eventual vanishing of the quasiparticle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For superconductivity away from a QCP, mediated by a massive boson, numerous earlier studies have found that the spectral function A(k, ω) at T = 0 has a δ-functional peak at ω = (∆2 + (ξk/Z)2)1/2, where ξk = vF (k − kF ) is a fermionic dispersion (vF is a Fermi velocity), ∆ is a superconducting gap, and Z is an inverse quasiparticle residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A δ-functional peak FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Three possible forms of the electronic spectral function A(k, ω) at T = 0 in a quantum critical superconductor at a small but finite k − kF and in the absence of impurity broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a): A(k, ω) vanishes at |ω| = ∆ and has a well-defined peak at ω > ∆, (b): A(k, ω) diverges at |ω| = ∆, but it non-monotonic at larger ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The peak in A(k, ω) at |ω| > ∆ broadens, but still exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (c): A(k, ω) diverges at |ω| = ∆, and monotonically decreases at larger ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In case (a) fermions can be viewed as well-defined quasiparticles, in case (c) the quasiparticle picture completely breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The case (b) is the intermediate one between (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' holds for momenta near the Fermi surface, as long as ω < ∆ + ω0, where ω0 is a mass of a pairing boson in energy units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At larger ω, fermionic damping kicks in, and the peak broadens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The same physics leads to peak-dip-hump behavior of A(k, ω) as a function of ω, observed most spectacularly in near-optimally doped cuprate Bi2Sr2CaCu2O8+δ (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='g, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [54, 55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At a QCP, the pairing boson becomes massless and ω0 vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This creates a singular behavior near the gap edge at ω = ∆, which holds even when ξk is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A simple experimentation shows that there are three possible forms of A(k, ω), which we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1: it (i) either vanishes at ω = ∆ and has a well-defined peak at ω > ∆ whose width at small ξk is parametrically smaller than its energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' or (ii) diverges at ω = ∆, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='13679v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='supr-con] 31 Jan 2023 Three possible forms of A( k,w) in quantum-critical superconductor (a) Quasiparticle picture (b) Partial breakdown (c) Complete breakdown of quasiparticle of quasiparticle (k,w) A △ 0 △ 0 △ 0 3 3 32 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 Leading exponent 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 Subleading exponent 8 c 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 2 4 6 8 10 D(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=" 8 ' 1:18 gap edge 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 (" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' )8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 D(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 10 5 0 5 10 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 1 2 3 4 5 DoS 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 100 (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' ")=7g 100 101 9 1=x0:5 9 1=x0:59 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8 (a) (b) (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a) Exponents ν and c for the leading and the subleading terms in the expansion D(ω) ≃ 1+α(∆−ω)ν +β(∆−ω)ν+c, where D(ω) = ∆(ω)/ω and the gap edge ∆ is the solution of D(ω = ∆) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (b) Numerical result for D(ω) for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Inset shows the power-law behavior near the gap edge with ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='18, consistent with (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (c) Fermionic DoS at T = 0 for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 (thick green line) and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 (thin pink line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In both cases, the DoS vanishes below the gap edge ∆ and has a power-law singularity above it N(ω) ∝ 1/(ω − ∆)ν/2, but the exponent ν is different in the two cases, as we show in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' but is non-monotonic at larger ω and displays a broad maximum at some ω > ∆, or (iii) diverges at ω = ∆ and monotonically decreases at larger ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In the first case, fermions in a quantum-critical superconductor can be viewed as well-defined quasiparticles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' in the last case the quasiparticle picture completely breaks down;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' the second case is the intermediate one between the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Our goal is to understand under what circumstances A(k, ω) of a quantum-critical superconductor has one of these forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For our study, we consider dispersion- full fermions, Yukawa-coupled to a massless boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We assume, like in earlier works (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [56]), that a boson is Landau overdamped, and its effective velocity is far smaller than vF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this situation, the interaction that gives rise to non-Fermi liquid in the normal state and to superconductivity, is a purely dynamical V (Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The fermionic self-energy and the pairing gap, tuned into a proper spatial pairing channel, are then determined by two coupled equations in the frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At a QCP, V (Ω) is singular at vanishing Ω in spatial dimension D ≤ 3, and behaves as V (Ω) ∝ (¯g/Ω)γ, where ¯g is the effective fermion-boson coupling, and the exponent γ is determined by the underlying microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The most studied models of this kind are of fermions near an Ising-nematic or Ising/ferromagnetic QCP (γ = 1/3) and near an antiferromagnetic or charge density wave QCP (γ = 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The same effective interaction emerges for dispersion-less fermions in a quantum dot coupled to Einstein bosons (the Yuakawa- SYK model) [57–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For this last case, the exponent γ is a continuous variable γ ∈ (0, 1), depending on the ratio of fermion and boson flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' An extension of the Yukawa- SYK model to γ ∈ (1, 2) has recently been proposed [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We follow these works and consider γ as a continuous variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We note that the value of γ is generally larger deep in a superconducting state because of feedback from superconductivity on the bosonic polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For simplicity, we neglect potential in-gap states associated with non-s-wave pairing symmetry and focus on the spectral function of fermions away from the nodal points and on features in the density of states (DoS) above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' An extension to models with in-gap states is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In previous studies of the γ-model, we focused on the novel superconducting behavior at γ > 1, when the pairing interaction is attractive on the Matsubara axis, while on the real axis ReV (Ω) is repulsive [62, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argued that this dichotomy gives rise to phase slips of the gap function on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Here, we restrict ourselves to γ ≤ 1, when this physics is not present and, hence, does not interfere with the analysis of the validity of a quasiparticle description in a superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Pairing gap and quasiparticle residue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For superconductivity mediated by a dynamical interaction, the paring gap ∆(ω) and the inverse quasiparticle residue Z(ω) are functions of the running real fermionic frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We define the gap edge ∆ (often called the gap) from the condition ∆(ω) = ω at ω = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For our purposes, it is convenient to introduce D(ω) = ∆(ω)/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The gap edge is at |D| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The equation for D(ω) that we need to solve is ωB(ω)D(ω) = A(ω) + C(ω), (1) where B(ω) and A(ω) are regular functions of ω (see [64, 65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The C(ω) term depends on the running D(ω), C(ω) = ¯gγ sin πγ 2 � ω 0 dΩ Ωγ D(ω − Ω) − D(ω) � D2(ω − Ω) − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (2) Its presence makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S3) an integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The inverse residue Z(ω) is expressed via D(ω′) as Z(ω) = B(ω) + ¯gγ sin πγ 2 ω � ω 0 dΩ Ωγ 1 � D2(ω − Ω) − 1 (3) 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Spectral function A(k, ω) at T = 0 for four representative γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The broadening in the plots is intrinsic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a-d): color- coded plot at negative ω, as measured by the ARPES intensity at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (e-f): constant-k cuts of A(k, ω) at ξk = 0 and at ξk = ±4¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ < 1/2, the spectral function has a sharp quasiparticle peak at ω + ∆ ∝ ξ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ > 1/2, the peak moves to ω + ∆ ∝ |ξk|1/(1−γ) and broadens up, which eventually disappears (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' and is readily obtained once D(ω) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At γ = 0, which models a BCS superconductor, C(ω) = 0 and D(ω) = A(ω)/(ωB(ω)) is a regular function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Near the gap edge at ω > 0, D(ω)−1 ∼ ω−∆ and Z(ω) ≈ Z(∆) ≡ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We assume and then verify that D(ω) remains regular in some range of γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Substituting D(ω)−1 ∼ ω−∆ into (S7) for γ > 0, we obtain C(ω)−C(∆) ∼ (ω−∆)3/2−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that C(ω) is non-analytic near the gap edge, but for γ < 1/2, the exponent 3/2−γ is larger than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this situation, the non-analytic term in C(ω) generates a non-analytic term in D(ω) of order (ω − ∆)3/2−γ, which is smaller than the regular ω−∆ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Evaluating the prefactors, we obtain slightly above the gap edge, at ω = ∆ + δ D′(∆ + δ) = 1 + αδ + A cos[π(3/2 − γ)]δ3/2−γ, D′′(∆ + δ) = −A sin[π(3/2 − γ)]δ3/2−γ, (4) where α ∼ 1/¯g, A = � α 2 ¯gγ sin(πγ/2) ∆B(∆) J(γ, 1) and J(γ, ν) is expressed via Beta functions: J(γ, ν) = B(1 − γ, γ − 1 − ν 2) − B(1 − γ, γ − 1 + ν 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (5) For γ > 1/2, 3/2 − γ > 1, and the calculation of D(ω) has to be done differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We find after straightforward analysis that the leading δ-dependent term in D(∆ + δ) is non-analytic and of order δν, where ν is the solution of J(γ, ν) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The exponent ν ≈ 1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='67(γ − 1/2) for γ ≈ 1/2 and ν ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 for γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The subleading term in D(∆ + δ) scales as δν+c, where c > 0 is approximately linear in γ − 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2, we plot ν(γ) and c(γ) along with the numerical results of D(ω) for a representative γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The exponent ν extracted from this numerical D(ω) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='18, which matches perfectly with the analytical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The behavior at γ = 1/2 is special, and we discuss it in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Substituting D(∆ + δ) into the formula for Z(ω), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S8), we obtain Z′(∆ + δ)=Z(∆)+B cos(π(γ + ν/2 − 1))δ1−γ−ν/2,(6) Z′′(∆ + δ)=B sin(π(γ + ν/2 − 1))δ1−γ−ν/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (7) where B = ¯gγ sin πγ 2 ∆ √ 2α B(1 − γ, ν 2 + γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ < 1/2, Z(ω) = Z(∆) + O(δ1/2−γ) is approximately a constant near the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ > 1/2, the inverse residue diverges at the gap edge, indicating a qualitative change in the system behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Spectral function and DoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The spectral function and the DoS per unit volume are given by A(k, ω) = − 1 π ImGR(k, ω), N(ω) = 1 V � k A(k, ω) = NF ωIm � 1 ∆2(ω) − ω2 , (8) where the retarded Green’s function GR(k, ω) = −(ωZ(ω) + ξk)/(ξ2 k + (∆2(ω) − ω2)Z2(ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' ARPES intensity is proportional to A(k, ω)nF (ω), which at T = 0 selects negative ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At γ = 0 (BCS limit), N(ω) ∼ 1/(ω−∆)1/2, and the spectral function has a δ-functional peak at ω = (∆2 + (ξk/Z)2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2 (c,d), we show the DoS N(ω), obtained from the numerical solution of the full gap equation (S3) for representative γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that in both cases the DoS describes a gapped continuum, but there is a qualitative difference in the behavior near the gap edge: for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35, N(ω) has the same 1/δ1/2 singularity as for γ = 0, and for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 (e) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 (f) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='45 1 1 A(k,w) Sk = 0 0> .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='. 0< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0 0 10 6- 8 7 6 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 w/g w/g(g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='65 (h) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 0 0 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 w/g w/g(a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='35 (b) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='45 2 1 4 2 19 6 3 8 3 10 4 12 10 5 0 5 10 10 5 0 5 10 Sk/g Sk/g 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='65 (d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 10 5 0 5 10 10 5 0 5 10 Sk/g Sk/g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5Max 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 the DOS behaves as 1/δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='59, which perfectly matches the analytical form δ−ν/2, given that ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='18 for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The spectral function A(k, ω) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For comparison with ARPES, we set ω to be negative: ω = −(∆ + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For any γ, there is no frequency range, where A(k, ω) is a δ-function, simply because the bosonic mass vanishes at a QCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Still, for γ < 1/2, D(−(∆+δ))−1 ∝ δ and Z(−(∆+δ)) ≈ Z(−∆) = Z(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this situation, the spectral weight on the Fermi surface, integrated over an infinitesimally small range around ω = −∆ immediately above the real axis, is finite, like in BCS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Away from the Fermi surface, the spectral function vanishes as |ω + ∆|1/2−γ at the gap edge and displays a quasiparticle peak at ω ≈ −(∆2 + (ξk/Z(∆))2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The peak is well defined at small δ as its width O(δ1/2−γ) is parametrically smaller than its frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This is the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ > 1/2, the situation is qualitatively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Now Z(−∆ − δ) diverges at δ → 0 and D(−∆ − δ) − 1 ∼ |δ|ν ≪ |δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this case, the integral of A(kF , ω) over an infinitesimally small range around ω = −∆ vanishes, which can be interpreted as a vanishing of a quasiparticle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At finite ξk, the spectral function diverges at the gap edge as 1/|ω + ∆|γ/2+γ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ slightly above 1/2, A(k, ω) is non-monotonic and possess a broad maximum at |ω + ∆| ∼ (ξk/¯gγ) 1 1−γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This is the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger γ, the maximum disappears, and A(k, ω) monotonically decreases at |ω| > ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This is the same behavior as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For small ξk, the maximum disappears at γ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger ξk, it disappears at smaller γ, first for positive ξk (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Comparison with ARPES The behavior shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4 is our result in some range of γ > 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For positive ξk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', outside the Fermi surface), the spectral function has a single non-dispersing maximum at the gap edge, except for the smallest ξk, while for negative ξk, A(k, ω) has a kink at the gap edge ω = −∆ and a dispersing maximum at ω = −∆ − O � |ξk|1/(1−γ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This behavior is consistent with the ARPES data for Bi2201, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The data shows that the spectral function near the antinode, where our analysis is valid, displays an almost non-dispersing maximum at positive ξk, while for negative ξk it displays a non-dispersing kink at the same energy and a dispersing maximum at larger |ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We associate the non-dispersing feature at both positive and negative ξk with the gap edge ∆, and associate the dispersing maximum, observed in [66] at ξk < 0, with the dispersing maximum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Discussion and summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this work, we analyzed the applicability of quasiparticle description of a superconducting state which emerges out of a non- Fermi liquid at a metallic QCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We considered the model with an effective dynamical 4-fermion interaction FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a) Spectral function A(k, ω) at positive and negative ξk = ±4¯g at γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' To account for impurity scattering, we convoluted the spectral function with a Lorentzian of width ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='03¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (b) Spectral function at a set of discrete momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' It displays a non-dispersing gap edge singularity (green dots) and a dispersing maximum (blue circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This theoretical A(k, ω) is consistent with the ARPES data for Bi2201, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [66] (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V (Ω) ∝ 1/Ωγ, mediated by a gapless boson at a QCP and analyzed the spectral function and the DoS for γ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Interaction V (Ω) gives rise to a non-Fermi liquid in the normal state with self-energy Σ(ω) ∝ ω1−γ and to pairing below some finite Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' A superconducting order gaps out low-energy excitations and, at a first glance, should restore fermionic coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We found, however, that this holds only for γ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For larger γ the spectral function and the DoS exhibit qualitatively different behavior than that in a superconductor with coherent quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (different power-laws).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argued that the quasiparticle peak broadens up and completely disappears for γ close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Away from a QCP, a pairing boson is massive and at the lowest energies a Fermi-liquid description holds already in the normal state and continue to hold in a superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In particular, in the immediate vicinity of the gap edge, the system displays a BCS- like behavior for all γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Still, the system behavior over a broad frequency range is governed by the physics at a QCP, as numerous experiments on the cuprates and other correlated systems indicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We argued that our results are quite consistent with the ARPES data for Bi2201 [11, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We acknowledge with thanks useful conversations with a number of our colleagues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This work was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Department of Energy, Office of Science, Basic Energy Sciences, under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' DE-SC0014402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Martin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fiory, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fleming, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schneemeyer, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Waszczak, Physical Review B 41, 846 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' L¨ohneysen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Pietrus, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Portisch, H.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Research 2, 013301 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [60] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Classen and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 104, 125120 (2021).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Abanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 103, 024522 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [63] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Abanov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Chubukov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 103, 184508 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [64] See the Supplementary Information for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [65] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Marsiglio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Schossmann, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Carbotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' B 37, 4965 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [66] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Hashimoto, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Karapetyan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Koralek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Hinton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Testaud, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Nathan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Yoshida, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Yao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Tanaka, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', Science 331, 1579 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 7 Supplementary information for “Density of states and spectral function of a superconductor out of a quantum-critical metal” by Shang-Shun Zhang and Andrey V Chubukov GAP EQUATION ALONG THE REAL-FREQUENCY AXIS AND ITS SOLUTION We will use the approach pioneered by Marsiglio, Shossmann, and Carbotte [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this approach, one first solves non-linear gap equation along the Matsubara axis, which can be done rather straightforwardly as the gap function ∆(ωm) can be chosen to be real for all frequencies and is a regular function of ωm even when the pairing boson is massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' One then uses this ∆(ωm) as an input for the equation for complex ∆(ω) along the real frequency axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The non-linear integral equation for D(ωm) = ωm∆(ωm) on the Matsubara axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' and the equation for the inverse quasiparticle residue Z(ωm) = 1 + Σ(ωm)/ωm (Σ(ωm) is the fermionic self-energy),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' have the form ωmD(ωm) = πT � ω′m (D(ω′ m) − D(ωm)) sgn(ω′ m) � 1 + D2(ω′m) V (ω − ω′ m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S1) Z(ωm) = 1 + 1 ωm πT � ω′m sgn(ω′ m) � 1 + D2(ω′m) V (ω − ω′ m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S2) where V (Ωm) = (¯g/|Ωm|)γ is the same as in the main text (¯g is an effective fermion-boson coupling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' and γ depends on the underlying microscopic model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='This set of equations has a non-zero solution D(ωm) below a finite pairing temperature Tp ∼ ¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S1 shows the numerical solution for ∆(ωm) at T = 10−6¯g ≪ Tp, for different γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see from the figure that ∆(ωm) approaches a constant value at small frequencies and decays as ω−γ m at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This behavior holds for all γ and can be easily verified analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The gap equation along the real frequency axis is ωB(ω)D(ω) = A(ω) + C(ω), (S3) (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (1) in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This equation is obtained by using the spectral representation of an analytic function on the upper frequency half-plane f(iωm) = 1 π � dx Imf(x) x − iωm (S4) and, where possible, keeping D(ωm) as an input function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This approach was pioneered for electron-phonon interaction in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' [3–6] for the electron-phonon problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Numerical resulls for the gap function ∆(ωm) along the Matsubara axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The calculation is performed at temperature T = 10−6¯g using the hybrid-frequency method [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 10 10° 6/(um)V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 10 102 100 104 wm /g8 20 40 60 80 100 120 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 D(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') (a) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:25 #5 D0 D00 20 40 60 80 100 120 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 2 Z(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=') (b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:25 Z0 Z00 1 2 3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 (c) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:5 1 2 3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 2 4 6 (d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:5 0 1 2 3 4 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 (e) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8 0 1 2 3 4 5 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='=7g 0 2 4 6 8 10 (f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = 0:8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Numerical results for D(ω) and Z(ω) for γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='25, γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 and γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For our case, the functions A(ω) and B(ω) are directly expressed via D(ωm) along the Matsubara axis as A(ω) = 1 2 � ∞ 0 dωm D(ωm) � 1 + D2(ωm) × � ¯gγ (ωm + iω)γ + ¯gγ (ωm − iω)γ � , (S5) B(ω) = 1 + i 2ω � ∞ 0 dωm 1 � 1 + D2(ωm) × � ¯gγ (ωm + iω)γ − ¯gγ (ωm − iω)γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S6) and C(ω) is given by C(ω) = ¯gγ sin πγ 2 � ω 0 dΩ Ωγ D(ω − Ω) − D(ω) � D2(ω − Ω) − 1 , (S7) (Eqn (3) in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This function depends on the running D(ω − Ω), which makes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S3) an integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The inverse residue Z(ω) is expressed via D(ω′) as Z(ω) = B(ω) + ¯gγ sin πγ 2 ω � ω 0 dΩ Ωγ 1 � D2(ω − Ω) − 1 (S8) (Eqn (4) in the main text) and is readily obtained once D(ω) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The gap equation along the real-frequency axis has an iterative structure in the sense that D(ω) depends on D(ω′) at ω′ < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This allows us to solve this equation iteratively, using the low-frequency form D(ω) ≃ ∆(0)/ω as an input, with ∆(0) ≡ ∆(ωm = πT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2 we show the results for D(ω) and Z(ω) for three representative values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In all cases, D(ω) and Z(ω) are real below the gap edge ω = ∆ and are complex above the gap edge, where ∆ is defined as ∆(ω) = 1 at ω = ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that for γ > 1/2, Z(ω) diverges at the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We use this fact in the main text in the analysis of the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The real and imaginary parts of the gap function near the gap edge ω = ∆ for γ = 1/2, obtained by solving the non-linear gap equation numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The leading term in D′(∆ − δ) − 1, shown in the inset of (a), is linear in δ∆ − ω, and the subleading scales as δ/| log |δ||, as is confirmed by the linear relation in panel (a), The imaginary part D′′ appears at negative δ above the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The numerical result in panel (b) clearly shows the scaling relation D′′ ∼ δ/ log2 (|δ|/¯g), expected from the Kramers-Kronig relation with D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' THE CASE OF γ = 1/2 In the main text we argued that for γ < 1/2, the function D(∆ − δ) − 1 ∝ δ, where δ = ∆ − ω, and the correction scales as δ3/2−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' More specifically, we found iteratively that D(∆ − δ) = 1 + δ ∞ � n=0 αnδnϵ (S9) where ϵ = 1/2−γ and α0 = O(1/¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The expression for α1 is presented in the main text, after Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' It is proportional to J(γ, 1) = B(1 − γ, γ − 3/2) − B(1 − γ, γ − 1/2), where B(a, b) is a Beta function (B(a, b) = Γ(a)Γ(b)/Γ(a + b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For small ϵ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', for γ ≤ 1/2), α1 ∼ J(γ, 1) ∼ 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For the next term in (S9) we find α2 ∼ 1/ϵ2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We see that the perturbative expansion in δϵ in (S9) holds for (δ/¯g)ϵ/ϵ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Outside this range, all terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S9) are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' As γ approaches 1/2 from below and ϵ decreases, the perturbative regime shrinks to exponentially small δ < ¯g exp(−| log ϵ|/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' To understand the form of D(ω) outside the perturbative regime, we express (δ/¯g)ϵ as eϵ log (δ/¯g) and expand (S10) in powers of log (δ/¯g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We obtain D(∆ − δ) = 1 + δ ∞ � n=0 ˜αn(log δ ¯g )n (S10) where ˜α0 = α0 + α1 + α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='., ˜α1 = ϵα1 + 2ϵα2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='., ˜α2 = 2ϵ2α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='. We see that each ˜αn is a series, in which the first term is independent on ϵ, and the others diverge as powers of 1/ϵ, because α1 ∼ 1/ϵ, α2 ∼ 1/ϵ2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We now argue that singular parts of ˜αn can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The argument is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' First, in the calculations, the 1/ϵ divergencies originate from the divergence of J(1/2 − ϵ, 1) ≈ 1/(2ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This divergence is regularized by a finite boson mass, such that strictly at ϵ = 0, one has J(1/2, 1) = 0 instead of infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Second, if we assume that ˜α0 in (S10) remains finite at ϵ = 0 and substitute the trial D(∆ − δ) = 1 + δ˜α0 in the gap equation at γ = 1/2 and compute iteratively the next term in D(∆−δ), we find it in the form ˜α1δ log (δ/¯g) with a finite ˜α1 = √¯g˜α0/(4∆B(∆)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Extending the iterative analysis, we find that all ˜αn are finite at γ = 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', ϵ = 0), as we anticipated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We didn’t manage to sum up analytically the logarithmic series in (S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The numerical solution for D(∆ − δ) for γ = 1/2 shows that D(∆ − δ) − 1 remains linear in δ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2 (c)), and the corrections scale as 1/| log (|δ|/¯g)| (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S3 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' By Kramers-Kronig relation, this implies that at negative δ, when ω > ∆ is above the gap edge, the imaginary part of D(ω) scales as D ′′(∆ + |δ|) ∝ δ/ log2 (|δ|/¯g) (the same form is obtained by just noticing that log(−|δ|) = log(−(ω − ∆ + i0)) = log |δ| − iπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This form of D ′′(∆ + |δ|) is consistent with our numerical solution above the gap edge, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The solution clearly shows that the ratio D ′′(∆ + |δ|)/δ decreases at the smallest δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' We next use the result for D(ω) to obtain the inverse quasi-particle residue near the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Substituting D(∆ − δ) ≈ 1 + ˜α0δ into (S8), we obtain at γ = 1/2 Z(∆ − δ) = 1 2∆ � ¯g ˜α0 | log δ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S11) 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' I log(8)|12 000 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 GO 8 5 10 1og2 [8]15(a) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 ×10-4 gap edge 1 5 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='843 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8435 / 3 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 G0000(b) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 13 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='510 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (a) The spectral function A(k, ω) for γ = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (b) Constant ξk cuts along the blue lines in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Analytically continuing this function to negative δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=', to ω above the threshold, we obtain Z(∆ + |δ|) = 1 2∆ � ¯g ˜α0 (| log |δ|| + iπ) , (S12) Note that the imaginary part of Z(ω) jumps to a finite value at ω infinitesimally above the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This behavior is consistent with the numerical solution for Z(ω), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S2 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Finally, we use the results for D(ω) and Z(ω) and compute the spectral function near the gap edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' On the Fermi surface, the spectral function at negative ω and |ω| > ∆ takes the form A(kF , ω) ∝ 1 |ω + ∆| log(¯g/|ω + ∆|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' (S13) Slightly away from the Fermi surface, the spectral function has a peak at |ω| = ∆+δk where δk ∼ (ξ2 k/¯g)/ log2(|¯g/ξk|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' The peak width scales as δk/ log(|¯g/ξk|) and is logarithmically smaller than the energy variation |ω| − ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Also, for any non-zero ξk, the spectral function jumps at the gap edge to a finite value of order 1/ξ2 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S4 we show the numerical result for the spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' It is consistent with the behavior we just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' UNIVERSAL FORM OF THE SPECTRAL FUNCTION AT 1/2 < γ < 1 For frequencies ω near the gap edge and for momenta near the Fermi surface, when ξk is much smaller than |ωZ(ω)|, a straightforward calculation shows that for γ > 1/2, the spectral function can be expressed as a scaling function of ξk/|ω + ∆|1−γ¯gγ (we set ω < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' Namely, A(k, ω) ∝ 1 |ω + ∆| ν 2 +1−γ Φ � ξk |ω + ∆|1−γ¯gγ � , (S14) where Φ(x) ≡ x2 + Q2 γ sin[π(ν − c)]/ sin(πc) � x2 + Q2γ cos(2πγ) �2 + Q4γ sin2(2πγ) (S15) with Qγ = sin(πγ/2)B(1 − γ, ν/2 + γ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S5, we plot the dimensionless function Φ(x) for different values of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At x ≫ 1, Φ(x) ∼ 1/x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' at x ≪ 1, Φ(x) ∼ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' For γ < γc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='9, function Φ(x) contains a local maximum at x2 ∗ ∼ � (u − v)2 + w2 − u, (S16) where u = Q2 γ sin[π(ν − c)]/ sin(πc), v = Q2 γ cos(2πγ), and w = Q2 γ sin(2πγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' This maximum can be interpreted as an over-damped, but still existing quasi-particle peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' At γ > γc, the function Φ(x) monotonically decreases with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' In this case, the quasiparticle description breaks down completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 10 5 0 5 10 Sk/g2 1 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='6 w/gMin(a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='5 19 3 2(b) 6 5 4 ntensity 3Max11 100 101 102 x 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='4 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='3 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='2 10!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content='1 )(x) 0:5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' 1:0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content="c ' 0:9 FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtFRT4oBgHgl3EQf7Tgp/content/2301.13679v1.pdf'} 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b/TdE0T4oBgHgl3EQfUwDn/content/tmp_files/2301.02256v1.pdf.txt @@ -0,0 +1,2672 @@ +arXiv:2301.02256v1 [astro-ph.GA] 5 Jan 2023 +MNRAS 000, 000–000 (2023) +Preprint 9 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Dynamical Data Mining Captures Disc-Halo Couplings +that Structure Galaxies +Alexander Johnson1⋆, Michael S. Petersen2, Kathryn V. Johnston1,3, Martin D. Weinberg4 +1Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA +2Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK +3Center for Computational Astrophysics, Flatiron Institute, 162 5th Av., New York City, NY 10010, USA +4Department of Astronomy, University of Massachusetts, Amherst MA 01003-9305, USA +9 January 2023 +ABSTRACT +Studying coupling between different galactic components is a challenging problem in galactic dynamics. Using basis +function expansions (BFEs) and multichannel singular spectrum analysis (mSSA) as a means of dynamical data +mining, we discover evidence for two multi-component disc-halo dipole modes in a Milky-Way-like simulated galaxy. +One of the modes grows throughout the simulation, while the other decays throughout the simulation. The multi- +component disc-halo modes are driven primarily by the halo, and have implications for the structural evolution of +galaxies, including observations of lopsidedness and other non-axisymmetric structure. In our simulation, the modes +create surface density features up to 10 per cent relative to the equilibrium model stellar disc. While the simulated +galaxy was constructed to be in equilibrium, BFE+mSSA also uncovered evidence of persistent periodic signals incited +by aphysical initial conditions disequilibrium, including rings and weak two-armed spirals, both at the 1 per cent level. +The method is sensitive to distinct evolutionary features at and even below the 1 per cent level of surface density +variation. The use of mSSA produced clean signals for both modes and disequilibrium, efficiently removing variance +owing to estimator noise from the input BFE time series. The discovery of multi-component halo-disc modes is strong +motivation for application of BFE+mSSA to the rich zoo of dynamics of multi-component interacting galaxies. +1 INTRODUCTION +The structures of galaxies are manifestations of how the laws +that govern dynamics combine with the nature of matter. Un- +derstanding galaxies strengthens our understanding of fun- +damental physics. There are tremendous opportunities to +deepen that understanding: a rich legacy of analytic descrip- +tions of galactic dynamics; community investment in high +resolution simulations; large scale, high dimensional surveys +of billions of stars and galaxies; and the emergence of the +vital field of data science to robustly mine and characterise +both simulated and real data sets. +Yet recent years have revealed the limits to our conception +of our home galaxy, long thought to be a quiet backwater in +the Universe. Maps of the positions and motions of billions of +stars from the Gaia satellite (Gaia Collaboration et al. 2016, +2018, 2022) have revealed a Milky Way in disarray, with abun- +dant signatures of action and reaction - past and ongoing +(e.g. Antoja et al. 2018; Trick et al. 2019; Friske & Sch¨onrich +2019; Helmi 2020). These represent significant departures +from the descriptions of equilibrium and mild perturbations +on which the field of Galactic Dynamics has been built +(Binney & Tremaine 2008). Simulations are capable of cap- +turing such complexities but robustly linking the features to +theoretical descriptions and identifying their physical origins +remains challenging. +Recent work by Weinberg & Petersen (2021) suggest one +approach to this challenge centred around two mathemat- +ical tools: Basis Function Expansions (BFE) and Multi- +Channel Singular Spectrum Analysis (mSSA). BFE rep- +resent a distribution as a linear combination of basis +functions, with half a century of application to galac- +tic +dynamics +(e.g. +Clutton-Brock +1972, +1973; +Kalnajs +1976; Polyachenko & Shukhman 1981; Weinberg 1989, 1999; +Petersen et al. 2022). When representing a simulation with a +fixed set of basis functions, one obtains time series of co- +efficients that encode the dynamics in a compressed rep- +resentation. mSSA is a method for identifying temporal +correlations. Together, one obtains a powerful analysis tool +for studying galaxy simulations. The method does not re- +quire prior information and thus can be considered a form of +unsupervised learning. Applying mSSA to BFE time-series, +Weinberg & Petersen (2021) analysed barred-galaxy simula- +tions. They found that BFE+mSSA could autonomously ex- +tract the dominant space and time correlated features and +disentangle different phase of bar formation and evolution +recovered through more traditional analysis (Petersen et al. +2021). +In +this +paper, +we +build +on +the +success +of +Weinberg & Petersen +(2021) +in +characterising +the +evo- +lution of a known feature and explore the use of BFE+mSSA +© 2023 The Authors + +2 +A. Johnson et al. +as a dynamical discovery tool. We do so through the analysis +of a model galaxy comprised of a stellar disc, stellar bulge, +and dark matter halo that is designed to be in equilibrium +and hence featureless (described in Section 2). Studying +such a galaxy serves as a ‘control’ sample for future work +with more feature-rich discs, with features from in situ (i.e. +spiral arms) or ex situ (i.e. minor mergers) sources. With a +control model, we want to answer the following questions +about BFE+mSSA as a dynamical data mining tool: +1) Can BFE+mSSA separate distinct features that overlap in +time and are not distinct by eye (real astrophysical signals, +phase mixing, and N-body noise)? +2) Can BFE+mSSA connect features within or across +components by identifying their shared spatial and temporal +structure? +The answer, as we shall see, is yes to both questions. +BFE+mSSA isolates features and allows them to be in- +terpreted independently, while also isolating interactions +between components independent of the presence of other +interactions. +While analysing the disc in the present study, it became +clear that the model was not the perfect featureless system we +intended. By applying BFE+mSSA to the disc, and then the +combination of disc+halo, we identify two dynamical causes +of features: phase-mixing from initial conditions, and interac- +tions between the disc and halo. We identify multiple distinct +dynamical signals in each, and examine the dynamical signals +in detail (Section 3). We find that the signals are likely to be +generic features in disc+halo systems, and can have real im- +pact on galaxies in the real Universe. +This study is a key step in understanding and exploring the +strengths and limitations of BFE+mSSA in multi-component +systems (see Section 4). In partnership, BFE+mSSA has +great potential beyond simulations analysis. Much of analytic +linear theory is also built on BFEs. Moreover BFEs may be +used to described observational data sets. Hence BFEs pro- +vide a common dynamical language to quantitatively con- +nect theory, simulations, observations and data science while +providing rigorous physical interpretations of dynamical pro- +cesses. We conclude in Section 5 with a discussion of how +our results impact galaxy evolution more generally, and how +BFE+mSSA fits in a larger program of dynamical data min- +ing. +2 METHODS +We first review the rationale and overarching goals for +BFE+mSSA analysis in dynamical systems in Section 2.1, +and then describe the construction of a model isolated +disc+bulge+halo galaxy in Section 2.2. Two appendices pro- +vide specifics of the expansions used in our analysis (Ap- +pendix B) and an overview of mSSA (Appendix C). +2.1 Rationale for BFE+mSSA analysis +All self-gravitating stellar systems, like ionised plasma, +have a spectrum of both continuous and point modes +(Krall & Trivelpiece +1973; +Ichimaru +1973; +Ikeuchi et al. +1974). Here, we define a mode to be a superposition of os- +Figure 1. Circular (black) and radial (red) frequency curves as a +function of radius for the T = 0 equilibrium model. Both frequen- +cies are computed using the epicyclic approximation, in the plane +of the disc (z = 0). Three frequency values have been marked to +guide the eye (Ω = 0.6, Ω = 1.5, and Ω = 6.6 cycles/Gyr), cor- +responding to spatial scales near the peak disc circular velocity +(2.2Rd = 7.7) and multiples of the halo scale length (a = 52 kpc). +cillations that lead to a self-similarly growing or damping +response to a perturbation1. +Continuous modes are excited by perturbations with +a continuous range of frequencies, for example a single en- +counter with a satellite. Other sources of disequilibrium, +whether physical or aphysical, also drive continuous response. +This continuous response appears as phase mixing in galax- +ies. These modes are also transient: since the response is not +dominated by a single frequency the mode quickly looses co- +herence and therefore is not self-sustaining. We expect that +mSSA will efficiently detect a plethora of signals owing to +continuous modes, of varying strength. These signals will ap- +pear with relatively broad frequency support. As the modes +are transient, few theoretical approaches exist capable of pre- +dicting the existence or evolution of these modes, making +BFE+mSSA an efficient tool to study them. +Point modes are excited by specific frequencies. They +have model-dependent self-similar shapes and well defined +frequencies and can therefore be reinforced by their own +gravity. The point modes are damped (growing) for sta- +ble (unstable) systems. The most commonly known point +mode is the Jeans’ instability in a homogeneous sea of stars +(e.g. Binney & Tremaine 2008). Fluctuations from environ- +mental disturbances such as satellite encounters or Poisson +noise from N-body distributions may excite these weakly +self-gravitating features. We expect that some of the results +recovered by mSSA will be the phase space manifestation +of these modes, appearing as distinct frequency peaks. Cal- +culations for unstable evolutionary modes in galactic discs +1 Mathematically, we are referring to the set of solutions to the col- +lisionless Boltzmann equation for at a specific complex frequency. +These are the solutions to the response operator that generalise +eigenfunctions in a finite vector space. In plasma physics, these +solutions are usually call ‘modes’ although there is some disagree- +ment. +MNRAS 000, 000–000 (2023) + +32 +16 +8 +Q=6.6 +cycles +Gyr +4 +QR +- +2 +Q +1.5 +1 +Q= 0.6 +- +0.5 +- +- +- +- +- +- +- +- +- +0.25 +1 +1 +2 +3 +5 +10 +20 +30 +50 +100 +R (kpc)Near-equilibrium disc-halo dynamics +3 +Figure 2. Disc coefficients over time for the first three harmonic +orders (m = 0, 1, 2) and all corresponding radial orders (n ∈ [0, 6]). +The coefficients have been detrended by subtracting the mean and +dividing out the variance. The coefficient series are dominated by +apparent noise, though some trends may be discerned: a steady +decrease in some m = 0 coefficients (upper panel), elevated am- +plitude towards the end of the simulation in m = 1, and some +periodicity in m = 2. The origin of these features is difficult to +interpret owing to the coefficient series’ noisy appearance across +multiple basis functions. Any spatial features encoded in the basis +are all but impossible to determine. +have found evidence for point modes supported in various +analytic geometries (e.g. Fouvry et al. 2015; De Rijcke et al. +2019). While we do not have explicit theoretical results for +damped modes at many azimuthal orders in discs, N-body +simulations seem to suggest that the amplitude is largest at +m = 2 and decreases for m > 2. Crucially for the problem at +hand (a disc+halo system), we have no analytic predictions +for the modal spectra, owing to the complexity of approaching +such a problem analytically. BFE+mSSA gives us a means to +detect these modes amongst a sea of other signals. +2.2 Model Galaxy +2.2.1 Simulation Overview +We design an isolated model Milky-Way-like galaxy for our +study of the compressive power2 of BFE and the dynamical +information one can extract with mSSA. We draw the model +from components in the merger simulation of Laporte et al. +(2018): a Hernquist profile dark matter halo with a mass of +1012M⊙ and a scale length of 52 kpc; an exponential stellar +disc with a mass of 6×1010M⊙, a scale length of 3.5 kpc, and +a sech2 scale height of 0.53 kpc; a Hernquist stellar bulge with +a mass of 1010M⊙ and a scale length of 0.7 kpc. The halo has +40×106 particles, the disc has 5×106 particles, and the bulge +has 106 particles. Unlike Laporte et al. (2018), we do not in- +troduce a satellite perturber so that our model galaxy evolves +in isolation. The initial circular and radial frequency curves in +2 Here, ‘compression’ refers to the amount of information one +needs to store. A straightforward metric is the total computer disk +space. We provide specifics to our simulation, but the scale of com- +pression should be similar in other simulations. +the disc plane are shown in Figure 1: as we shall see below, we +are able to use these frequencies to inform our mSSA analy- +sis. We evolve the model with Gadget-4 (Springel et al. 2021) +for 5.49 Gyr, saving snapshots every 0.01 Gyr, for a total of +549 snapshots. The total simulation requires approximately +800 GB of computer disk storage. +2.2.2 BFE representation +To compactly describe the simulation, we represent each com- +ponent in each snapshot with a BFE designed to provide +compression and create a continuous representation from the +particles. Further information regarding the BFEs used may +be found in Appendix B. In a BFE, a target distribution is +represented as the linear sum of some chosen basis functions, +with weighting on each of the basis functions (coefficients). +If the basis functions are selected well, the distribution will +be described by a small number of functions and correspond- +ing coefficients, Cµ, where µ is a tag that indexes each basis +function. The coefficients then are a measure of the impor- +tance of each basis function to representing the overall dis- +tribution. To facilitate representing the distribution with the +smallest number of functions, we choose expansions whose +lowest-order function resembles the target equilibrium. +For a principally two-dimensional structure, the stellar +disc, we use a Fourier-Laguerre expansion3. The Fourier- +Laguerre basis for expanding disc surface density was intro- +duced in Weinberg & Petersen (2021). Given the exponential +weighting of Laguerre polynomials, they serve as a natural +radial basis element for exponential discs. If the scale lengths +are chosen to match, the equilibrium disc is well-represented +by the lowest-order Laguerre polynomials. The scale length +of our Fourier-Laguerre expansion is 3.5 kpc, matching the +scale length of the modelled disc. To capture angular struc- +ture, we expand in Fourier terms cos φ and sin φ. We in- +dex the Fourier azimuthal with m, and the Laguerre radial +terms with n, creating (2m − 1) × n total coefficients, each +tagged with a unique (m, n), written Cmn. We find that as +expected, C00 dominates by multiple orders of magnitude as +desired. We expand the disc to mmax = 6, nmax = 6, making +2 × (mmax + 1) × nmax = 84 coefficients for the disc. The +choice of maximum radial order is motivated by a desire to +probe specific spatial scales. The n = 6 radial Laguerre den- +sity function has nodes at 0.9, 3.1, 6.8, 12.1, 19.7, and 30.9 +kpc, thus ensuring that the majority of the nodes are within +18 kpc of the disc centre (where 90% of the particles are +located). +The dark matter halo4 is efficiently described through the +empirical orthogonal function basis approach introduced in +Weinberg (1999) and most recently updated in Petersen et al. +3 Another option is presented in Weinberg & Petersen (2021): the +use of 3d basis functions designed to resemble the exponential +disc. In this work, we use the 2d Fourier-Laguerre expansion owing +to the straightforward generalisation to the expansion of velocity +fields, which will be the subject of future works. +4 We also tested bulge expansions, using a similar basis to the +dark matter halo. Tests indicated that information contained in +the bulge basis was redundant with the dark matter halo: this +makes sense for two spherical components. Therefore, we omit the +bulge expansion from the analysis in the rest of the paper. +MNRAS 000, 000–000 (2023) + +10 +Conl +n +一 +(detrended) +n +0 +-10 +Coefficient Amplitudes +Cin +2.5 +0.0 +-2.5 +2.5 +0.0 +-2.5 +0 +2 +3 +5 +Time (Gyr)4 +A. Johnson et al. +mSSA +DFT peak +contrast +SV +name +decomposition +PCs +(Gyr−1) +(R < Rd) +fraction +Disequilibrium Signal 1: halo profile readjustment (slow decay) +Group m0-1 +disc m = 0 +0,1 +0.2 +0.031 +0.641 +Group l0-1 +halo l = 0 +0,1,2,3 +0.4 +- +0.944 +Group m0l0-1 +disc m = 0, halo l = 0 +0,1,2,3 +0.2 +0.054 +0.832 +Disequilibrium Signal 2: phase mixing of disc initial conditions (fast decay) +Group m0-2 +disc m = 0 +2,3,4,5 +6.4 +0.006 +0.084 +Group l0-2 +halo l = 0 +4,5 +6.6 +- +0.028 +Group m0l0-2 +disc m = 0, halo l = 0 +4,5 +6.6 +0.007 +0.037 +Group m1-3 +disc m = 1 +4,5 +6.9 +0.002 +0.057 +Group m2-1 +disc m = 2 +0,1 +6.6 +0.006 +0.201 +Group m4-1 +disc m = 4 +0,1 +14.2 +0.004 +0.086 +Group m6-1 +disc m = 6 +0,1 +20.2 +0.001 +0.036 +Group m2m4m6-1 +disc m = 2, 4, 6 +0,1 +6.6 +0.010 +0.072 +Group m1l1-3 +disc m = 1, halo l = 1 +6,7 +6.9 +0.002 +0.040 +Group m1m2l1-2 +disc m = 1, 2, halo l = 1 +2,3 +6.6 +0.003 +0.082 +Table 1. Summary of two different signals identified in our mSSA decompositions as associated with initial disequilibrium. The first +signal results from halo disequilibrium, and the appearance in the disc is primarily manifest in the central surface density. The second +signal is present in myriad decompositions, but appears to be seeded first by disequilibrium in the disc m = 0, which then persists in other +harmonics. Disc feature strengths are reported in surface density to give a measure of ‘visual contrast’, defined as max (|∆Σ|) within a disc +scale length (see equation C9). Contrasts have an approximate error of 0.001, estimated from grid size adjustments. Owing to simulation +sampling rates (0.01 Gyr), the DFT peak is only accurate to 0.1. +(2022). Beginning with the equilibrium distributions, we de- +sign a 1d radial model that matches the initial spherically +symmetric density profile. From this one-dimensional model, +we construct an empirical orthogonal function basis whose +lowest-order member perfectly matches the input initial den- +sity profile. Higher-order terms are generated as eigenfunc- +tions of the Sturm-Liouville equation with the input equi- +librium potential-density model and appropriate boundary +conditions. The three-dimensional structure of the spherical +components is described by a spherical harmonic expansion +in the angular coordinates. Each term in the expansion is rep- +resented by three numbers: the spherical harmonic indices ℓ +and |m| ≤ ℓ and the index of the radial basis function n. In +total, we have (ℓmax + 1)2 × nmax coefficients per snapshot. +For the halo, we expand to ℓmax = 2, nmax = 11. The expan- +sions, for the entire simulation, only require approximately 12 +MB of storage: a more than 60000× compression, with the +benefit of encoding the dynamics. In practice, we will often +consolidate the same-integer positive and negative spherical +harmonic m indices when describing the coefficient ampli- +tudes such that a quoted (ℓ, m) tag contains both ±m. As +expected, the Cℓmn = C000 term is the largest by multiple +orders of magnitude, with C generally decreasing as either +(ℓ, m) or n increases. +3 EVOLUTION OF A NEAR-EQUILIBRIUM +GALAXY +Our isolated disc+bulge+halo galaxy was constructed to be +in a completely stable equilibrium. However, the model is +not in equilibrium, for reasons both physical and unphysical. +Figure 2 shows the raw BFE coefficients for the low-order +disc harmonics derived from the simulation snapshots. While +it is clear that the coefficient time-series are noisy, inspection +by eye suggests that there exists lower frequency coherent +signals buried in the higher frequency noise: early evolution +in m = 0; modestly elevated power at late times in m = 1; +and a periodic signal in m = 2. +To explore dynamical evolution in our simulation, we per- +formed mSSA decompositions of various combinations of +BFE coefficients. These decompositions revealed clean, per- +sistent features in the individual low-order disc harmonics +(m = 0, 1, 2), which we concentrate on understanding in this +section. We also augment the analysis of the low-order disc +harmonics with mSSA analysis of halo coefficients, joins of +disc and halo coefficients, and higher-order disc harmonics +(m > 2). These multi-component mSSA analyses prove to +be the most fruitful in identifying the causes of different fea- +tures. The full results of all our analyses are presented in +Appendix A. +Section 3.1 describes how the results of the mSSA analysis +can be used to group coefficients into separate dynamical fea- +tures, characterise the properties of these features and come +to a physical understanding of their nature. The following +subsections illustrate these ideas by dividing our own anal- +ysis of the disc+bulge+halo simulation into three classifica- +tions: initial conditions disequilibrium (Section 3.2), secular +evolution signals (Section 3.3), and fluctuations and other +uninterpretable features (Section 3.4). +3.1 Interpreting the results of the mSSA analysis +We use several diagnostics (denoted below in slanted text) +to describe the character and understand the nature of the +features identified in the mSSA analysis. Each diagnostic has +a corresponding section in Appendix C describing the math- +ematical details. +Applied to BFE multiple series, mSSA identifies temporally +correlated signals in the BFE coefficients series as an ensem- +ble. Briefly, mSSA uses the autocorrelation of time lagged ma- +trix of the input series and performs an eigenanalysis to find +MNRAS 000, 000–000 (2023) + +Near-equilibrium disc-halo dynamics +5 +Figure 3. An analysis of two monopole signals resulting from distinct sources of initial disequilibrium. The left panels show the recon- +structed coefficient amplitudes over time for each signal (identified as Groups 1 and 2 in both disc-only, halo-only, and disc+halo analyses). +The right panels show the power spectra of the reconstructed coefficients for each group. The first signal is a slow rearrangement owing to +the halo settling in the presence of the disc, manifest by eye in the disc primarily as a change in the central surface density (cf. Figure 4). +We show the appearance of this signal in the disc and halo as the upper two rows. The second signal is ringing in the disc resulting +from the initial velocity disequilibrium of the disc. While the signal decays rapidly in the monopole component, the disequilibrium seeds +long-lasting persistent periodic features in other harmonics: see entries under ‘Disequilibrium Signal 2’ in Table 1. We show the appearance +of this signal in the disc and halo as the lower two rows. In each left-hand panel, we show two thicknesses of curves: the thick lines are for +the components when analysed separately and the thin lines are for the components when analysed jointly. That the different thicknesses +of lines, for the same radial order, are not particularly different, is strong evidence that the features are correlated between the disc and +halo. +mSSA +DFT peak +contrast +singular value +name +decomposition +PCs +(Gyr−1) +(R < Rd) +fraction +Point Mode 1: slow growth +Group m1-1 +disc m = 1 +0,1 +0.6 +0.007 +0.201 +Group l1-1 +halo l = 1 +0,1,2,3 +0.4 +- +0.272 +Group m1l1-1 +disc m = 1, halo l = 1 +0,1,2,3 +0.6 +0.008 +0.244 +Point Mode 2: slow decay +Group m1-2 +disc m = 1 +2,3 +1.7 +0.003 +0.064 +Group l1-2 +halo l = 1 +4,5 +1.5 +- +0.035 +Group m1l1-2 +disc m = 1, halo l = 1 +4,5 +1.5 +0.003 +0.048 +Table 2. The coupled disc+halo dipole modes appearing in different mSSA decompositions. Both modes appear in multiple mSSA +decompositions, and that they both appear in disc-only, halo-only, and disc-halo decompositions strongly suggests that they ar both joint +modes. In the table, disc harmonics are denoted with m, halo harmonics are denoted with l. Columns are the same as in Table 1. +dominant trends. Each time series is detrended by its mean +and variance to intercompare the variations in each coeffi- +cient series with. These eigenvectors describing these trends +are usually called principal components (PCs). As we always +find multiple PCs contribute to a single dynamical feature in +our analysis (see ‘PCs’ column in Tables), we will refer to each +feature as a ‘Group’ (of PCs), labelling the strongest group +(ordered by PC variance) as the first group. We also denote +the particular decomposition by the input coefficient har- +monic in the group name. For example, the strongest group +in the m = 0 disc analysis will be labelled ‘Group m0-1’, +and the strongest group in the l = 1 halo analysis will be +MNRAS 000, 000–000 (2023) + +420２ +Group m0-1 (disc) +Group m0-1 (disc) +n=0n=4 +1 +n= 2 +n=6 +n=3 +4 +Reconstructed Coefficients (detrended) +0 +４２０２ +Group /0-1 (halo) +Group /0-1 (halo) +Reconstruction DFT (normalised) +4 +Group m0-2 (disc) +Groupm0-2(disc) +20 +2 +4 +0 +4 +Group /0-2 (halo) +Group /0-2 (halo) +20 +.4 +0 +1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0 +5 +10 +15 +Time (Gyr) +Frequency +1 +Gvr6 +A. Johnson et al. +Figure 4. Disc monopole (m = 0) surface density as a function +of radius and time, computed from the full coefficient series (up- +per panel), showing a largely featureless disc. The surface density +has been normalised by the central surface density. The remaining +panels show the contribution to the surface density deviations for +two groups of m = 0 principal components, identified as two dis- +equilibrium signals (see Table 1). The surface density deviations +are computed relative to the m = 0, n = 0 background, and are of +the order a few per cent (excepting the outer disc, where the low +densities mean a variations naturally result in a larger per cent +variation). +labelled ‘Group l1-1’. As PC groups capture trends in basis +function coefficients that are correlated over snapshots, PC +groups capture how spatial features dynamically evolve. +Mathematically descriptive (but often difficult to interpret +beyond the most significant few), the mSSA decomposition +returns singular values (SVs) as measurements of the contri- +bution of each PC to the total decomposition. Larger SVs +indicate which PCs represent more of the net change in time +of the distribution. This property greatly helps the robust +identification of features that represent true dynamical evo- +lution. PCs which correspond to random fluctuations due to +(e.g.) numerical noise are by nature uncorrelated. They have +very low SV even as they may be the dominant source of +variations in the surface density. Conversely, PCs which de- +scribe evolution in coefficient series that are coherent over +time will have high SV even though they may be (orders of +magnitude) below the inherent noise. We report the singular +value fraction5 attributable to a given group in the Tables. +We examine the coefficient reconstructions from a group of +PCs for physical insight. From the coefficient reconstructions, +we can also construct power spectra from a Discrete Fourier +Transform (DFTs) of the reconstructed coefficients from a +group of PCs give insight into frequencies (and time scales) +5 To compute the relative contribution, we normalise each singular +value corresponding to a particular principal value to the sum of +all singular values. Then, we can say that some per cent of the +signal is represented by the principal component (or group). We +will call this the contribution of a principal component (or group), +and may be interpreted as a measure of signal robustness. +that characterise the time evolution of a feature. Approx- +imately equal values of dominant frequencies in the power +spectra of the coefficient reconstructions between different +PCs from mSSA of the same component suggest they are +describing different aspects of the same feature and may be +grouped together. If equal values occur across different com- +ponents they may be mutually interacting. See the ‘DFT +peak’ entry in Tables, which reports the frequency value +where the DFT is maximised. +We can also calculate contrast in the disc from the recon- +structions6. Calculating the average of the fractional devi- +ation in surface density within one disc scalelength gives a +measure of the ‘detectability’ of a feature (by eye or algo- +rithm). See the ‘contrast’ entry in Tables. Related, the in- +ferred location in the galaxy is where the dominant frequen- +cies found in the power spectrum match the circular velocity +of the unperturbed galaxy can indicate the spatial scales of +any interactions taking place. Refer to Figure 1. +In general, identified features evolve as one of the following +types of evolution (noted in Tables): decaying, where a fea- +ture peaks at the beginning of the simulation and decays in +importance; growing, where the feature grows and then satu- +rates in amplitude with later maximum times therefore hav- +ing slower growth rates; or consistent with no evolution. By +comparing the evolution type across different components, +one may also infer causality. The relative growth or decay +may indicate when one component is driving another. +3.2 Initial Conditions Disequilibrium Uncovered +Through Disc m = 0 Analysis +We start our investigation with perhaps the most striking +feature in the raw coefficients apparent in the top panel of +Figure 2 which shows the evolution of the m = 0 (monopole) +disc coefficients. The figure suggests the simulation suffers +from a disequilibrium that is typical in disc-halo initial con- +ditions: outwardly propagating rings in surface density. This +section reports the insights into this apparent evolution af- +forded by mSSA, starting from its application to the m = 0 +disc coefficients alone (3.2.1). The properties of the features +identified in this preliminary analysis provide a template for +further applications of mSSA both to the halo (separately +and combined with the disc, see 3.2.2) and higher order disc +terms (see 3.2.3). Table 1 summarises the properties of all +these analyses. +3.2.1 Grouping into Dynamical Features +The mSSA analysis of the m = 0 disc reconstructed coeffi- +cients reveals that PCs (0,1) and PCs (2,3,4,5) had distinct +power spectra, suggesting natural groupings. This also sug- +gested the presence of two distinct dynamical features with +the signal in Figure 2. The properties of these two groups +that are quoted below are summarised in Table 1, with the +rows labelled ‘Group m0-1’ and ‘Group m0-2’ corresponding +to this first mSSA analysis. +Two more figures illustrate our results. Figure 4 shows the +6 We do not look at the contrast in the halo, as this is not +straightforwardly measured in real galaxies. Therefore, the con- +trast columns do not contain entries for halo-only mSSA analyses. +MNRAS 000, 000–000 (2023) + +20 +0 +UnprocessedCoefficients +(Z/Zo) +10 +-2 +601 +4 +Radius (kpc) +5 +Group m0-1 +△Z/Zo (per cent) +10 +20 +Group m0-2 +10 +-5 +0.2 +0.4 +0.6 +0.8 +1.0 +Time (Gyr)Near-equilibrium disc-halo dynamics +7 +Figure 5. An analysis of two groups obtained from the disc-only m = 1 mSSA decomposition. Each group corresponds to a distinct point +mode, discussed in the text as ‘Mode 1’ and ‘Mode 2’. The left panels show the reconstructed m = 1 coefficient amplitudes over time for +Groups m1-1 and m1-2. The right panels show the power spectra of the reconstructed m = 1 coefficients for each group. Both modes have +well-defined slow patterns – significantly slower than any frequency associated with stars in the disc – and show evolving behaviour: the +first mode is unstable and grows with time, while the second mode is damped and decays with time. The mode summaries are listed in +Table 2. +Figure 6. Normalised face-on (x, y) disc surface density deviation determined for two groups in the m = 1 decomposition. Each group +corresponds to a distinct point mode, discussed in the text as ‘Mode 1’ and ‘Mode 2’. The panels shows a reconstruction of snapshots +for either Group m1-1 (upper row) or Group m1-2 (lower row) in the disc-only m = 1 decomposition (cf. Figure 5). Both groups are +retrograde with respect to the disc rotation (rotation direction of the pattern is marked with an arrow). The mode shown in the upper +panels grows in amplitude over the course of the simulation; the mode shown in the lower panels decays in amplitude over the course of +the simulation, evident from the surface density features. Neither pattern strongly winds; both are a largely self-similar evolution, despite +being fairly tightly wound. +amplitude (left hand panel) and DFTs of the coefficient re- +constructions for Groups m0-1 and m0-2, revealing their dis- +tinct temporal characteristics. In Figure 3, we show the m = 0 +surface density amplitude reconstruction as a function of disc +radius (y-axis) and time (x-axis) from the unprocessed coef- +ficients (top panel), as well as the surface density deviations +relative to a smooth monopole background, constructed from +the two m = 0 PC groups. +Overall, we find the following characteristics. +Group m0-1 represents a dynamical feature that shows weak +evolution over the entire simulations with a surface den- +sity contrast of approximately 3 per cent. The slow decay +of Group m0-1 produces power at a range of very low fre- +MNRAS 000, 000–000 (2023) + +1.0 +10 +Group ml-1 +5 +0 +0.5 +5 +△Z/20 (×10-2) +(kpc) +-10 +=0.1 +=5.20 +0.0 +10 +Group ml-2 +y +5 +0 +-0.5 +-5 +-10 +-1.0 +-10 +0 +10-10 +0 +10 -10 +0 +10 -10 +0 +10 +x (kpc)Group m1-1 +Group m1-1 +n=0n=4 +Reconstructed Coefficients +2 +n=1 n=5 +2 +n= +n三6 +Reconstruction DFT +0 +n=3 +(detrended) +(normalised) +2 +0 +Group m1-2 +Group m1-2 +2 +0 +-2 +1 +0 +0 +1 +2 +3 +4 +5 +0 +2 +4 +6 +8 +10 +Time (Gyr) +Frequency (r)8 +A. Johnson et al. +Figure 7. Amplitude and phase as a function of radius and time +for the disc-only m = 1 decomposition for the first two groups iden- +tified in the mSSA analysis. Each group corresponds to a distinct +point mode. From top to bottom, we show the amplitude and phase +for the unprocessed m = 1 coefficient streams, the reconstructed +coefficients of Group m1-1, and the reconstructed coefficients of +Group m1-2. The density is shown as the log of the absolute value +of the density. Both groups show coherent phases identifiable in +the seemingly random phase information of the unprocessed co- +efficients. The growing (decaying) nature of Group m1-1 (Group +m1-2) is also evident in the amplitudes. +Figure 8. Description of the strongest principal component group +for halo and disc decompositions: a growing multi-component point +mode. The upper panel shows the detrended and normalised am- +plitude of the reconstructed cosine component of the m = 1 (disc; +grey curves) or l = 1 (halo; black curves) n = 0 coefficient ver- +sus time. The solid curves are for mSSA decompositions run on +each component alone (Group m1-1 and Group l1-1). The dashed +curves are for the joint halo+disc mSSA decomposition (Group +m1l1-1). The lower panel shows the power spectrum (DFT ampli- +tude vs frequency), for the four series shown in the upper panel. +The relative similarity of the curves and power spectra suggests +that the patterns are correlated between the disc and halo. The +slow growth of the disc amplitude over time relative to the larger +halo amplitude at the outset of the simulation suggests that the +halo is responsible for driving the mode. +quencies, peaked at 0.2Gyr−1. +Group m0-2 shows outwardly propagating rings in surface +density that start at the beginning of the simulation and dis- +appear after ≈ 1 Gyr, losing speed as they move to larger +radii. While this is a sub-1 per cent effect within a disc scale +length, at larger radii, the surface density deviation is obvi- +ous by eye as ringing features. The periodic nature of Group +m0-2 corresponds to a frequency peak at 6.4Gyr−1. +We conclude that mSSA has cleanly separated two distinct +evolutionary processes operating simultaneously within one +harmonic term. The next two subsections explore the nature +of both of these features. +3.2.2 Group 1: Halo-driven disequilibrium? +The appearance of Group m0-1, at low frequency, suggests +that its origin may be connected to the halo, where timescales +are naturally long. Specifically, the frequency 0.2 Gyr−1 cor- +responds to a circular orbit at R +∼ +50 kpc (see Figure +1). This motivated us to apply mSSA to the l = 0 coeffi- +cients representing the halo component in the simulation to +explore this connection further. We run analyses of both the +halo l = 0 alone and in combination with the disc m = 0 +coefficients. +The results of the analysis of the halo alone is shown in +lower panels of Figure 3 and summarised in the second row +of Table 1. These demonstrate that the readjustment of the +halo component’s radial profile is even more significant than +MNRAS 000, 000–000 (2023) + +Reconstructed Amplitude +(detrended,normalised) +Group m1/1-1 (cosine only) +0 +-1 +1 +0 +1 +2 +3 +4 +5 +Time (Gyr) +Reconstructed DFT +solid:individual;dashed:correlated +(normalised) +1.0 +black:halo;grey:disc +0.5 +0.0 +0 +1 +2 +3 +4 +Frequency20 +UnprocessedCoefficients +20 +[≥/60 +-4 +28 +Group m1-1 +Radius (kpc) +5 +20 +TT +phase (radians) +20 +Group m1-2 +20 +-T +0 +0 +2 +4 +Time (Gyr)Near-equilibrium disc-halo dynamics +9 +Figure 9. Normalised face-on (x, y) halo z = 0 plane density deviation reconstruction snapshots for Group m1l1-1 (upper panels) and +Group m1l1-2 (lower panels) in the halo-and-disc l = 1 + m = 1 decomposition. Each group corresponds to a distinct point mode. +The patterns extends to large radii in the halo and are retrograde with respect to the disc rotation. The halo reconstructions exhibit +significantly less ordered behaviour compared to the disc owing to the three-dimensional nature of the mode, which also tips relative to +the z = 0 plane. However, the bulk properties are similar to the disc (cf. Figure 6). The mode summaries are listed in Table 2. That the +joint decomposition of the halo and disc returns the same groups, with similar behaviour, is strong evidence for the mutual mode nature +of the features. The large spatial scale of the modes in the halo, coupled with their relatively early coherence, is suggestive that the modes +are induced by the halo. +the disc radial profile, with a signal amplitude twice as strong +as the disc (compare detrended amplitudes in Figure 3). Such +halo-driven disequilibrium is also a common feature for nu- +merical realisations of multi-component galaxies as their com- +bined equilibrium properties have been approximated, for ex- +ample through Jeans modelling or adiabatic contraction cor- +rections. Thus the mass distribution of the halo adjusts to +full equilibrium in the presence of the disc, and vice versa. +In Table 1, a comparison of rows 1 (analysis disc coeffi- +cients alone), 2 (halo coefficients alone) and 3 (disc and halo +coefficients combined) confirms: (i) all three mSSA analyses +have similar temporal structures, corresponding to the dy- +namical timescales at several tens of kpc in the system; (ii) +the joint disc/halo analysis actually identifies the same co- +herent features in the disc and at greater contrast (0.054 vs +0.031) than the disc analysis alone; (iii) that the driver for the +combined evolution is likely the halo given the larger ampli- +tude of its coherent changes relative to random fluctuations +for that component. +The above results demonstrate the ability of mSSA to suc- +cessfully identified the mutual readjustment of the coupled +disc-halo system from a mild disequilibrium state. +3.2.3 Group 2: Disc-driven disequilibrium +The strength of Group m0-2 in the analysis inspired an in- +vestigation as to whether this disequilibrium could also seed +other features in the simulation. Examination of other mSSA +decompositions for different coefficient combinations finds +many similar-frequency signals (see lower rows of Table 1). +Even disc harmonics (m = 2, 4, 6) show a persistent signal in +the most important PCs (0 and 1) with a pattern speed of +∼ 3.3 cycles/Gyr that is equal to the half the Group m0-2 +frequency peak of 6.6 cycles/Gyr7. Note that the joint analy- +sis of all even disc harmonics (m = 2, 4, 6) returns essentially +the same results as the m = 2 only decomposition. In the +case of harmonic orders m > 2, this result likely owes to the +need for higher order harmonics to fully represent the feature +being described. +The remaining rows of Table 1 demonstrate that the Group +m0-2 disc disequilibrium signal is also evident at a lower +level (i.e. higher PC numbers, lower contrast in the disc and +smaller SV) in both the disc m = 1 and halo l = 1 decom- +positions when comparing frequency structure of the groups. +While the peak surface density deviation is near the outset of +the simulation for m = 0, in higher harmonic orders the sig- +nal does not completely fade over the simulation, with peak +measured contrasts coming at later times. Our findings show +the utility of mSSA in detecting evolution incited across dif- +ferent harmonic orders. +7 The pattern speed of a harmonic is the number of cycles per +Gyr divided by the harmonic number. That is, the pattern speed +of the disc-only decomposition of Group 2 m harmonic coefficients +is Ωm = ΩDFT/m cycles/Gyr. +MNRAS 000, 000–000 (2023) + +1.0 +Group /1-1 +20 +0 +0.5 +Ap/po (×10-2) +-20 +(kpc) +[=0.1 +=1.8 +=3.5 +T=5.20 +0.0 +Group /1-2 +y +20 +0 +-0.5 +-20 +-1.0 +-25 +0 +25 +-25 +0 +25 +-25 +0 +25 +-25 +0 +25 +x (kpc)10 +A. Johnson et al. +3.2.4 Key insights +In this section, BFE+mSSA has been used to increase our +understanding of a dynamical simulation by: +(i) separating distinct evolutionary pathways within a single +harmonic; +(ii) identifying coupling between multiple components; +(iii) detecting features across different harmonics within a +single component. +These results emphasise that initial conditions for near +equilibrium studies of galaxy evolution need to be dynam- +ically relaxed (or virialised) by evolving in isolation for tens +of halo dynamical times (i.e. much longer than than the +equivalent timescale in the disc) prior to an studies of in- +teractions in order to truly isolate signatures of the external +perturbation. While the perturbation in our study is a nu- +merical artifact, the distinct adjustments to density profiles +and couplings within and across components uncovered by +BFE+mSSA represent the drivers of the evolution of galax- +ies seeded by any perturbation. +3.3 Secular evolution signals Uncovered Through +Disc m = 1 Analysis +Examination of the PCs from the mSSA decomposition of the +dipole disc harmonic (m = 1) revealed two groups, with prop- +erties summarised in Table 2 and contributing coefficients +and power spectra visualised in the left and right panels of +Figure 5. Examination of the power spectra show that these +features are distinct in nature to the disequilibrium-seeded +m = 0-dominated Groups m0-1 and m0-2 described in the +previous section in that they have clear, well-defined frequen- +cies, rather than a broad spectrum. This indicates that each +of these groups may be a point mode present in the system. As +discussed in Section 2.1, point modes are a result of the fun- +damental properties of the underlying phase-space distribu- +tion. They have single-valued real and imaginary frequencies +(hence the descriptive point) that describe the periodicity and +growth or decay of the features they support. These modes +drive secular, self-sustained evolution distinct from that of +a transient response to an external driver (e.g. the disequi- +librium initial conditions in the previous section) that phase +mixes away. Hence we refer to these groups as ‘Mode 1’ and +‘Mode 2’, and examine their nature in the following subsec- +tions. In the disc m = 1 analysis, these are Groups m1-1 and +m1-2. +3.3.1 Appearance of modes in the disc +We augment the information about the two modes sum- +marised in Figure 5 and Table 2 with visualisations of their +appearance in Figures 6 and 7. Figure 6 shows selected face- +on disc surface density reconstructions to demonstrate that +both modes create spiral patterns that are retrograde relative +to the rotation of the disc. Figure 7 illustrates the radial (y- +axis) and time (x-axis) evolution of the surface density (upper +panel in each pair) and phase over (lower panel in each pair) +for the full time sequence, indicating both the growth/decay +and periodicity. Inspection of these figures and the table pro- +vide the full characterisation of the modes. +Mode 1 groups m = 1 PCs 0 and 1, reconstructing a +slowly rotating, growing mode. Referring to Figure 1, the +frequency of the signal (Ω = 0.6 cycles/Gyr) is located near +the scale radius of the halo, well outside the disc8. Mode 1 +grows significantly in amplitude over the simulation, with the +peak surface density signal coming near the end of the simu- +lation. Computing the contrast in the outer, low-density disc +(r > 12 kpc), the surface density deviation amplitude reaches +10 per cent, detectable as lopsidedness in deep imaging of disc +galaxies. +Mode 2 groups m = 1 PCs 2 and 3, reconstructing a +slowly rotating, slowly decaying mode. The frequency of the +signal (Ω = 1.7 cycles/Gyr) is located closer to the Galactic +centre, but also beyond the bulk of the disc mass. Mode 2 +decays from the outset of the simulation, and is significantly +weaker than the first mode, with a peak contrast of order 0.1 +per cent within a scale length. +3.3.2 Connection between the disc and halo +Since the frequencies of the two modes are consistent with +halo frequencies we naturally suspect that the halo is sup- +porting the modes. To test this, we perform additional mSSA +decompositions: first with the l = 1 halo coefficients alone, +and then with the l = 1 halo coefficients jointly with the +m = 1 disc coefficients9. The results of the runs are sum- +marised in Table 2. We find sets of PCs in the halo-only +decompositions corresponding to Modes 1 and 2, which we +associate by means of their similar frequencies. We also find +corresponding PCs in the joint disc-halo decomposition. The +joint analysis in particular suggests that the modes are multi- +component in nature, owing to the similar properties between +all decompositions. +Figure 8 provides an example visualisation for a single ra- +dial coefficient (n = 0) contributing to Mode 1 to verify +this interpretation. Comparing the coefficients reconstructed +from identified in the independent analyses of the disc and +halo (solid lines), as well as the joint disc-halo decomposition +(dashed lines), we find the same features are identified in both +the combined and independent analyses: the curves in the +upper panel of Figure 8 are unchanged whether the decom- +position is performed on a per-component basis, or jointly. +This implies that the same principal component can describe +the evolution in both the disc and halo, and that the sig- +nal is strong enough in both components to be identified in +per-component analyses. This is a strong indication of a cor- +related multi-component signal. In general the same features +will not be recovered from combined analysis of different com- +ponents because the inter-component decomposition need not +match the intra-component decomposition. In contrast, our +joint analysis finds a single PC group may be used to recon- +struct the modes in both the disc and the halo, identifying +them as a mutual mode. +8 For m > 0 harmonics, PC groupings frequently occur in pairs +that describe both the amplitude and phase of a feature. In the left +hand panels of Figure 5 only the cosine terms in the coefficients are +plotted to allow the reader to infer both amplitude and periodicity. +9 To find correlated features between the halo and disc we choose +halo coefficients that can describe features with meaningful projec- +tions into the disc plane. To this end we choose only the Y m +l += Y 1 +1 +terms of the halo expansion, excluding the Y 0 +1 term. In addition +we use the same number of coefficients from each component to +avoid introducing the prior of unequal representation. +MNRAS 000, 000–000 (2023) + +Near-equilibrium disc-halo dynamics +11 +For both modes, we can examine and compare timescales +and amplitudes to try to understand the driver of the evolu- +tion. Comparing between components, the feature strength +is higher in the halo at earlier times in each mode (of order +1% density contrast in the halo, but well below that in the +disc), implying that the halo is responsible for starting each +mode at large radii (compare Figures 6 and 9). For the grow- +ing Mode 1, estimating the growth rate from the modulus of +the coefficients at early times also reveals the growth of the +halo feature to be twice that of the disc. The saturation point +of the halo is also measurably earlier than the disc (T = 2.2 +Gyr in the halo vs T = 3.2 Gyr in the halo). +The comparison of the disc and halo features in the pre- +vious paragraph suggest that the modes may arise from +a fundamental dynamical property of the halo component. +Figure 9 shows snapshots of the halo feature during the +simulation at times corresponding to Figure 6. The fea- +tures are both slow retrograde pattern which build and/or +damp over time. They bear hallmarks – a slow dipole pat- +tern at relatively large scales – of the weakly damped l = +1 modes in spherical systems that have been studied in +using linear perturbation theory. These were first identi- +fied by Weinberg (1994), and later additionally reported by +Heggie et al. (2020), Fouvry & Prunet (2021), and Weinberg +(2022). +We conclude that BFE+mSSA has allowed us to detect and +characterise slow, secular evolution of our isolated simulated +galaxy due to the nature of the underlying equilibrium. +3.3.3 Key insights +The results in this section provide additional illustrations of +the ability of BFE+mSSA to separate evolutionary pathways +in a single harmonic and to detect coupling across compo- +nents. +Most significantly, BFE+mSSA allowed the detection of +slow, low-level secular evolution in our simulation that +had been predicted in analytic work, (Weinberg 1994; +Fouvry & Prunet 2021) and recently observed in star cluster +and dark-matter-halo-only simulations (Heggie et al. 2020; +Weinberg 2022). The analytic work suggests that spherical +systems, such as dark matter halos, generically exhibit dipole +point modes. The common existence of these modes has im- +portant implications for understanding lopsidedness in galax- +ies: the halo and disc mutually open dynamical avenues that +cannot be taken by either component independently; there- +fore many dynamical features are simply inexplicable without +an understanding of the interplay between components. How- +ever, making a clear connection between the theory and ob- +served galaxies has been hampered by the technical challenge +of applying analytic work to multi-component systems. More- +over, while numerical simulations routinely represent multiple +component systems, the description of the results is typically +limited to visualisations and statistical analyses that can only +qualitatively be connected to dynamical drivers. +BFE+mSSA has bridged this gap by clearly showing an +l = 1 mode in our simulated halo driving lopsidedness in +our simulated disc. These results speak to the promise of +BFE+mSSA for forging the missing connection between the- +ory, simulations, and observations needed to interpret galac- +tic properties in terms of our fundamental dynamical under- +standing secular evolution. +3.4 Fluctuations and other uninterpretable features +In the two previous sections, we identified interpretable sig- +nals in various harmonics of both the disc and halo coeffi- +cients in groups of low-order PCs using BFE+mSSA. How- +ever, inspection of the last column of Tables 1 and 2 shows +that these PC groups only contain a fraction of the total sin- +gular values (which are normalised to total unity): most of +the groups represent less than 20 per cent of the variance +in the coefficients being analysed10. The rest of the signal +spread over many (many!) higher-order PCs with lower SVs. +These are PCs with very weak self-gravity. We refer to these +remaining terms as the nullity, owing to its uninterpretable +nature: it will contain numerical noise, but may also contain +signals too weak to be included in our analysis. +To understand the properties of the nullity, we collect all +uninterpretable PCs for a given mSSA decomposition and +analyse their reconstructions, summarising the results for +low-order disc harmonics in Figure 10 and for all decompo- +sitions in Table 3. Figure 10 shows the reconstructed coef- +ficients and corresponding power spectrum for the PCs as- +signed to the nullity for low-order disc harmonics. Compar- +ing this to the corresponding Figures 4 and 5 for lower or- +der PCs, the difference is clear. The bottom panels for the +m = 2 nullity do have hints of a signal in the form of low- +level systematic evolution in the left hand panel and some +clear peaks in the right panel. We discuss future strategies +to hunt for weak signals in Section 4.1. However, in general, +there is a lack of periodic or systematic evolution in the left +hand panels and flat spectra of frequencies in the right hand +panel, characteristic of noise. A comparison of the contrast +columns of Tables 3, A1 and 2 shows that the fluctuations +in the surface density derived from the nullity are mostly +stronger than the coherent signals in this particular simula- +tion: our BFE+mSSA analysis has supported insights that +would otherwise be inaccessible. +4 LOOKING AHEAD +4.1 Essential Future Work - assessment of weak +feature significance +Our +analyses +of +simulations +of +bar +formation +(Weinberg & Petersen +2021; +Petersen et al. +2022) +and +an isolated disc galaxy (this paper) amply illustrate the +facility of BFE+mSSA to learn about both significant and +expected as well as subtle and unanticipated dynamical evo- +lution. The results are very promising for general applications +to a wide variety of dynamical systems. However, our work +so far has been involved close supervision of BFE+mSSA +to both interpret and understand the significance of what +features it has identified. +In particular, the interpretative ambiguity we encountered +in the higher order terms in this paper outlines the current +limit of BFE+mSSA. This limit motivates the need for a rig- +orous statistical analysis of significance for mSSA-identified +signals. Many of the well-known approaches from statistical +10 The exception are some of the PCs associated with the +monopole, which encode the equilibrium. These PCs are respon- +sible for upwards of 60 per cent of the singular value signal, cf. +Table 1. +MNRAS 000, 000–000 (2023) + +12 +A. Johnson et al. +Figure 10. An analysis of the content in the nullity for m = 0 (upper panels), m = 1 (middle panels), and m = 2) lower panels. The +left panels show the reconstructed nullity coefficient amplitudes over time for m = 0, 1, 2 (top to bottom). The right panels show power +spectrum of the reconstructed nullity coefficients for each harmonic. Both m = 0 and m = 1 show no discernible signals. The m = 2 +harmonic shows some periodicity, but the power spectrum suggests the frequencies are broad and not strongly coherent. Therefore, we +are confident that we are not throwing away interpretable signal in the nullity in any harmonics. These reconstructions may be compare +to the unprocessed coefficients, Figure 2, for a quantitative analysis of what signals are part of coherent signal groups. +mSSA +PCs +DFT peak +contrast +SV +decomposition +(Gyr−1) +(R < Rd) +fraction +disc m = 0 +6+ +- +0.005 +0.275 +disc m = 1 +6+ +- +0.008 +0.678 +disc m = 2 +2+ +- +0.017 +0.799 +disc m = 3 +2+ +- +0.012 +0.936 +disc m = 4 +2+ +- +0.010 +0.914 +disc m = 5 +2+ +- +0.006 +0.954 +disc m = 6 +2+ +- +0.005 +0.964 +disc m = 1, 3, 5 +2+ +- +0.027 +0.933 +disc m = 2, 4, 6 +2+ +- +0.037 +0.928 +halo l = 0 +6+ +- +- +0.028 +halo l = 1 +6+ +- +- +0.693 +disc m = 0, halo l = 0 +6+ +- +0.021 +0.130 +disc m = 1, halo l = 1 +8+ +- +0.010 +0.667 +disc m = 1, 2, halo l = 1 +4+ +- +0.012 +0.801 +Table 3. Summary of principal components assigned the nullity in our decompositions. We refer to each collection of PCs here as the +‘Nullity’, rather than a PC group. Disc harmonics are denoted with m, halo harmonics are denoted with l. Columns are the same as in +Table 1. +analysis would be suitable for this purpose. For example, let +us take the hypothesis that the signal observed at m = 2, 3, 4 +is consistent with background noise as a test case. That is, our +null hypothesis is that our simulation can generate with the +same properties of the signal in question without inherent self +gravity. To do this, we need to generate a simulation with the +same noise spectrum as the full simulation but without any +self-gravitating features on the spatial and temporal scales of +our putative signal. Let us assume that we know how to per- +form such simulations (we propose an exp-enabled approach +below). An ensemble of these null-hypothesis simulations can +be run and analysed using mSSA. From the ensemble of sim- +ulations, one may construct prediction intervals for singular +values under the null hypothesis. Then, if the singular value +corresponding to the signal in question is beyond the pre- +diction intervals, the corresponding principal component is +considered significant. In such a case, the signal can be re- +liably reconstructed. This approach is often called Markov +Chain SSA (MC-SSA, see Allen & Smith 1996). +Analyses of this sort are particularly well-suited to the exp +framework described in Petersen et al. (2022). We can use the +mSSA analysis to construct a realistic reconstruction of the +coefficients series from the self-gravitating simulation with- +out the self-gravitating features of interest by removing the +groups corresponding to the signal in question. In the study +presented here, this would be akin to retaining only the nul- +lity reconstructions of the coefficients. We can generate new +MNRAS 000, 000–000 (2023) + +(detrended) +m= 0 Nullity +m = 0 Nullity +(normalised) +0 +2 +Coefficients +m = 1 Nullity +m= 1 Nullity +N +2 +Reconstructed +m = 2 Nullity +2 +0 +0 +1 +2 +3 +4 +5 +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +Time (Gyr) +Frequency +GyrNear-equilibrium disc-halo dynamics +13 +coefficient series from an autoregressive model11 consistent +with the coefficient covariance from the mSSA reconstruc- +tion. Then, exp allows initial potential fields from the re- +constructed coefficients to be replayed for a new ensemble +of particles with very little computational effort. The result- +ing expansion coefficient series are gathered automatically +for analysis by mSSA, and can be analysed for significance +of detected features. A detailed description of the MC-SSA +approach in the exp context will be described in a later con- +tribution. +4.2 Prospects for applications to simulations +Despite the limitations, there are multitude of prospects for +immediate, supervised applications of BFE+mSSA to simu- +lations of galaxies, whether isolated, interacting or evolving +in the full cosmological context. +Dynamical analyses of simulations of galactic evolution. +Recent +surveys +(Majewski et al. +2017; +Steinmetz et al. +2020; Gaia Collaboration et al. 2022) demonstrate that the +Milky Way continues to evolve through satellite interaction. +N-body simulations have explained some key observational +signatures +(Laporte et al. +2019; +Petersen & Pe˜narrubia +2020; Garavito-Camargo et al. 2021a; Vasiliev et al. 2021; +Hunt et al. +2022a). +However, +interpretation +of +these +simulations is challenging since many actors contribute +simultaneously. The BFE+mSSA knowledge discovery ap- +proach is capable of separating, characterising and dissecting +the signatures of the mutual interactions of each component +in simulations by separating features by correlating tem- +poral and spatial scales non-parametrically. BFE+mSSA +promise detailed predictions and identification of features +in current stellar data sets (see Petersen & Pe˜narrubia +2021; Garavito-Camargo et al. 2021b; Lilleengen et al. 2022, +for some recent results) and confident mapping the the +dark matter halo’s global structure and distortions to that +structure. This goal was unimaginable even 5 years ago. +Structural characterisation and correlation of fields. This +paper demonstrated the discovery of two-dimensional fea- +tures in disc density resulting from internal (disequilibrium- +related) +dynamics +and +halo +interactions. +However, +BFE+mSSA can be applied to any field in any num- +ber +of +dimensions. +For +example, +Weinberg & Petersen +(2021) illustrated a three-dimensional disc BFE. The exp +library already enables joint BFE+mSSA investigations +of any number of three-dimensional density and potential +fields. These may be augmented by kinematic fields as in +Weinberg & Petersen (2021) or some other field such as +star formation rates and implied local metallicity. If the +additional fields encode spatial information (e.g. they are +BFE coefficients or even radial and azimuthal bins), their +temporal and spatial scales will be correlated with the +density and potential fields. The BFE+mSSA can adapt to +new observational tools and windows as new surveys become +available. +11 Autoregressive noise models are typically used for null hypothe- +ses in MC-SSA because SSA provides good estimates for frequen- +cies and exponential factors processes generated by the related lin- +ear recurrence relations (Golyandina & Zhigljavsky 2013, Section +3). +Understanding of noise. There have been many years of de- +bate on the effect of noise in conclusions drawn from dynami- +cal simulations, from bar-halo interactions (Weinberg & Katz +2007), through dynamical friction (Weinberg 2001), to satel- +lite disruption (Errani & Pe˜narrubia 2020). BFE+mSSA +clearly separates the correlated, quasi-periodic signals result- +ing from dynamical interaction and coupling from the fluctu- +ating forces resulting from finite particle number stochastic +effects. We expect that couplings in orbital dynamics have +frequencies near or smaller than the characteristic orbital +frequencies. Since the individual PCs describe the tempo- +ral behaviour of components assigned to the noise field and +the power spectrum describes their characteristic frequencies, +mSSA provides a natural classification of signal and noise. +Investigations of test-particle orbits with and without the +noise component provide a diagnostic tool for the reliability +of features in simulations and the role of fluctuations more +generally. +5 CONCLUSIONS +5.1 Near-equilibrium evolution: the importance of +multi-component modes +We applied BFE+mSSA to a simulation of an isolated Milky +Way like galaxy. The BFE+mSSA combination allows us +to automatically identify the main features in the model +galaxy and their origins. Most remarkably, BFE+mSSA +achieved this in the challenging case of an isolated, multi- +component galaxy that had specifically been constructed to +not evolve and where the dynamical signatures were below +the level of the noise. Our work complements a prior investi- +gation (Weinberg & Petersen 2021) which used BFE+mSSA +to characterise significant evolution of known nature in a sim- +ulation which formed a galactic bar. +In our near-equilibrium model, we identified – for the +first time – two multi-component (disc-halo) dipole point +modes (Figures 5-8) which evolve over time (one growing, +one damped; Table 2). This discovery is enabled by the +BFE+mSSA methodology; such dynamical effects are at a +level such that other methods, such as Fourier analyses, +will not be able to recover the signals. Halo modes are +expected from linear perturbation theory (Weinberg 1994; +Fouvry & Prunet 2021), and are observed in simulations +of star clusters (Heggie et al. 2020) and dark matter halos +(Weinberg 2022), but the coupling of a spheroid to a disc +has not been discovered to date. The BFE+mSSA methodol- +ogy makes the identification of point modes straightforward, +and provides several avenues for corroboration. We employed +several different mSSA decompositions to validate our find- +ings. The existence of point modes in these isolated simula- +tions demonstrates the fundamental contribution of compo- +nent interactions to the dynamical evolution of galaxies. We +expect that the existence of such multi-component disc-halo +modes is a generic feature of such systems, possibly including +the Milky Way. These modes likely have influence over the +structural evolution of disc galaxies. For instance, our results +immediately suggest that low dipole modes will be most de- +tectable at large radii (e.g. R > 20 kpc in the Milky Way), +where density contrasts can exceed 10 per cent relative to a +smooth disc. +MNRAS 000, 000–000 (2023) + +14 +A. Johnson et al. +In addition to point modes, we identified the long-lived +results of initial conditions disequilibrium, resulting from in- +dividual halo and disc disequilibrium features. Starting with +rings encoded in the monopole m = 0 coefficients, we found +correlations with many other harmonics, including a persis- +tent m = 2 signal. We uncovered an aphysical ‘settling’ of +the halo in response to the presence of the disc at the outset +of the simulation. Future work modelling idealised galaxies +must take care to ensure that disequilibria in the initial con- +ditions, and the resulting persistent features, are not treated +as real dynamics. +Finally, we quantified the remaining signal that we classi- +fied as the nullity, and put limits on the magnitude of un- +explained surface density fluctuations in the disc. The desire +to push even deeper in the decomposition of simulations mo- +tivates the essential future work, but also inspires prospects +for future applications. +5.2 Dynamical Data Mining as the future of +Galactic Dynamics +Galactic Dynamics is a mature field with elegant descriptions +of equilibrium systems, estimates for scales of the processes +involved in the interactions that are known affect them and +sophisticated analytic methods that describe evolution in the +linear regime. While we can understand the basic governing +principles with detailed mathematical models from Hamilto- +nian perturbation theory, the inter-component and environ- +mental interactions that produce this morphology are hard +if not impossible to study from modal analysis alone. BFE +methods both underpin our dynamical data mining tech- +nique and are often used in analytic perturbation work. Thus +they provide a natural bridge between theoretical work and +numerical simulation. The combination of BFE representa- +tion of the possibly unknown dynamics in simulations with +a machine-based knowledge acquisition tool such as mSSA +allows for identification of couplings that may be too hard +to predict otherwise. This natural synergy between mathe- +matical theory and simulation is the main motivation for our +approach. Series of BFE can also be used to characterise ob- +served fields in galaxies. +A galaxy’s picturesque morphological structure is a his- +torical summary of its evolution. Cosmological predictions +for the frequency of galactic interactions explain the abun- +dant signatures of disequilibrium observed. The detail of our +picture of disequilibrium is rapidly advancing, in terms of +resolution in the Galaxy (e.g. Hunt et al. 2022b) in galax- +ies and occurrence rate in others (e.g. Pearson et al. 2022). +Simulated realisations of galaxies in disequilibrium are simi- +larly advancing in resolution and scale. However, the tools to +take full advantage of this twin onslaught of data, simulated +and real, are currently lacking. Such tools must be capable of +modelling galaxies in disequilibrium, make quantitative and +dynamically meaningful connections between simulated and +observed galaxies and connect with analytic work in the lin- +ear regime. +As outlined in Section 4.2, our results suggest the tremen- +dous promise of BFE+mSSA for the field of Galactic Dynam- +ics, with a myriad of envisioned applications. Many of these +applications can be undertaken now by adopting the super- +vised learning approach to using BFE+mSSA. These include +detailed analyses of galactic components for galaxies in both +isolated and cosmological settings. +Nor is there any reason to limit the BFE+mSSA analysis +to galaxies. BFE+mSSA can be used for the characterisa- +tion and dynamical evolution of self gravitating, interacting +systems more generally and in any context, from binary aster- +oids in the solar system (Quillen et al. 2022), through proto- +planetary discs (Cadman et al. 2021), to nuclear star clusters +in the centres of galaxies (Fouvry et al. 2022) . +The remaining and key challenge to be solved is to un- +derstand how to confidently assess the significance of all the +features that BFE+mSSA recovers in an unsupervised way. +Once this is developed it will be possible to broadly apply +BFE+mSSA to large samples of systems: both simulated and +real. +ACKNOWLEDGEMENTS +We warmly thank Chervin Laporte for sharing the initial con- +ditions from his simulation with us. We acknowledge support +from the Center for Computational Astrophysics (CCA) at +the Flatiron Institute in the form of access to their compu- +tational resources which allowed us to create our simulation +and generate the associated data. In addition, we thank CCA +leadership and staff for hosting the Beyond-BFE collabora- +tion meetings. We thank members of the B-BFE collabora- +tion and the Dynamics Group at CCA for numerous conversa- +tions during development of this paper. MSP’s contributions +were partially supported by grant Segal ANR-19-CE31-0017 +of the French Agence Nationale de la Recherche as well as a +UKRI Stephen Hawking Fellowship. KVJ and AJ’s contribu- +tions were supported by NSF grant AST-1715582. +DATA AVAILABILITY +The code, data, and simulation used to generate the results +in this article will be made available upon reasonable request +to the appropriate author. +REFERENCES +Allen M. R., Smith L. A., 1996, J. 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H., Coronado J., Rix H.-W., 2019, MNRAS, 484, 3291 +Vasiliev E., Belokurov V., Erkal D., 2021, MNRAS, 501, 2279 +Weinberg M. D., 1989, MNRAS, 239, 549 +Weinberg M. D., 1994, ApJ, 421, 481 +Weinberg M. D., 1999, AJ, 117, 629 +Weinberg M. D., 2001, MNRAS, 328, 321 +Weinberg M. D., 2022, arXiv e-prints, p. arXiv:2209.06846 +Weinberg M. D., Katz N., 2007, MNRAS, 375, 425 +Weinberg M. D., Petersen M. S., 2021, MNRAS, 501, 5408 +APPENDIX A: ALL MSSA COMBINATIONS +TESTED +Table A1 summarises all tested combinations in mSSA. All +tests include all radial orders, i.e. n ∈ [0, 6]. All mSSA de- +compositions use a window length L = 250, which is ap- +proximately half of the input time series (N = 549). This +creates sets of 250 PCs. The PCs are sorted by singular value +magnitude, which does not guarantee that PCs with physi- +cal similarity are consecutive. Therefore, we determined the +grouping of the PCs through direct examination, which in +practice was straightforward. The information in this table +is repeated from Tables 1-3 in text, consolidated here and +reorganized by mSSA decomposition for ease of comparison. +APPENDIX B: BASIS FUNCTION EXPANSION +DETAILS +In this Appendix, we briefly describe the basis function ex- +pansion technical implementation for completeness and clar- +ity. Full details may be found in Petersen et al. (2022) for the +spherical expansions, and in Weinberg & Petersen (2021) for +the Fourier-Laguerre disc expansion. +B1 Spherical expansions: Empirical Orthogonal +Functions +For an initially spherically-symmetric model (such as a dark +matter halo or stellar bulge) we use spherically symmetric +basis functions derived using the machinery in exp. The po- +tential and density take the form +φnlm = φ0(r)unl(r)Ylm(θ, φ) +(B1) +ρnlm = ρ0(r)unl(r)Ylm(θ, φ) +(B2) +where unl are eigenfunctions determined by exp, Ylm(θ, φ) +are the usual spherical harmonics12, and ρ0(r) and φ0(r) are +input unperturbed model density and potential. The model +density and potential are typically chosen to be the initial +conditions. The functions unl(r) are eigenfunctions of a the +Sturm-Liouville equation. Each function is a solution to the +Poisson equation and has n nodes with increasingly tighter +spacing (as n increases). The u00(r) function is a constant, +which makes the ℓ = 0, n = 0 potential and density terms +exactly proportional to the input model. Other terms are then +perturbations on top of the model potential and density. The +functions are biorthogonal, satisfying two conditions: +� +d3v φnlm ρn′l′m′ ∝ δnn′δll′δmm′ +(B3) +∇2φnlm = 4πGρnlm. +(B4) +The +derivation +of +these +functions +is +described +in +Petersen et al. +(2022). +For +the +purposes +of +this +work, +the BFE representation of density is given by the projection +of the coefficients onto the BFE: +ˆρhalo(r, θ, φ; t) = +� +l +� +m +� +n +Almn(t)ρ0(r)unl(r)Ylm(θ, φ) +(B5) +where Almn(t) is the coefficient amplitude for a given function +indexed by (l, m, n), and possibly is a function of time. An +analogous expression may also be written for the potential. +The ˆ· notation indicates that the quantity is reconstructed +from the coefficients. +B2 Disc surface density expansion: +Fourier-Laguerre +A spherical expansion is not appropriate for a strongly flat- +tened stellar disc. While exp supports a three-dimensional +12 For ease of numerical implementation, exp uses the real +spherical harmonics: Ylm,cos = +1 +2 (Ylm + Yl−m) and Ylm,sin = +1 +2i (Ylm − Yl−m). +MNRAS 000, 000–000 (2023) + +16 +A. Johnson et al. +Fourier +visual +singular +mSSA +peak +contrast +evolution +value +decomposition +Group +PCs +(Gyr−1) +(R < Rd) +type +fraction +interpretation +disc-only decompositions +m = 0 +1 +0,1 +0.2 +0.031 +slow decay +0.641 +phase mixing of halo initial conditions +2 +2,3,4,5 +6.4 +0.006 +fast decay +0.084 +phase mixing of disc initial conditions +nullity +6+ +- +0.005 +no evolution +0.275 +- +m = 1 +1 +0,1 +0.6 +0.007 +slow growth +0.201 +coupling with halo +2 +2,3 +1.7 +0.003 +slow decay +0.064 +coupling with halo +3 +4,5 +6.9 +0.002 +fast decay +0.057 +phase mixing of disc initial conditions +nullity +6+ +- +0.008 +no evolution +0.678 +- +m = 2 +1 +0,1 +6.6 +0.006 +slow growth +0.201 +phase mixing of disc initial conditions +nullity +2+ +- +0.017 +no evolution +0.799 +- +m = 3 +1 +0,1 +13.8 +0.005 +slow growth +0.064 +projection of m = 1 Group 1 +nullity +2+ +- +0.012 +no evolution +0.936 +- +m = 4 +1 +0,1 +14.2 +0.004 +slow growth +0.086 +projection of m = 2 Group 1 +nullity +2+ +- +0.010 +no evolution +0.914 +- +m = 5 +1 +0,1 +23.1 +0.001 +slow growth +0.046 +projection of m = 1 Group 1 +nullity +2+ +- +0.006 +no evolution +0.954 +- +m = 6 +1 +0,1 +20.2 +0.001 +slow growth +0.036 +projection of m = 2 Group 1 +nullity +2+ +- +0.005 +no evolution +0.964 +- +m = 1, 3, 5 +1 +0,1 +0.6 +0.008 +slow growth +0.067 +projection of m = 1 Group 1 +nullity +2+ +- +0.027 +no evolution +0.933 +- +m = 2, 4, 6 +1 +0,1 +6.6 +0.010 +slow growth +0.072 +projection of m = 2 Group 1 +nullity +2+ +- +0.037 +no evolution +0.928 +- +halo-only decompositions +l = 0 +1 +0,1,2,3 +0.4 +- +slow decay +0.944 +phase mixing of halo initial conditions +2 +4,5 +6.6 +- +fast decay +0.028 +phase mixing of disc initial conditions +nullity +6+ +- +- +no evolution +0.028 +- +l = 1 +1 +0,1,2,3 +0.4 +- +slow growth +0.272 +weakly self-gravitating mode +2 +4,5 +1.5 +- +slow decay +0.035 +weakly self-gravitating mode +nullity +6+ +- +- +no evolution +0.693 +- +disc-halo decompositions +m = 0,l = 0 +1 +0,1,2,3 +0.2 +0.054 +slow decay +0.832 +phase mixing of halo initial conditions +2 +4,5 +6.6 +0.007 +fast decay +0.037 +phase mixing of disc initial conditions +nullity +6+ +- +0.021 +no evolution +0.130 +- +m = 1,l = 1 +1 +0,1,2,3 +0.6 +0.008 +slow growth +0.244 +weakly self-gravitating mode and coupling +2 +4,5 +1.5 +0.003 +slow decay +0.048 +weakly self-gravitating mode and coupling +3 +6,7 +6.9 +0.002 +fast decay +0.040 +phase mixing of initial conditions and coupling +nullity +8+ +- +0.010 +no evolution +0.667 +- +m = 1, 2,l = 1 +1 +0,1 +0.6 +0.010 +slow growth +0.117 +weakly self-gravitating mode and coupling +2 +2,3 +6.6 +0.003 +fast decay +0.082 +phase mixing of disc initial conditions and coupling +nullity +4+ +- +0.012 +no evolution +0.801 +- +Table A1. Summary of different modes identified in our MSSA decompositions. Disc harmonics are denoted with m, halo harmonics +are denoted with l. Disc feature strengths are reported in surface density to give a measure of ‘visual contrast’, defined as max (|∆Σ|) +within a disc scale length (see equation C9). Contrasts have an approximate error of 0.001, estimated from grid size adjustments. Owing +to simulation sampling rates (0.01 Gyr), the DFT peak is only accurate to 0.1. The group name can be derived for each row by combining +the decomposition harmonic(s) and concatenating with the group number. For example, the name of the first row would be Group m0-1. +empirical orthogonal function basis that may be used to rep- +resent the potential and density, we choose in this work to +project the disc to the two-dimensional plane. We work in +polar coordinates R and φ for a two-dimensional expansion. +For the radial coordinate we use the Laguerre basis func- +tions described in Weinberg & Petersen (2021), which were +created to match the exponential profile of a typical stellar +disc. This choice minimises the number of required functions. +MNRAS 000, 000–000 (2023) + +Near-equilibrium disc-halo dynamics +17 +The Laguerre polynomials are defined as: +Gn(R) = +1 +a√n + 1 exp +� +−R +a +� +L1 +n +�2R +a +� +, +(B6) +where L1 +n is the associated Laguerre polynomial of order 1 +and degree n, and a is the scale length of the disc. Then +G0(R) = 2 +ae−R/a closely approximates the disc density and +the majority of the reconstruction power resides in a single +term. Moreover, the polynomials satisfy the the orthogonality +condition +� +Gn(R)Gn′(R)RdR = δnn′, +(B7) +and thus Gn(R) can be used to reconstruct the radial struc- +ture of the disc with a small number of expansion terms. +The azimuthal dependence is described with a Fourier se- +ries. Combining this with the radial Laguerre basis func- +tions we obtain a set of two-dimensional basis functions +Gn(R) cos mφ and Gn(R) sin mφ. We call these Fourier- +Laguerre functions. The coefficients are naturally determined +by the projection of f(R, φ) into the basis functions. In our +case we are accumulating discrete particles so rather than a +continuous integral we obtain the discrete summations: +Cmn = 1 +2π +� +i +f(Ri, φi)eimφiGn(Ri) +(B8) +where (Ri, φi) is the position of particle i. The reconstruction +of the surface density field is then +ˆΣdisc(R, φ; t) = +� +m +� +n +Cmn(t)eimφGn(R), +(B9) +where the coefficients Cmn(t) may have some time depen- +dence. +APPENDIX C: MULTICHANNEL SINGULAR +SPECTRAL ANALYSIS +Singular Spectral Analysis (SSA) is a method of non- +parametrically decomposing a time series into a sum of com- +ponents that ideally capture different aspects of the series. +Multichannel Singular Spectral Analysis (mSSA) extends +SSA to include multiple series, such that one may identify co- +herent signals between series. A full description and pedagog- +ical examples may be found in Weinberg & Petersen (2021). +In this appendix, we briefly describe the important elements +of SSA and describe the implementation used in this paper. +C1 A conceptual introduction to mSSA +In this section, we present a brief and intuitive introduction +to mSSA through the conceptual relationship with principal +component analysis (PCA). PCA characterises correlations +within a data set by transforming to coordinates where most +of the correlation is represented in a small number of di- +mensions in space. mSSA itself is a generalisation of PCA: +mSSA performs computations analogous to PCA on a grand +trajectory matrix which is constructed to represent different +variations in time intervals. The extension of PCA to mul- +tiple times allows for the decomposition of different series +of samples of variables over time. In our case, the variables +are the individual BFE coefficients for each snapshot in the +simulation. We can then simultaneously decompose struc- +ture in space and time. We refer the interested reader to +Golyadina et al. (2001) for a more thorough description of +SSA. +C1.1 The goals of Principal Component Analysis +The goal of PCA is the reduction of the data set’s dimen- +sionality while retaining as much as possible of the variation +present in the data set. This is achieved by a linear transfor- +mation or rotation to a new set of axes, the principal com- +ponents (PCs), which are uncorrelated and ordered by their +contribution to the total variance. For a brief introduction, +consider a data sample of N random variables with M chan- +nels (or dimensions). PCA is mathematically equivalent to +the eigen-analysis of the covariance matrix: +C ∝ X⊺ · X +(C1) +where X is a M ×N matrix whose (i, j)th element is the data +with zero-mean and unit variance for each of the M chan- +nels: (xij − µi)/σi where µi, σ2 +i is the mean and variance for +channel i, respectively13. The sums over covariance between +channels is effected by the matrix multiplication in equation +(C1). Then, the leading eigenvector or PC is the direction +that maximises the variance. The next PC is the direction +that maximises the variance, uncorrelated to the first, and so +on. +C1.2 Relationship of mSSA and Principal Component +Analysis +For our analysis, we have time series of N samples for each +of M BFE coefficients that have been reduced to zero mean +and unit variance: {ai,j : i = 1, . . . , M; j = 1, ...., N}. Each of +the N samples corresponds to a phase-space snapshot in our +simulation, providing a two-dimensional grid in coefficient +channel (the i index) and time (the j index). A PCA anal- +ysis of this matrix, as described above, identifies the sets of +weighted coefficients (the principal components) which con- +tain the highest proportion of the variance in each of the +N temporal views. Each of these sets represents a coherent +spatial pattern that is present in the data. +The mSSA algorithm adds the ability to simultaneously +find dominant shapes in space that also evolve similarly in +time14. For our case, this is done by constructing a new ma- +trix where each row contains information from a sequence of +L < N snapshots rather than a single one, effectively creat- +ing a time window. Each of K = N − L + 1 rows is of length +M × L. The ith row is a concatenation of the coefficients de- +rived from snapshots i - i + L − 1, so each contains the same +coefficients, but lagged by systematic amounts in time. The +resulting (M × L) × K matrix is known as the grand trajec- +tory matrix (Ghil et al. 2002). Constructing grand covariance +matrix and performing a PCA would now compare not one, +13 The covariance matrix is often normalised in PCA as C = 1 +d X⊺ · +X, where if the mean is determined from the data, i.e., the sample +variance, we have d = N − 1. +14 SSA is a unique case where M = 1. That is, we only have +one input channel. In this work, we advocate for use of mSSA +specifically, but one could also apply the discussion here to SSA. +MNRAS 000, 000–000 (2023) + +18 +A. Johnson et al. +but L snapshots at once to find repeating patterns at differ- +ent lag times, effectively sliding a window of length L over +the simulation. The resulting PCs maximise the variance in +the K = N − L + 1 overlapping views of the M time series +simultaneously. +For intuition, consider a set of coefficients that all have si- +nusoidal variation with a single period. At time lags that are +multiples of the period, the variance will be large as the sig- +nals reinforce each other coherently. At incommensurate lags, +the covariance will tend to zero. This allows mSSA to natu- +rally find the coherent temporal signals in the data. For each +oscillatory signal in our series, we will find a pair of eigen- +vectors that represent the same frequency, just as in Fourier +analysis. A sinusoid was simply an example; the method will +work any temporally coherent signal including exponential +growth or decay. mSSA is purely non-parametric in this sense. +C2 Details of mSSA implementation +SSA is principal component analysis (PCA) of sequentially +lagged L-length windows of a time series (where L is a user- +specified window length). The key is that distinct features +have different projections into L-lagged space and so PCA +separates the different features. When working optimally on a +dynamical system input, the different features will correspond +to different dynamical phenomena, such as a galactic bar (as +in Weinberg & Petersen 2021). The SSA procedure involves +three main steps: embedding, singular value decomposition, +and grouping/reconstruction. +C2.1 Embedding +In this step of SSA one forms a matrix which represents the +sequence of L-lagged windows of the time series. This matrix +is called the trajectory matrix. Consider an input time series +⃗s = {s1 . . . sN} and window length L. The trajectory matrix +is formed as +T = + + +s1 +s2 +. . . +sN−L+1 +s2 +s3 +. . . +sN−L+2 +... +... +... +sL +sL+1 +. . . +sN + + . +(C2) +It is common to denote N − L + 1 as K. The anti-diagonals +of T are equal by construction. Matrices with this property +are called Hankel matrices. +C2.2 Singular Value Decomposition: Principal Components +and Singular Values +After forming the trajectory matrix, the next step is to per- +form a singular value decomposition (SVD). We begin by in- +specting the dimensions of the trajectory matrix, T ∈ RL×K. +To maximise computational efficiency, we construct the co- +variance matrix such that it is min(K, L) × min(K, L). +Consider the SVD of the trajectory matrix +T = UΛ1/2V⊺. +(C3) +The covariance matrix for L < K, C = TT⊺ = UΛU⊺. When +K < L, we can write the covariance as C = T⊺T = VΛV⊺. +The choice in constructing the covariance matrix is to de- +crease the computational complexity by choosing the rep- +resentation with the smallest rank nullity15. We may then +obtain the SSA principal components either directly from +U ∈ RL×K, or as U = TV. The SSA empirical orthogonal +functions either come directly as V ∈ RL×K, or may be com- +puted as V = (U⊺T)⊺. The singular values are given along +the diagonal of Λ: that is, the ith singular value is σi = √λi. +We sort the singular values from highest to lowest and sort +the columns of U and V accordingly. +Given the ith empirical orthogonal function Vi and a prin- +cipal component Ui, we can form the matrix Ai = σiUiV ⊺ +i . +Summed over all i, these matrices are the set of matrices +which best approximate T and have orthogonal column/row +spaces. In this sense the SVD of C gives us a set of matrices +which each carry independent information. We refer to the +tuples (σi, Ui, Vi) as eigentriples. +C2.3 Grouping +The various input coefficient series are often not independent. +This, combined with sampling noise, may lead to informa- +tion smearing between PCs. Of practical importance, there +are often multiple eigenvectors and PCs that correspond to a +particular dynamical signal. The mSSA practitioner will need +to group eigenvectors that describe the same or related sig- +nals together. For example, even for a pure sinusoidal signal, +the basic group will contain two eigenvectors, with identical +eigenvalues. The pair together describe the amplitude and +phase. +For typical features found in disc galaxies, such as arms +and bars, the relevant groups often contain multiple eigen- +vectors. There are a variety of ways to motivate groupings. +As a starting point, a group of PCs describing a single dy- +namical features will often have similar eigenvalues. Next, +plotting the PCs over time by eye often reveals whether their +evolution is similar. This can be quantified by (for example) +taking a Fourier transform. In particular, power spectra can +clearly indicate which PCs are describing noise as they will +have a broad frequency spectrum. The understanding of the +noise can also be used to modify the window length to get +better noise separation properties. +w-correlation matrices are a good grouping diagnostic +(Weinberg & Petersen 2021). These matrices quantify the +correlation between PCs and can be used to guide group- +ing by using the correlation as an indication of which PCs +belong in groups. Though we have listed two approaches to +grouping, it is important to note that these alone do not es- +tablish the groups. The grouping is determined by the content +of the PCs. We find grouping to be important primarily be- +cause misgrouping can spread out the correlated signal and +lead to an underestimate of a particular correlated degree of +freedom. +As an example, examination of PCs may indicate that two +long duration trends are present in the same PC. In this +case, the window length could be increased to separate those +15 The information content is the same in both, since the “wrong” +choice simply increases the rank of the nullity which has no useful +information. +MNRAS 000, 000–000 (2023) + +Near-equilibrium disc-halo dynamics +19 +trends. However, it is also common that PCs may demon- +strate the existence of multiple behaviours present in single +PCs leading to a modification of the window length in the +way described in the previous paragraph to achieve better +separation. The converse situation can also occur, where sin- +gle processes are being split into many PCs, in which case +the window length can be shortened to simplify grouping. +The goal of grouping is a partition of the indices i into m +distinct subsets such that each set of m subsets corresponds +to a distinct feature. The trajectory matrix reconstructed +from select i indices is +˜T +k = +� +i∈Im +σiUiV ⊺ +i +(C4) +where Im is a list of i indices in the group. After segmenting +the eigen-triples into groups, each ˜T +k ideally corresponds to +a distinct feature. +C2.4 Coefficient Reconstruction +Given that the matrices ˜T +k need not be (and almost certainly +will not be) Hankel, we must convert them into the closest +Hankel matrices that preserve the decomposition. The step +of SSA in which this is done is referred to as reconstruction, +resulting in the coefficient reconstructions. +The final step is the reconstruction of the original series +from the PC groups which is referred to as the reconstruc- +tion stage. In order for a trajectory matrix to unambiguously +correspond to a time series it must be Hankel. However, in +general ˜T +k need not be Hankel. It is thus necessary to “Han- +kelise” each ˜Tk by setting each anti-diagonal to the average +value along the anti-diagonal. The “Hankelised” matrix is the +best Hankel representation of the original matrix in terms of +the Frobenius norm (see Weinberg & Petersen 2021). This +antidiagonal averaging procedure results in reconstructed co- +efficients for a set of pre-selected PCs, Im: +˜sk +j += + + + + + + + + + + + + + + + + + + + + + +1 +j +j +� +n=1 +U k +n−j+1V k +n +if 1 ≤ j < L − 1, +1 +L +L +� +n=1 +U k +n−j+1V k +n +if L ≤ j ≤ N − L + 1 +1 +N − j + 1 +N +� +n=N−L+1 +U k +n−j+1V k +l +if N − L + 2 ≤ j ≤ N. += +{˜s1,k . . . ˜sN,k}. +(C5) +The series ˜sk, where k is the group label (typically an integer), +is the reconstructed coefficient series. +C2.5 Power Spectra from Discrete Fourier Transforms +We use the Discrete Fourier Transform (DFT) to estimate +primary frequencies in reconstructed coefficients for the pur- +poses of grouping. The DFT of a reconstructed coefficient is +defined as +˜sk(ω) = F[˜sk(t)] = +N−1 +� +j=0 +e−i2πωtj/N ˜sk(tj) +(C6) +where C(tk) is a reconstructed coefficient time series of inter- +est. One may also compute the DFT of a PC directly for an +alternate grouping strategy. Plotting the resulting frequency +and DFT values results in the power spectrum. +C2.6 Contrast measurement +Using the reconstructed coefficients, one can construct the +field of interest using a reduced version of either equation +(B5) or equation (B9), depending on the component. The +reduced version of each equation will sum over a selected +set of indices corresponding to the input series to mSSA. For +example, if one were to use disc coefficients Cm=0,n∈[0,5] as the +input series ⃗s for the group indexed by k (cf. equation C5), the +surface density representation given by equation B9 would +become +˜Σk +disc(R, φ; t) = +� +m=0 +� +n∈[0,5] +˜sk +mn(t)eimφGn(R), +(C7) +where the notation ˜· indicates an approximated and trun- +cated field representation (in this case surface density) from +a select PC group. +To compute the contrast ∆Σ, we divide the selected field +representation by the unperturbed m = 0, n = 0 representa- +tion +ˆΣdisc,00(R; t) = C00(t)G0(R), +(C8) +giving +∆Σ(R, φ; t) = +˜Σk +disc +ˆΣdisc,00 +. +(C9) +Analogous expressions exist for computing halo density con- +trast. In this work we consider only disc surface density con- +trasts. +C2.7 Implementation of mSSA +mSSA involves the exact same procedure as SSA, but with +a grand-trajectory matrix (Ghil et al. 2002) that is the con- +catenation of the trajectory matrices for the individual series. +All other procedures are the same except for diagonal averag- +ing. The choice of L is no longer symmetric around N +2 because +only one of the dimensions is correlated. +We form the grand trajectory matrix by concatenating tra- +jectory matrices: +H = +� +T0, T1, . . . , TM +� +∈ RL×MK +(C10) +where M is the total number of different coefficient series. +The next step is to find the SVD of H such that we may write +the grand-trajectory matrix as the sum of principal compo- +nents, H = �r +i=1 σiUiV ⊺ +i . This could be computed directly +but it is unnecessarily expensive and could be prohibitive for +a large number of series. We therefore solve the smaller L×L +eigenvalue problem instead. +First note that H ∈ RL×MK and L ≤ N, thus HH⊺ ∈ RL×L +is smaller than RN×N. If N is sufficiently modest (as is the +case in this work with N = 599), we may find U by performing +the eigendecomposition +HH⊺ = UΛU⊺, +(C11) +which only requires the eigen-decomposition of the L × L +matrix HH⊺. We can then find ΣV⊺ as +U⊺H = U⊺(UΣV⊺) = ΣV⊺. +(C12) +We then have everything required to write H = �L +i UiσiV ⊺ +i +doing only the eigen-decomposition of an L × L matrix. We +MNRAS 000, 000–000 (2023) + +20 +A. Johnson et al. +found that this optimisation, in the modest N limit, allowed +the correlation of a much greater number of input series M. +After obtaining H = � +i σiUiV ⊺ +i , we do the same group- +ing described in 3.1.3 and obtain H = �m +k=1 ˜Tk where +˜Tk = � +i∈Ik σiUiV ⊺ +i . We also must Hankelize each ˜Tk by +applying the Hankelization algorithm from equation (C5) to +each block of the grand trajectory matrix independently. Af- +ter this procedure has been applied, each Hankelized PC +group ideally corresponds to a different feature. +MNRAS 000, 000–000 (2023) + diff --git a/TdE0T4oBgHgl3EQfUwDn/content/tmp_files/load_file.txt b/TdE0T4oBgHgl3EQfUwDn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..05650b5d3adbe5d47082410631a7544e70ccd8b2 --- /dev/null +++ b/TdE0T4oBgHgl3EQfUwDn/content/tmp_files/load_file.txt @@ -0,0 +1,1317 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf,len=1316 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='02256v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='GA] 5 Jan 2023 MNRAS 000, 000–000 (2023) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 Dynamical Data Mining Captures Disc-Halo Couplings that Structure Galaxies Alexander Johnson1⋆, Michael S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Petersen2, Kathryn V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnston1,3, Martin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Weinberg4 1Department of Astronomy, Columbia University, 550 West 120th Street, New York, NY 10027, USA 2Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK 3Center for Computational Astrophysics, Flatiron Institute, 162 5th Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=', New York City, NY 10010, USA 4Department of Astronomy, University of Massachusetts, Amherst MA 01003-9305, USA 9 January 2023 ABSTRACT Studying coupling between different galactic components is a challenging problem in galactic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Using basis function expansions (BFEs) and multichannel singular spectrum analysis (mSSA) as a means of dynamical data mining, we discover evidence for two multi-component disc-halo dipole modes in a Milky-Way-like simulated galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' One of the modes grows throughout the simulation, while the other decays throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The multi- component disc-halo modes are driven primarily by the halo, and have implications for the structural evolution of galaxies, including observations of lopsidedness and other non-axisymmetric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In our simulation, the modes create surface density features up to 10 per cent relative to the equilibrium model stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While the simulated galaxy was constructed to be in equilibrium, BFE+mSSA also uncovered evidence of persistent periodic signals incited by aphysical initial conditions disequilibrium, including rings and weak two-armed spirals, both at the 1 per cent level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The method is sensitive to distinct evolutionary features at and even below the 1 per cent level of surface density variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The use of mSSA produced clean signals for both modes and disequilibrium, efficiently removing variance owing to estimator noise from the input BFE time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The discovery of multi-component halo-disc modes is strong motivation for application of BFE+mSSA to the rich zoo of dynamics of multi-component interacting galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 1 INTRODUCTION The structures of galaxies are manifestations of how the laws that govern dynamics combine with the nature of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Un- derstanding galaxies strengthens our understanding of fun- damental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' There are tremendous opportunities to deepen that understanding: a rich legacy of analytic descrip- tions of galactic dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' community investment in high resolution simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' large scale, high dimensional surveys of billions of stars and galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' and the emergence of the vital field of data science to robustly mine and characterise both simulated and real data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Yet recent years have revealed the limits to our conception of our home galaxy, long thought to be a quiet backwater in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Maps of the positions and motions of billions of stars from the Gaia satellite (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2016, 2018, 2022) have revealed a Milky Way in disarray, with abun- dant signatures of action and reaction - past and ongoing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Antoja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Trick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Friske & Sch¨onrich 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Helmi 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These represent significant departures from the descriptions of equilibrium and mild perturbations on which the field of Galactic Dynamics has been built (Binney & Tremaine 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Simulations are capable of cap- turing such complexities but robustly linking the features to theoretical descriptions and identifying their physical origins remains challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Recent work by Weinberg & Petersen (2021) suggest one approach to this challenge centred around two mathemat- ical tools: Basis Function Expansions (BFE) and Multi- Channel Singular Spectrum Analysis (mSSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE rep- resent a distribution as a linear combination of basis functions, with half a century of application to galac- tic dynamics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Clutton-Brock 1972, 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Kalnajs 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Polyachenko & Shukhman 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Weinberg 1989, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' When representing a simulation with a fixed set of basis functions, one obtains time series of co- efficients that encode the dynamics in a compressed rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' mSSA is a method for identifying temporal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Together, one obtains a powerful analysis tool for studying galaxy simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The method does not re- quire prior information and thus can be considered a form of unsupervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Applying mSSA to BFE time-series, Weinberg & Petersen (2021) analysed barred-galaxy simula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' They found that BFE+mSSA could autonomously ex- tract the dominant space and time correlated features and disentangle different phase of bar formation and evolution recovered through more traditional analysis (Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this paper, we build on the success of Weinberg & Petersen (2021) in characterising the evo- lution of a known feature and explore the use of BFE+mSSA © 2023 The Authors 2 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' as a dynamical discovery tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We do so through the analysis of a model galaxy comprised of a stellar disc, stellar bulge, and dark matter halo that is designed to be in equilibrium and hence featureless (described in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Studying such a galaxy serves as a ‘control’ sample for future work with more feature-rich discs, with features from in situ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' spiral arms) or ex situ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' minor mergers) sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' With a control model, we want to answer the following questions about BFE+mSSA as a dynamical data mining tool: 1) Can BFE+mSSA separate distinct features that overlap in time and are not distinct by eye (real astrophysical signals, phase mixing, and N-body noise)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2) Can BFE+mSSA connect features within or across components by identifying their shared spatial and temporal structure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The answer, as we shall see, is yes to both questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE+mSSA isolates features and allows them to be in- terpreted independently, while also isolating interactions between components independent of the presence of other interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While analysing the disc in the present study, it became clear that the model was not the perfect featureless system we intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' By applying BFE+mSSA to the disc, and then the combination of disc+halo, we identify two dynamical causes of features: phase-mixing from initial conditions, and interac- tions between the disc and halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We identify multiple distinct dynamical signals in each, and examine the dynamical signals in detail (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We find that the signals are likely to be generic features in disc+halo systems, and can have real im- pact on galaxies in the real Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This study is a key step in understanding and exploring the strengths and limitations of BFE+mSSA in multi-component systems (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In partnership, BFE+mSSA has great potential beyond simulations analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Much of analytic linear theory is also built on BFEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Moreover BFEs may be used to described observational data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Hence BFEs pro- vide a common dynamical language to quantitatively con- nect theory, simulations, observations and data science while providing rigorous physical interpretations of dynamical pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We conclude in Section 5 with a discussion of how our results impact galaxy evolution more generally, and how BFE+mSSA fits in a larger program of dynamical data min- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2 METHODS We first review the rationale and overarching goals for BFE+mSSA analysis in dynamical systems in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1, and then describe the construction of a model isolated disc+bulge+halo galaxy in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Two appendices pro- vide specifics of the expansions used in our analysis (Ap- pendix B) and an overview of mSSA (Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Rationale for BFE+mSSA analysis All self-gravitating stellar systems, like ionised plasma, have a spectrum of both continuous and point modes (Krall & Trivelpiece 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Ichimaru 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Ikeuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Here, we define a mode to be a superposition of os- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Circular (black) and radial (red) frequency curves as a function of radius for the T = 0 equilibrium model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Both frequen- cies are computed using the epicyclic approximation, in the plane of the disc (z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Three frequency values have been marked to guide the eye (Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6, Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5, and Ω = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 cycles/Gyr), cor- responding to spatial scales near the peak disc circular velocity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2Rd = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7) and multiples of the halo scale length (a = 52 kpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' cillations that lead to a self-similarly growing or damping response to a perturbation1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Continuous modes are excited by perturbations with a continuous range of frequencies, for example a single en- counter with a satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Other sources of disequilibrium, whether physical or aphysical, also drive continuous response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This continuous response appears as phase mixing in galax- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These modes are also transient: since the response is not dominated by a single frequency the mode quickly looses co- herence and therefore is not self-sustaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We expect that mSSA will efficiently detect a plethora of signals owing to continuous modes, of varying strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These signals will ap- pear with relatively broad frequency support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As the modes are transient, few theoretical approaches exist capable of pre- dicting the existence or evolution of these modes, making BFE+mSSA an efficient tool to study them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Point modes are excited by specific frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' They have model-dependent self-similar shapes and well defined frequencies and can therefore be reinforced by their own gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The point modes are damped (growing) for sta- ble (unstable) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The most commonly known point mode is the Jeans’ instability in a homogeneous sea of stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Binney & Tremaine 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Fluctuations from environ- mental disturbances such as satellite encounters or Poisson noise from N-body distributions may excite these weakly self-gravitating features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We expect that some of the results recovered by mSSA will be the phase space manifestation of these modes, appearing as distinct frequency peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Cal- culations for unstable evolutionary modes in galactic discs 1 Mathematically, we are referring to the set of solutions to the col- lisionless Boltzmann equation for at a specific complex frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These are the solutions to the response operator that generalise eigenfunctions in a finite vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In plasma physics, these solutions are usually call ‘modes’ although there is some disagree- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 32 16 8 Q=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 cycles Gyr 4 QR 2 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 1 Q= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='25 1 1 2 3 5 10 20 30 50 100 R (kpc)Near-equilibrium disc-halo dynamics 3 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Disc coefficients over time for the first three harmonic orders (m = 0, 1, 2) and all corresponding radial orders (n ∈ [0, 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The coefficients have been detrended by subtracting the mean and dividing out the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The coefficient series are dominated by apparent noise, though some trends may be discerned: a steady decrease in some m = 0 coefficients (upper panel), elevated am- plitude towards the end of the simulation in m = 1, and some periodicity in m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The origin of these features is difficult to interpret owing to the coefficient series’ noisy appearance across multiple basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Any spatial features encoded in the basis are all but impossible to determine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' have found evidence for point modes supported in various analytic geometries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Fouvry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' De Rijcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While we do not have explicit theoretical results for damped modes at many azimuthal orders in discs, N-body simulations seem to suggest that the amplitude is largest at m = 2 and decreases for m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Crucially for the problem at hand (a disc+halo system), we have no analytic predictions for the modal spectra, owing to the complexity of approaching such a problem analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE+mSSA gives us a means to detect these modes amongst a sea of other signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Model Galaxy 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Simulation Overview We design an isolated model Milky-Way-like galaxy for our study of the compressive power2 of BFE and the dynamical information one can extract with mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We draw the model from components in the merger simulation of Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2018): a Hernquist profile dark matter halo with a mass of 1012M⊙ and a scale length of 52 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' an exponential stellar disc with a mass of 6×1010M⊙, a scale length of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 kpc, and a sech2 scale height of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='53 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' a Hernquist stellar bulge with a mass of 1010M⊙ and a scale length of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The halo has 40×106 particles, the disc has 5×106 particles, and the bulge has 106 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Unlike Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2018), we do not in- troduce a satellite perturber so that our model galaxy evolves in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The initial circular and radial frequency curves in 2 Here, ‘compression’ refers to the amount of information one needs to store.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A straightforward metric is the total computer disk space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We provide specifics to our simulation, but the scale of com- pression should be similar in other simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' the disc plane are shown in Figure 1: as we shall see below, we are able to use these frequencies to inform our mSSA analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We evolve the model with Gadget-4 (Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2021) for 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='49 Gyr, saving snapshots every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='01 Gyr, for a total of 549 snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The total simulation requires approximately 800 GB of computer disk storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 BFE representation To compactly describe the simulation, we represent each com- ponent in each snapshot with a BFE designed to provide compression and create a continuous representation from the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Further information regarding the BFEs used may be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In a BFE, a target distribution is represented as the linear sum of some chosen basis functions, with weighting on each of the basis functions (coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' If the basis functions are selected well, the distribution will be described by a small number of functions and correspond- ing coefficients, Cµ, where µ is a tag that indexes each basis function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The coefficients then are a measure of the impor- tance of each basis function to representing the overall dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To facilitate representing the distribution with the smallest number of functions, we choose expansions whose lowest-order function resembles the target equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For a principally two-dimensional structure, the stellar disc, we use a Fourier-Laguerre expansion3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The Fourier- Laguerre basis for expanding disc surface density was intro- duced in Weinberg & Petersen (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Given the exponential weighting of Laguerre polynomials, they serve as a natural radial basis element for exponential discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' If the scale lengths are chosen to match, the equilibrium disc is well-represented by the lowest-order Laguerre polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The scale length of our Fourier-Laguerre expansion is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 kpc, matching the scale length of the modelled disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To capture angular struc- ture, we expand in Fourier terms cos φ and sin φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We in- dex the Fourier azimuthal with m, and the Laguerre radial terms with n, creating (2m − 1) × n total coefficients, each tagged with a unique (m, n), written Cmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We find that as expected, C00 dominates by multiple orders of magnitude as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We expand the disc to mmax = 6, nmax = 6, making 2 × (mmax + 1) × nmax = 84 coefficients for the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The choice of maximum radial order is motivated by a desire to probe specific spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The n = 6 radial Laguerre den- sity function has nodes at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='8, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7, and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='9 kpc, thus ensuring that the majority of the nodes are within 18 kpc of the disc centre (where 90% of the particles are located).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The dark matter halo4 is efficiently described through the empirical orthogonal function basis approach introduced in Weinberg (1999) and most recently updated in Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3 Another option is presented in Weinberg & Petersen (2021): the use of 3d basis functions designed to resemble the exponential disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this work, we use the 2d Fourier-Laguerre expansion owing to the straightforward generalisation to the expansion of velocity fields, which will be the subject of future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 4 We also tested bulge expansions, using a similar basis to the dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Tests indicated that information contained in the bulge basis was redundant with the dark matter halo: this makes sense for two spherical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Therefore, we omit the bulge expansion from the analysis in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 10 Conl n 一 (detrended) n 0 10 Coefficient Amplitudes Cin 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0 2 3 5 Time (Gyr)4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' mSSA DFT peak contrast SV name decomposition PCs (Gyr−1) (R < Rd) fraction Disequilibrium Signal 1: halo profile readjustment (slow decay) Group m0-1 disc m = 0 0,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='641 Group l0-1 halo l = 0 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='944 Group m0l0-1 disc m = 0, halo l = 0 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='832 Disequilibrium Signal 2: phase mixing of disc initial conditions (fast decay) Group m0-2 disc m = 0 2,3,4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='084 Group l0-2 halo l = 0 4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='028 Group m0l0-2 disc m = 0, halo l = 0 4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='037 Group m1-3 disc m = 1 4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='057 Group m2-1 disc m = 2 0,1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='201 Group m4-1 disc m = 4 0,1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='086 Group m6-1 disc m = 6 0,1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='036 Group m2m4m6-1 disc m = 2, 4, 6 0,1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='072 Group m1l1-3 disc m = 1, halo l = 1 6,7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='040 Group m1m2l1-2 disc m = 1, 2, halo l = 1 2,3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='082 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Summary of two different signals identified in our mSSA decompositions as associated with initial disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The first signal results from halo disequilibrium, and the appearance in the disc is primarily manifest in the central surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The second signal is present in myriad decompositions, but appears to be seeded first by disequilibrium in the disc m = 0, which then persists in other harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Disc feature strengths are reported in surface density to give a measure of ‘visual contrast’, defined as max (|∆Σ|) within a disc scale length (see equation C9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Contrasts have an approximate error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='001, estimated from grid size adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Owing to simulation sampling rates (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='01 Gyr), the DFT peak is only accurate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Beginning with the equilibrium distributions, we de- sign a 1d radial model that matches the initial spherically symmetric density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' From this one-dimensional model, we construct an empirical orthogonal function basis whose lowest-order member perfectly matches the input initial den- sity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Higher-order terms are generated as eigenfunc- tions of the Sturm-Liouville equation with the input equi- librium potential-density model and appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The three-dimensional structure of the spherical components is described by a spherical harmonic expansion in the angular coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each term in the expansion is rep- resented by three numbers: the spherical harmonic indices ℓ and |m| ≤ ℓ and the index of the radial basis function n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In total, we have (ℓmax + 1)2 × nmax coefficients per snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For the halo, we expand to ℓmax = 2, nmax = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The expan- sions, for the entire simulation, only require approximately 12 MB of storage: a more than 60000× compression, with the benefit of encoding the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In practice, we will often consolidate the same-integer positive and negative spherical harmonic m indices when describing the coefficient ampli- tudes such that a quoted (ℓ, m) tag contains both ±m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As expected, the Cℓmn = C000 term is the largest by multiple orders of magnitude, with C generally decreasing as either (ℓ, m) or n increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3 EVOLUTION OF A NEAR-EQUILIBRIUM GALAXY Our isolated disc+bulge+halo galaxy was constructed to be in a completely stable equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, the model is not in equilibrium, for reasons both physical and unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 2 shows the raw BFE coefficients for the low-order disc harmonics derived from the simulation snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While it is clear that the coefficient time-series are noisy, inspection by eye suggests that there exists lower frequency coherent signals buried in the higher frequency noise: early evolution in m = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' modestly elevated power at late times in m = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' and a periodic signal in m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To explore dynamical evolution in our simulation, we per- formed mSSA decompositions of various combinations of BFE coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These decompositions revealed clean, per- sistent features in the individual low-order disc harmonics (m = 0, 1, 2), which we concentrate on understanding in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We also augment the analysis of the low-order disc harmonics with mSSA analysis of halo coefficients, joins of disc and halo coefficients, and higher-order disc harmonics (m > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These multi-component mSSA analyses prove to be the most fruitful in identifying the causes of different fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The full results of all our analyses are presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 describes how the results of the mSSA analysis can be used to group coefficients into separate dynamical fea- tures, characterise the properties of these features and come to a physical understanding of their nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The following subsections illustrate these ideas by dividing our own anal- ysis of the disc+bulge+halo simulation into three classifica- tions: initial conditions disequilibrium (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2), secular evolution signals (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3), and fluctuations and other uninterpretable features (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Interpreting the results of the mSSA analysis We use several diagnostics (denoted below in slanted text) to describe the character and understand the nature of the features identified in the mSSA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each diagnostic has a corresponding section in Appendix C describing the math- ematical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Applied to BFE multiple series, mSSA identifies temporally correlated signals in the BFE coefficients series as an ensem- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Briefly, mSSA uses the autocorrelation of time lagged ma- trix of the input series and performs an eigenanalysis to find MNRAS 000, 000–000 (2023) Near-equilibrium disc-halo dynamics 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' An analysis of two monopole signals resulting from distinct sources of initial disequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The left panels show the recon- structed coefficient amplitudes over time for each signal (identified as Groups 1 and 2 in both disc-only, halo-only, and disc+halo analyses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The right panels show the power spectra of the reconstructed coefficients for each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The first signal is a slow rearrangement owing to the halo settling in the presence of the disc, manifest by eye in the disc primarily as a change in the central surface density (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We show the appearance of this signal in the disc and halo as the upper two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The second signal is ringing in the disc resulting from the initial velocity disequilibrium of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While the signal decays rapidly in the monopole component, the disequilibrium seeds long-lasting persistent periodic features in other harmonics: see entries under ‘Disequilibrium Signal 2’ in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We show the appearance of this signal in the disc and halo as the lower two rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In each left-hand panel, we show two thicknesses of curves: the thick lines are for the components when analysed separately and the thin lines are for the components when analysed jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' That the different thicknesses of lines, for the same radial order, are not particularly different, is strong evidence that the features are correlated between the disc and halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' mSSA DFT peak contrast singular value name decomposition PCs (Gyr−1) (R < Rd) fraction Point Mode 1: slow growth Group m1-1 disc m = 1 0,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='201 Group l1-1 halo l = 1 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='272 Group m1l1-1 disc m = 1, halo l = 1 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='244 Point Mode 2: slow decay Group m1-2 disc m = 1 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='064 Group l1-2 halo l = 1 4,5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='035 Group m1l1-2 disc m = 1, halo l = 1 4,5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='048 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The coupled disc+halo dipole modes appearing in different mSSA decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Both modes appear in multiple mSSA decompositions, and that they both appear in disc-only, halo-only, and disc-halo decompositions strongly suggests that they ar both joint modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In the table, disc harmonics are denoted with m, halo harmonics are denoted with l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Columns are the same as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' dominant trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each time series is detrended by its mean and variance to intercompare the variations in each coeffi- cient series with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These eigenvectors describing these trends are usually called principal components (PCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As we always find multiple PCs contribute to a single dynamical feature in our analysis (see ‘PCs’ column in Tables), we will refer to each feature as a ‘Group’ (of PCs), labelling the strongest group (ordered by PC variance) as the first group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We also denote the particular decomposition by the input coefficient har- monic in the group name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For example, the strongest group in the m = 0 disc analysis will be labelled ‘Group m0-1’, and the strongest group in the l = 1 halo analysis will be MNRAS 000, 000–000 (2023) 420２ Group m0-1 (disc) Group m0-1 (disc) n=0n=4 1 n= 2 n=6 n=3 4 Reconstructed Coefficients (detrended) 0 ４２０２ Group /0-1 (halo) Group /0-1 (halo) Reconstruction DFT (normalised) 4 Group m0-2 (disc) Groupm0-2(disc) 20 2 4 0 4 Group /0-2 (halo) Group /0-2 (halo) 20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 0 5 10 15 Time (Gyr) Frequency 1 Gvr6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Disc monopole (m = 0) surface density as a function of radius and time, computed from the full coefficient series (up- per panel), showing a largely featureless disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The surface density has been normalised by the central surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The remaining panels show the contribution to the surface density deviations for two groups of m = 0 principal components, identified as two dis- equilibrium signals (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The surface density deviations are computed relative to the m = 0, n = 0 background, and are of the order a few per cent (excepting the outer disc, where the low densities mean a variations naturally result in a larger per cent variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' labelled ‘Group l1-1’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As PC groups capture trends in basis function coefficients that are correlated over snapshots, PC groups capture how spatial features dynamically evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Mathematically descriptive (but often difficult to interpret beyond the most significant few), the mSSA decomposition returns singular values (SVs) as measurements of the contri- bution of each PC to the total decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Larger SVs indicate which PCs represent more of the net change in time of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This property greatly helps the robust identification of features that represent true dynamical evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' PCs which correspond to random fluctuations due to (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=') numerical noise are by nature uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' They have very low SV even as they may be the dominant source of variations in the surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Conversely, PCs which de- scribe evolution in coefficient series that are coherent over time will have high SV even though they may be (orders of magnitude) below the inherent noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We report the singular value fraction5 attributable to a given group in the Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We examine the coefficient reconstructions from a group of PCs for physical insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' From the coefficient reconstructions, we can also construct power spectra from a Discrete Fourier Transform (DFTs) of the reconstructed coefficients from a group of PCs give insight into frequencies (and time scales) 5 To compute the relative contribution, we normalise each singular value corresponding to a particular principal value to the sum of all singular values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Then, we can say that some per cent of the signal is represented by the principal component (or group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We will call this the contribution of a principal component (or group), and may be interpreted as a measure of signal robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' that characterise the time evolution of a feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Approx- imately equal values of dominant frequencies in the power spectra of the coefficient reconstructions between different PCs from mSSA of the same component suggest they are describing different aspects of the same feature and may be grouped together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' If equal values occur across different com- ponents they may be mutually interacting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' See the ‘DFT peak’ entry in Tables, which reports the frequency value where the DFT is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We can also calculate contrast in the disc from the recon- structions6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Calculating the average of the fractional devi- ation in surface density within one disc scalelength gives a measure of the ‘detectability’ of a feature (by eye or algo- rithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' See the ‘contrast’ entry in Tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Related, the in- ferred location in the galaxy is where the dominant frequen- cies found in the power spectrum match the circular velocity of the unperturbed galaxy can indicate the spatial scales of any interactions taking place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Refer to Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In general, identified features evolve as one of the following types of evolution (noted in Tables): decaying, where a fea- ture peaks at the beginning of the simulation and decays in importance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' growing, where the feature grows and then satu- rates in amplitude with later maximum times therefore hav- ing slower growth rates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' or consistent with no evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' By comparing the evolution type across different components, one may also infer causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The relative growth or decay may indicate when one component is driving another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Initial Conditions Disequilibrium Uncovered Through Disc m = 0 Analysis We start our investigation with perhaps the most striking feature in the raw coefficients apparent in the top panel of Figure 2 which shows the evolution of the m = 0 (monopole) disc coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The figure suggests the simulation suffers from a disequilibrium that is typical in disc-halo initial con- ditions: outwardly propagating rings in surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This section reports the insights into this apparent evolution af- forded by mSSA, starting from its application to the m = 0 disc coefficients alone (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The properties of the features identified in this preliminary analysis provide a template for further applications of mSSA both to the halo (separately and combined with the disc, see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2) and higher order disc terms (see 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Table 1 summarises the properties of all these analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Grouping into Dynamical Features The mSSA analysis of the m = 0 disc reconstructed coeffi- cients reveals that PCs (0,1) and PCs (2,3,4,5) had distinct power spectra, suggesting natural groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This also sug- gested the presence of two distinct dynamical features with the signal in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The properties of these two groups that are quoted below are summarised in Table 1, with the rows labelled ‘Group m0-1’ and ‘Group m0-2’ corresponding to this first mSSA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Two more figures illustrate our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 4 shows the 6 We do not look at the contrast in the halo, as this is not straightforwardly measured in real galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Therefore, the con- trast columns do not contain entries for halo-only mSSA analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 20 0 UnprocessedCoefficients (Z/Zo) 10 2 601 4 Radius (kpc) 5 Group m0-1 △Z/Zo (per cent) 10 20 Group m0-2 10 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 Time (Gyr)Near-equilibrium disc-halo dynamics 7 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' An analysis of two groups obtained from the disc-only m = 1 mSSA decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each group corresponds to a distinct point mode, discussed in the text as ‘Mode 1’ and ‘Mode 2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The left panels show the reconstructed m = 1 coefficient amplitudes over time for Groups m1-1 and m1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The right panels show the power spectra of the reconstructed m = 1 coefficients for each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Both modes have well-defined slow patterns – significantly slower than any frequency associated with stars in the disc – and show evolving behaviour: the first mode is unstable and grows with time, while the second mode is damped and decays with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The mode summaries are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Normalised face-on (x, y) disc surface density deviation determined for two groups in the m = 1 decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each group corresponds to a distinct point mode, discussed in the text as ‘Mode 1’ and ‘Mode 2’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The panels shows a reconstruction of snapshots for either Group m1-1 (upper row) or Group m1-2 (lower row) in the disc-only m = 1 decomposition (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Both groups are retrograde with respect to the disc rotation (rotation direction of the pattern is marked with an arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The mode shown in the upper panels grows in amplitude over the course of the simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' the mode shown in the lower panels decays in amplitude over the course of the simulation, evident from the surface density features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Neither pattern strongly winds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' both are a largely self-similar evolution, despite being fairly tightly wound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' amplitude (left hand panel) and DFTs of the coefficient re- constructions for Groups m0-1 and m0-2, revealing their dis- tinct temporal characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In Figure 3, we show the m = 0 surface density amplitude reconstruction as a function of disc radius (y-axis) and time (x-axis) from the unprocessed coef- ficients (top panel), as well as the surface density deviations relative to a smooth monopole background, constructed from the two m = 0 PC groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Overall, we find the following characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Group m0-1 represents a dynamical feature that shows weak evolution over the entire simulations with a surface den- sity contrast of approximately 3 per cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The slow decay of Group m0-1 produces power at a range of very low fre- MNRAS 000, 000–000 (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 10 Group ml-1 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 5 △Z/20 (×10-2) (kpc) 10 =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 10 Group ml-2 y 5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 5 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 10 0 10-10 0 10 -10 0 10 -10 0 10 x (kpc)Group m1-1 Group m1-1 n=0n=4 Reconstructed Coefficients 2 n=1 n=5 2 n= n三6 Reconstruction DFT 0 n=3 (detrended) (normalised) 2 0 Group m1-2 Group m1-2 2 0 2 1 0 0 1 2 3 4 5 0 2 4 6 8 10 Time (Gyr) Frequency (r)8 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Amplitude and phase as a function of radius and time for the disc-only m = 1 decomposition for the first two groups iden- tified in the mSSA analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each group corresponds to a distinct point mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' From top to bottom, we show the amplitude and phase for the unprocessed m = 1 coefficient streams, the reconstructed coefficients of Group m1-1, and the reconstructed coefficients of Group m1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The density is shown as the log of the absolute value of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Both groups show coherent phases identifiable in the seemingly random phase information of the unprocessed co- efficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The growing (decaying) nature of Group m1-1 (Group m1-2) is also evident in the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Description of the strongest principal component group for halo and disc decompositions: a growing multi-component point mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The upper panel shows the detrended and normalised am- plitude of the reconstructed cosine component of the m = 1 (disc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' grey curves) or l = 1 (halo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' black curves) n = 0 coefficient ver- sus time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The solid curves are for mSSA decompositions run on each component alone (Group m1-1 and Group l1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The dashed curves are for the joint halo+disc mSSA decomposition (Group m1l1-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The lower panel shows the power spectrum (DFT ampli- tude vs frequency), for the four series shown in the upper panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The relative similarity of the curves and power spectra suggests that the patterns are correlated between the disc and halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The slow growth of the disc amplitude over time relative to the larger halo amplitude at the outset of the simulation suggests that the halo is responsible for driving the mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' quencies, peaked at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2Gyr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Group m0-2 shows outwardly propagating rings in surface density that start at the beginning of the simulation and dis- appear after ≈ 1 Gyr, losing speed as they move to larger radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While this is a sub-1 per cent effect within a disc scale length, at larger radii, the surface density deviation is obvi- ous by eye as ringing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The periodic nature of Group m0-2 corresponds to a frequency peak at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4Gyr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We conclude that mSSA has cleanly separated two distinct evolutionary processes operating simultaneously within one harmonic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The next two subsections explore the nature of both of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Group 1: Halo-driven disequilibrium?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The appearance of Group m0-1, at low frequency, suggests that its origin may be connected to the halo, where timescales are naturally long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Specifically, the frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Gyr−1 cor- responds to a circular orbit at R ∼ 50 kpc (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This motivated us to apply mSSA to the l = 0 coeffi- cients representing the halo component in the simulation to explore this connection further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We run analyses of both the halo l = 0 alone and in combination with the disc m = 0 coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The results of the analysis of the halo alone is shown in lower panels of Figure 3 and summarised in the second row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These demonstrate that the readjustment of the halo component’s radial profile is even more significant than MNRAS 000, 000–000 (2023) Reconstructed Amplitude (detrended,normalised) Group m1/1-1 (cosine only) 0 1 1 0 1 2 3 4 5 Time (Gyr) Reconstructed DFT solid:individual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='dashed:correlated (normalised) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 black:halo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='grey:disc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 0 1 2 3 4 Frequency20 UnprocessedCoefficients 20 [≥/60 4 28 Group m1-1 Radius (kpc) 5 20 TT phase (radians) 20 Group m1-2 20 T 0 0 2 4 Time (Gyr)Near-equilibrium disc-halo dynamics 9 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Normalised face-on (x, y) halo z = 0 plane density deviation reconstruction snapshots for Group m1l1-1 (upper panels) and Group m1l1-2 (lower panels) in the halo-and-disc l = 1 + m = 1 decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each group corresponds to a distinct point mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The patterns extends to large radii in the halo and are retrograde with respect to the disc rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The halo reconstructions exhibit significantly less ordered behaviour compared to the disc owing to the three-dimensional nature of the mode, which also tips relative to the z = 0 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, the bulk properties are similar to the disc (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The mode summaries are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' That the joint decomposition of the halo and disc returns the same groups, with similar behaviour, is strong evidence for the mutual mode nature of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The large spatial scale of the modes in the halo, coupled with their relatively early coherence, is suggestive that the modes are induced by the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' the disc radial profile, with a signal amplitude twice as strong as the disc (compare detrended amplitudes in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Such halo-driven disequilibrium is also a common feature for nu- merical realisations of multi-component galaxies as their com- bined equilibrium properties have been approximated, for ex- ample through Jeans modelling or adiabatic contraction cor- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Thus the mass distribution of the halo adjusts to full equilibrium in the presence of the disc, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In Table 1, a comparison of rows 1 (analysis disc coeffi- cients alone), 2 (halo coefficients alone) and 3 (disc and halo coefficients combined) confirms: (i) all three mSSA analyses have similar temporal structures, corresponding to the dy- namical timescales at several tens of kpc in the system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (ii) the joint disc/halo analysis actually identifies the same co- herent features in the disc and at greater contrast (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='054 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='031) than the disc analysis alone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (iii) that the driver for the combined evolution is likely the halo given the larger ampli- tude of its coherent changes relative to random fluctuations for that component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The above results demonstrate the ability of mSSA to suc- cessfully identified the mutual readjustment of the coupled disc-halo system from a mild disequilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3 Group 2: Disc-driven disequilibrium The strength of Group m0-2 in the analysis inspired an in- vestigation as to whether this disequilibrium could also seed other features in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Examination of other mSSA decompositions for different coefficient combinations finds many similar-frequency signals (see lower rows of Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Even disc harmonics (m = 2, 4, 6) show a persistent signal in the most important PCs (0 and 1) with a pattern speed of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3 cycles/Gyr that is equal to the half the Group m0-2 frequency peak of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 cycles/Gyr7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Note that the joint analy- sis of all even disc harmonics (m = 2, 4, 6) returns essentially the same results as the m = 2 only decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In the case of harmonic orders m > 2, this result likely owes to the need for higher order harmonics to fully represent the feature being described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The remaining rows of Table 1 demonstrate that the Group m0-2 disc disequilibrium signal is also evident at a lower level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' higher PC numbers, lower contrast in the disc and smaller SV) in both the disc m = 1 and halo l = 1 decom- positions when comparing frequency structure of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While the peak surface density deviation is near the outset of the simulation for m = 0, in higher harmonic orders the sig- nal does not completely fade over the simulation, with peak measured contrasts coming at later times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Our findings show the utility of mSSA in detecting evolution incited across dif- ferent harmonic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 7 The pattern speed of a harmonic is the number of cycles per Gyr divided by the harmonic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' That is, the pattern speed of the disc-only decomposition of Group 2 m harmonic coefficients is Ωm = ΩDFT/m cycles/Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 Group /1-1 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 Ap/po (×10-2) 20 (kpc) [=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='8 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 T=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 Group /1-2 y 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 25 0 25 25 0 25 25 0 25 25 0 25 x (kpc)10 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 Key insights In this section, BFE+mSSA has been used to increase our understanding of a dynamical simulation by: (i) separating distinct evolutionary pathways within a single harmonic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (ii) identifying coupling between multiple components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (iii) detecting features across different harmonics within a single component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These results emphasise that initial conditions for near equilibrium studies of galaxy evolution need to be dynam- ically relaxed (or virialised) by evolving in isolation for tens of halo dynamical times (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' much longer than than the equivalent timescale in the disc) prior to an studies of in- teractions in order to truly isolate signatures of the external perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While the perturbation in our study is a nu- merical artifact, the distinct adjustments to density profiles and couplings within and across components uncovered by BFE+mSSA represent the drivers of the evolution of galax- ies seeded by any perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3 Secular evolution signals Uncovered Through Disc m = 1 Analysis Examination of the PCs from the mSSA decomposition of the dipole disc harmonic (m = 1) revealed two groups, with prop- erties summarised in Table 2 and contributing coefficients and power spectra visualised in the left and right panels of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Examination of the power spectra show that these features are distinct in nature to the disequilibrium-seeded m = 0-dominated Groups m0-1 and m0-2 described in the previous section in that they have clear, well-defined frequen- cies, rather than a broad spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This indicates that each of these groups may be a point mode present in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1, point modes are a result of the fun- damental properties of the underlying phase-space distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' They have single-valued real and imaginary frequencies (hence the descriptive point) that describe the periodicity and growth or decay of the features they support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These modes drive secular, self-sustained evolution distinct from that of a transient response to an external driver (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' the disequi- librium initial conditions in the previous section) that phase mixes away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Hence we refer to these groups as ‘Mode 1’ and ‘Mode 2’, and examine their nature in the following subsec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In the disc m = 1 analysis, these are Groups m1-1 and m1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Appearance of modes in the disc We augment the information about the two modes sum- marised in Figure 5 and Table 2 with visualisations of their appearance in Figures 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 6 shows selected face- on disc surface density reconstructions to demonstrate that both modes create spiral patterns that are retrograde relative to the rotation of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 7 illustrates the radial (y- axis) and time (x-axis) evolution of the surface density (upper panel in each pair) and phase over (lower panel in each pair) for the full time sequence, indicating both the growth/decay and periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Inspection of these figures and the table pro- vide the full characterisation of the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Mode 1 groups m = 1 PCs 0 and 1, reconstructing a slowly rotating, growing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Referring to Figure 1, the frequency of the signal (Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 cycles/Gyr) is located near the scale radius of the halo, well outside the disc8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Mode 1 grows significantly in amplitude over the simulation, with the peak surface density signal coming near the end of the simu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Computing the contrast in the outer, low-density disc (r > 12 kpc), the surface density deviation amplitude reaches 10 per cent, detectable as lopsidedness in deep imaging of disc galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Mode 2 groups m = 1 PCs 2 and 3, reconstructing a slowly rotating, slowly decaying mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The frequency of the signal (Ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7 cycles/Gyr) is located closer to the Galactic centre, but also beyond the bulk of the disc mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Mode 2 decays from the outset of the simulation, and is significantly weaker than the first mode, with a peak contrast of order 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 per cent within a scale length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Connection between the disc and halo Since the frequencies of the two modes are consistent with halo frequencies we naturally suspect that the halo is sup- porting the modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To test this, we perform additional mSSA decompositions: first with the l = 1 halo coefficients alone, and then with the l = 1 halo coefficients jointly with the m = 1 disc coefficients9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The results of the runs are sum- marised in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We find sets of PCs in the halo-only decompositions corresponding to Modes 1 and 2, which we associate by means of their similar frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We also find corresponding PCs in the joint disc-halo decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The joint analysis in particular suggests that the modes are multi- component in nature, owing to the similar properties between all decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 8 provides an example visualisation for a single ra- dial coefficient (n = 0) contributing to Mode 1 to verify this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Comparing the coefficients reconstructed from identified in the independent analyses of the disc and halo (solid lines), as well as the joint disc-halo decomposition (dashed lines), we find the same features are identified in both the combined and independent analyses: the curves in the upper panel of Figure 8 are unchanged whether the decom- position is performed on a per-component basis, or jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This implies that the same principal component can describe the evolution in both the disc and halo, and that the sig- nal is strong enough in both components to be identified in per-component analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This is a strong indication of a cor- related multi-component signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In general the same features will not be recovered from combined analysis of different com- ponents because the inter-component decomposition need not match the intra-component decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In contrast, our joint analysis finds a single PC group may be used to recon- struct the modes in both the disc and the halo, identifying them as a mutual mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 8 For m > 0 harmonics, PC groupings frequently occur in pairs that describe both the amplitude and phase of a feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In the left hand panels of Figure 5 only the cosine terms in the coefficients are plotted to allow the reader to infer both amplitude and periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 9 To find correlated features between the halo and disc we choose halo coefficients that can describe features with meaningful projec- tions into the disc plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To this end we choose only the Y m l = Y 1 1 terms of the halo expansion, excluding the Y 0 1 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In addition we use the same number of coefficients from each component to avoid introducing the prior of unequal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) Near-equilibrium disc-halo dynamics 11 For both modes, we can examine and compare timescales and amplitudes to try to understand the driver of the evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Comparing between components, the feature strength is higher in the halo at earlier times in each mode (of order 1% density contrast in the halo, but well below that in the disc), implying that the halo is responsible for starting each mode at large radii (compare Figures 6 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For the grow- ing Mode 1, estimating the growth rate from the modulus of the coefficients at early times also reveals the growth of the halo feature to be twice that of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The saturation point of the halo is also measurably earlier than the disc (T = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Gyr in the halo vs T = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Gyr in the halo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The comparison of the disc and halo features in the pre- vious paragraph suggest that the modes may arise from a fundamental dynamical property of the halo component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 9 shows snapshots of the halo feature during the simulation at times corresponding to Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The fea- tures are both slow retrograde pattern which build and/or damp over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' They bear hallmarks – a slow dipole pat- tern at relatively large scales – of the weakly damped l = 1 modes in spherical systems that have been studied in using linear perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These were first identi- fied by Weinberg (1994), and later additionally reported by Heggie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2020), Fouvry & Prunet (2021), and Weinberg (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We conclude that BFE+mSSA has allowed us to detect and characterise slow, secular evolution of our isolated simulated galaxy due to the nature of the underlying equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3 Key insights The results in this section provide additional illustrations of the ability of BFE+mSSA to separate evolutionary pathways in a single harmonic and to detect coupling across compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Most significantly, BFE+mSSA allowed the detection of slow, low-level secular evolution in our simulation that had been predicted in analytic work, (Weinberg 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Fouvry & Prunet 2021) and recently observed in star cluster and dark-matter-halo-only simulations (Heggie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Weinberg 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The analytic work suggests that spherical systems, such as dark matter halos, generically exhibit dipole point modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The common existence of these modes has im- portant implications for understanding lopsidedness in galax- ies: the halo and disc mutually open dynamical avenues that cannot be taken by either component independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' there- fore many dynamical features are simply inexplicable without an understanding of the interplay between components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' How- ever, making a clear connection between the theory and ob- served galaxies has been hampered by the technical challenge of applying analytic work to multi-component systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' More- over, while numerical simulations routinely represent multiple component systems, the description of the results is typically limited to visualisations and statistical analyses that can only qualitatively be connected to dynamical drivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE+mSSA has bridged this gap by clearly showing an l = 1 mode in our simulated halo driving lopsidedness in our simulated disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These results speak to the promise of BFE+mSSA for forging the missing connection between the- ory, simulations, and observations needed to interpret galac- tic properties in terms of our fundamental dynamical under- standing secular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 Fluctuations and other uninterpretable features In the two previous sections, we identified interpretable sig- nals in various harmonics of both the disc and halo coeffi- cients in groups of low-order PCs using BFE+mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' How- ever, inspection of the last column of Tables 1 and 2 shows that these PC groups only contain a fraction of the total sin- gular values (which are normalised to total unity): most of the groups represent less than 20 per cent of the variance in the coefficients being analysed10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The rest of the signal spread over many (many!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=') higher-order PCs with lower SVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These are PCs with very weak self-gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We refer to these remaining terms as the nullity, owing to its uninterpretable nature: it will contain numerical noise, but may also contain signals too weak to be included in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To understand the properties of the nullity, we collect all uninterpretable PCs for a given mSSA decomposition and analyse their reconstructions, summarising the results for low-order disc harmonics in Figure 10 and for all decompo- sitions in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 10 shows the reconstructed coef- ficients and corresponding power spectrum for the PCs as- signed to the nullity for low-order disc harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Compar- ing this to the corresponding Figures 4 and 5 for lower or- der PCs, the difference is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The bottom panels for the m = 2 nullity do have hints of a signal in the form of low- level systematic evolution in the left hand panel and some clear peaks in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We discuss future strategies to hunt for weak signals in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, in general, there is a lack of periodic or systematic evolution in the left hand panels and flat spectra of frequencies in the right hand panel, characteristic of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A comparison of the contrast columns of Tables 3, A1 and 2 shows that the fluctuations in the surface density derived from the nullity are mostly stronger than the coherent signals in this particular simula- tion: our BFE+mSSA analysis has supported insights that would otherwise be inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 4 LOOKING AHEAD 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Essential Future Work - assessment of weak feature significance Our analyses of simulations of bar formation (Weinberg & Petersen 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022) and an isolated disc galaxy (this paper) amply illustrate the facility of BFE+mSSA to learn about both significant and expected as well as subtle and unanticipated dynamical evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The results are very promising for general applications to a wide variety of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, our work so far has been involved close supervision of BFE+mSSA to both interpret and understand the significance of what features it has identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In particular, the interpretative ambiguity we encountered in the higher order terms in this paper outlines the current limit of BFE+mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This limit motivates the need for a rig- orous statistical analysis of significance for mSSA-identified signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Many of the well-known approaches from statistical 10 The exception are some of the PCs associated with the monopole, which encode the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These PCs are respon- sible for upwards of 60 per cent of the singular value signal, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' An analysis of the content in the nullity for m = 0 (upper panels), m = 1 (middle panels), and m = 2) lower panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The left panels show the reconstructed nullity coefficient amplitudes over time for m = 0, 1, 2 (top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The right panels show power spectrum of the reconstructed nullity coefficients for each harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Both m = 0 and m = 1 show no discernible signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The m = 2 harmonic shows some periodicity, but the power spectrum suggests the frequencies are broad and not strongly coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Therefore, we are confident that we are not throwing away interpretable signal in the nullity in any harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These reconstructions may be compare to the unprocessed coefficients, Figure 2, for a quantitative analysis of what signals are part of coherent signal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' mSSA PCs DFT peak contrast SV decomposition (Gyr−1) (R < Rd) fraction disc m = 0 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='275 disc m = 1 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='678 disc m = 2 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='799 disc m = 3 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='936 disc m = 4 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='914 disc m = 5 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='954 disc m = 6 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='964 disc m = 1, 3, 5 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='933 disc m = 2, 4, 6 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='928 halo l = 0 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='028 halo l = 1 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='693 disc m = 0, halo l = 0 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='130 disc m = 1, halo l = 1 8+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='667 disc m = 1, 2, halo l = 1 4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='801 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Summary of principal components assigned the nullity in our decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We refer to each collection of PCs here as the ‘Nullity’, rather than a PC group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Disc harmonics are denoted with m, halo harmonics are denoted with l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Columns are the same as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' analysis would be suitable for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For example, let us take the hypothesis that the signal observed at m = 2, 3, 4 is consistent with background noise as a test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' That is, our null hypothesis is that our simulation can generate with the same properties of the signal in question without inherent self gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To do this, we need to generate a simulation with the same noise spectrum as the full simulation but without any self-gravitating features on the spatial and temporal scales of our putative signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Let us assume that we know how to per- form such simulations (we propose an exp-enabled approach below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' An ensemble of these null-hypothesis simulations can be run and analysed using mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' From the ensemble of sim- ulations, one may construct prediction intervals for singular values under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Then, if the singular value corresponding to the signal in question is beyond the pre- diction intervals, the corresponding principal component is considered significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In such a case, the signal can be re- liably reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This approach is often called Markov Chain SSA (MC-SSA, see Allen & Smith 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Analyses of this sort are particularly well-suited to the exp framework described in Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We can use the mSSA analysis to construct a realistic reconstruction of the coefficients series from the self-gravitating simulation with- out the self-gravitating features of interest by removing the groups corresponding to the signal in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In the study presented here, this would be akin to retaining only the nul- lity reconstructions of the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We can generate new MNRAS 000, 000–000 (2023) (detrended) m= 0 Nullity m = 0 Nullity (normalised) 0 2 Coefficients m = 1 Nullity m= 1 Nullity N 2 Reconstructed m = 2 Nullity 2 0 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 Time (Gyr) Frequency GyrNear-equilibrium disc-halo dynamics 13 coefficient series from an autoregressive model11 consistent with the coefficient covariance from the mSSA reconstruc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Then, exp allows initial potential fields from the re- constructed coefficients to be replayed for a new ensemble of particles with very little computational effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The result- ing expansion coefficient series are gathered automatically for analysis by mSSA, and can be analysed for significance of detected features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A detailed description of the MC-SSA approach in the exp context will be described in a later con- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Prospects for applications to simulations Despite the limitations, there are multitude of prospects for immediate, supervised applications of BFE+mSSA to simu- lations of galaxies, whether isolated, interacting or evolving in the full cosmological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Dynamical analyses of simulations of galactic evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Recent surveys (Majewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Steinmetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022) demonstrate that the Milky Way continues to evolve through satellite interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' N-body simulations have explained some key observational signatures (Laporte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Petersen & Pe˜narrubia 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Garavito-Camargo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Vasiliev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, interpretation of these simulations is challenging since many actors contribute simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The BFE+mSSA knowledge discovery ap- proach is capable of separating, characterising and dissecting the signatures of the mutual interactions of each component in simulations by separating features by correlating tem- poral and spatial scales non-parametrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE+mSSA promise detailed predictions and identification of features in current stellar data sets (see Petersen & Pe˜narrubia 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Garavito-Camargo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Lilleengen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022, for some recent results) and confident mapping the the dark matter halo’s global structure and distortions to that structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This goal was unimaginable even 5 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Structural characterisation and correlation of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This paper demonstrated the discovery of two-dimensional fea- tures in disc density resulting from internal (disequilibrium- related) dynamics and halo interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, BFE+mSSA can be applied to any field in any num- ber of dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For example, Weinberg & Petersen (2021) illustrated a three-dimensional disc BFE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The exp library already enables joint BFE+mSSA investigations of any number of three-dimensional density and potential fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These may be augmented by kinematic fields as in Weinberg & Petersen (2021) or some other field such as star formation rates and implied local metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' If the additional fields encode spatial information (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' they are BFE coefficients or even radial and azimuthal bins), their temporal and spatial scales will be correlated with the density and potential fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The BFE+mSSA can adapt to new observational tools and windows as new surveys become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 11 Autoregressive noise models are typically used for null hypothe- ses in MC-SSA because SSA provides good estimates for frequen- cies and exponential factors processes generated by the related lin- ear recurrence relations (Golyandina & Zhigljavsky 2013, Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Understanding of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' There have been many years of de- bate on the effect of noise in conclusions drawn from dynami- cal simulations, from bar-halo interactions (Weinberg & Katz 2007), through dynamical friction (Weinberg 2001), to satel- lite disruption (Errani & Pe˜narrubia 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE+mSSA clearly separates the correlated, quasi-periodic signals result- ing from dynamical interaction and coupling from the fluctu- ating forces resulting from finite particle number stochastic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We expect that couplings in orbital dynamics have frequencies near or smaller than the characteristic orbital frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Since the individual PCs describe the tempo- ral behaviour of components assigned to the noise field and the power spectrum describes their characteristic frequencies, mSSA provides a natural classification of signal and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Investigations of test-particle orbits with and without the noise component provide a diagnostic tool for the reliability of features in simulations and the role of fluctuations more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 5 CONCLUSIONS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Near-equilibrium evolution: the importance of multi-component modes We applied BFE+mSSA to a simulation of an isolated Milky Way like galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The BFE+mSSA combination allows us to automatically identify the main features in the model galaxy and their origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Most remarkably, BFE+mSSA achieved this in the challenging case of an isolated, multi- component galaxy that had specifically been constructed to not evolve and where the dynamical signatures were below the level of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Our work complements a prior investi- gation (Weinberg & Petersen 2021) which used BFE+mSSA to characterise significant evolution of known nature in a sim- ulation which formed a galactic bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In our near-equilibrium model, we identified – for the first time – two multi-component (disc-halo) dipole point modes (Figures 5-8) which evolve over time (one growing, one damped;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This discovery is enabled by the BFE+mSSA methodology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' such dynamical effects are at a level such that other methods, such as Fourier analyses, will not be able to recover the signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Halo modes are expected from linear perturbation theory (Weinberg 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Fouvry & Prunet 2021), and are observed in simulations of star clusters (Heggie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2020) and dark matter halos (Weinberg 2022), but the coupling of a spheroid to a disc has not been discovered to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The BFE+mSSA methodol- ogy makes the identification of point modes straightforward, and provides several avenues for corroboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We employed several different mSSA decompositions to validate our find- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The existence of point modes in these isolated simula- tions demonstrates the fundamental contribution of compo- nent interactions to the dynamical evolution of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We expect that the existence of such multi-component disc-halo modes is a generic feature of such systems, possibly including the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These modes likely have influence over the structural evolution of disc galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For instance, our results immediately suggest that low dipole modes will be most de- tectable at large radii (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' R > 20 kpc in the Milky Way), where density contrasts can exceed 10 per cent relative to a smooth disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 14 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In addition to point modes, we identified the long-lived results of initial conditions disequilibrium, resulting from in- dividual halo and disc disequilibrium features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Starting with rings encoded in the monopole m = 0 coefficients, we found correlations with many other harmonics, including a persis- tent m = 2 signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We uncovered an aphysical ‘settling’ of the halo in response to the presence of the disc at the outset of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Future work modelling idealised galaxies must take care to ensure that disequilibria in the initial con- ditions, and the resulting persistent features, are not treated as real dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Finally, we quantified the remaining signal that we classi- fied as the nullity, and put limits on the magnitude of un- explained surface density fluctuations in the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The desire to push even deeper in the decomposition of simulations mo- tivates the essential future work, but also inspires prospects for future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Dynamical Data Mining as the future of Galactic Dynamics Galactic Dynamics is a mature field with elegant descriptions of equilibrium systems, estimates for scales of the processes involved in the interactions that are known affect them and sophisticated analytic methods that describe evolution in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While we can understand the basic governing principles with detailed mathematical models from Hamilto- nian perturbation theory, the inter-component and environ- mental interactions that produce this morphology are hard if not impossible to study from modal analysis alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE methods both underpin our dynamical data mining tech- nique and are often used in analytic perturbation work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Thus they provide a natural bridge between theoretical work and numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The combination of BFE representa- tion of the possibly unknown dynamics in simulations with a machine-based knowledge acquisition tool such as mSSA allows for identification of couplings that may be too hard to predict otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This natural synergy between mathe- matical theory and simulation is the main motivation for our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Series of BFE can also be used to characterise ob- served fields in galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A galaxy’s picturesque morphological structure is a his- torical summary of its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Cosmological predictions for the frequency of galactic interactions explain the abun- dant signatures of disequilibrium observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The detail of our picture of disequilibrium is rapidly advancing, in terms of resolution in the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Hunt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022b) in galax- ies and occurrence rate in others (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Simulated realisations of galaxies in disequilibrium are simi- larly advancing in resolution and scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, the tools to take full advantage of this twin onslaught of data, simulated and real, are currently lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Such tools must be capable of modelling galaxies in disequilibrium, make quantitative and dynamically meaningful connections between simulated and observed galaxies and connect with analytic work in the lin- ear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As outlined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2, our results suggest the tremen- dous promise of BFE+mSSA for the field of Galactic Dynam- ics, with a myriad of envisioned applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Many of these applications can be undertaken now by adopting the super- vised learning approach to using BFE+mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These include detailed analyses of galactic components for galaxies in both isolated and cosmological settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Nor is there any reason to limit the BFE+mSSA analysis to galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' BFE+mSSA can be used for the characterisa- tion and dynamical evolution of self gravitating, interacting systems more generally and in any context, from binary aster- oids in the solar system (Quillen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022), through proto- planetary discs (Cadman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2021), to nuclear star clusters in the centres of galaxies (Fouvry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2022) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The remaining and key challenge to be solved is to un- derstand how to confidently assess the significance of all the features that BFE+mSSA recovers in an unsupervised way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Once this is developed it will be possible to broadly apply BFE+mSSA to large samples of systems: both simulated and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We warmly thank Chervin Laporte for sharing the initial con- ditions from his simulation with us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We acknowledge support from the Center for Computational Astrophysics (CCA) at the Flatiron Institute in the form of access to their compu- tational resources which allowed us to create our simulation and generate the associated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In addition, we thank CCA leadership and staff for hosting the Beyond-BFE collabora- tion meetings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We thank members of the B-BFE collabora- tion and the Dynamics Group at CCA for numerous conversa- tions during development of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MSP’s contributions were partially supported by grant Segal ANR-19-CE31-0017 of the French Agence Nationale de la Recherche as well as a UKRI Stephen Hawking Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' KVJ and AJ’s contribu- tions were supported by NSF grant AST-1715582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} 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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=', Katz N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=', 2007, MNRAS, 375, 425 Weinberg M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=', Petersen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=', 2021, MNRAS, 501, 5408 APPENDIX A: ALL MSSA COMBINATIONS TESTED Table A1 summarises all tested combinations in mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' All tests include all radial orders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' n ∈ [0, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' All mSSA de- compositions use a window length L = 250, which is ap- proximately half of the input time series (N = 549).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This creates sets of 250 PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The PCs are sorted by singular value magnitude, which does not guarantee that PCs with physi- cal similarity are consecutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Therefore, we determined the grouping of the PCs through direct examination, which in practice was straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The information in this table is repeated from Tables 1-3 in text, consolidated here and reorganized by mSSA decomposition for ease of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' APPENDIX B: BASIS FUNCTION EXPANSION DETAILS In this Appendix, we briefly describe the basis function ex- pansion technical implementation for completeness and clar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Full details may be found in Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2022) for the spherical expansions, and in Weinberg & Petersen (2021) for the Fourier-Laguerre disc expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' B1 Spherical expansions: Empirical Orthogonal Functions For an initially spherically-symmetric model (such as a dark matter halo or stellar bulge) we use spherically symmetric basis functions derived using the machinery in exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The po- tential and density take the form φnlm = φ0(r)unl(r)Ylm(θ, φ) (B1) ρnlm = ρ0(r)unl(r)Ylm(θ, φ) (B2) where unl are eigenfunctions determined by exp, Ylm(θ, φ) are the usual spherical harmonics12, and ρ0(r) and φ0(r) are input unperturbed model density and potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The model density and potential are typically chosen to be the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The functions unl(r) are eigenfunctions of a the Sturm-Liouville equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each function is a solution to the Poisson equation and has n nodes with increasingly tighter spacing (as n increases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The u00(r) function is a constant, which makes the ℓ = 0, n = 0 potential and density terms exactly proportional to the input model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Other terms are then perturbations on top of the model potential and density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The functions are biorthogonal, satisfying two conditions: � d3v φnlm ρn′l′m′ ∝ δnn′δll′δmm′ (B3) ∇2φnlm = 4πGρnlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (B4) The derivation of these functions is described in Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For the purposes of this work, the BFE representation of density is given by the projection of the coefficients onto the BFE: ˆρhalo(r, θ, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' t) = � l � m � n Almn(t)ρ0(r)unl(r)Ylm(θ, φ) (B5) where Almn(t) is the coefficient amplitude for a given function indexed by (l, m, n), and possibly is a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' An analogous expression may also be written for the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The ˆ· notation indicates that the quantity is reconstructed from the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' B2 Disc surface density expansion: Fourier-Laguerre A spherical expansion is not appropriate for a strongly flat- tened stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' While exp supports a three-dimensional 12 For ease of numerical implementation, exp uses the real spherical harmonics: Ylm,cos = 1 2 (Ylm + Yl−m) and Ylm,sin = 1 2i (Ylm − Yl−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Fourier visual singular mSSA peak contrast evolution value decomposition Group PCs (Gyr−1) (R < Rd) type fraction interpretation disc-only decompositions m = 0 1 0,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='031 slow decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='641 phase mixing of halo initial conditions 2 2,3,4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='006 fast decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='084 phase mixing of disc initial conditions nullity 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='005 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='275 m = 1 1 0,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='007 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='201 coupling with halo 2 2,3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='003 slow decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='064 coupling with halo 3 4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='002 fast decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='057 phase mixing of disc initial conditions nullity 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='008 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='678 m = 2 1 0,1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='006 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='201 phase mixing of disc initial conditions nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='017 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='799 m = 3 1 0,1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='005 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='064 projection of m = 1 Group 1 nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='012 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='936 m = 4 1 0,1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='004 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='086 projection of m = 2 Group 1 nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='914 m = 5 1 0,1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='001 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='046 projection of m = 1 Group 1 nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='006 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='954 m = 6 1 0,1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='001 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='036 projection of m = 2 Group 1 nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='005 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='964 m = 1, 3, 5 1 0,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='008 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='067 projection of m = 1 Group 1 nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='027 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='933 m = 2, 4, 6 1 0,1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='072 projection of m = 2 Group 1 nullity 2+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='037 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='928 halo-only decompositions l = 0 1 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 slow decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='944 phase mixing of halo initial conditions 2 4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 fast decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='028 phase mixing of disc initial conditions nullity 6+ no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='028 l = 1 1 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='272 weakly self-gravitating mode 2 4,5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 slow decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='035 weakly self-gravitating mode nullity 6+ no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='693 disc-halo decompositions m = 0,l = 0 1 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='054 slow decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='832 phase mixing of halo initial conditions 2 4,5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='007 fast decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='037 phase mixing of disc initial conditions nullity 6+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='021 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='130 m = 1,l = 1 1 0,1,2,3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='008 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='244 weakly self-gravitating mode and coupling 2 4,5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='003 slow decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='048 weakly self-gravitating mode and coupling 3 6,7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='002 fast decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='040 phase mixing of initial conditions and coupling nullity 8+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='667 m = 1, 2,l = 1 1 0,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='010 slow growth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='117 weakly self-gravitating mode and coupling 2 2,3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='003 fast decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='082 phase mixing of disc initial conditions and coupling nullity 4+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='012 no evolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='801 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Summary of different modes identified in our MSSA decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Disc harmonics are denoted with m, halo harmonics are denoted with l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Disc feature strengths are reported in surface density to give a measure of ‘visual contrast’, defined as max (|∆Σ|) within a disc scale length (see equation C9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Contrasts have an approximate error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='001, estimated from grid size adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Owing to simulation sampling rates (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='01 Gyr), the DFT peak is only accurate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The group name can be derived for each row by combining the decomposition harmonic(s) and concatenating with the group number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For example, the name of the first row would be Group m0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' empirical orthogonal function basis that may be used to rep- resent the potential and density, we choose in this work to project the disc to the two-dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We work in polar coordinates R and φ for a two-dimensional expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For the radial coordinate we use the Laguerre basis func- tions described in Weinberg & Petersen (2021), which were created to match the exponential profile of a typical stellar disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This choice minimises the number of required functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) Near-equilibrium disc-halo dynamics 17 The Laguerre polynomials are defined as: Gn(R) = 1 a√n + 1 exp � −R a � L1 n �2R a � , (B6) where L1 n is the associated Laguerre polynomial of order 1 and degree n, and a is the scale length of the disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Then G0(R) = 2 ae−R/a closely approximates the disc density and the majority of the reconstruction power resides in a single term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Moreover, the polynomials satisfy the the orthogonality condition � Gn(R)Gn′(R)RdR = δnn′, (B7) and thus Gn(R) can be used to reconstruct the radial struc- ture of the disc with a small number of expansion terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The azimuthal dependence is described with a Fourier se- ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Combining this with the radial Laguerre basis func- tions we obtain a set of two-dimensional basis functions Gn(R) cos mφ and Gn(R) sin mφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We call these Fourier- Laguerre functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The coefficients are naturally determined by the projection of f(R, φ) into the basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In our case we are accumulating discrete particles so rather than a continuous integral we obtain the discrete summations: Cmn = 1 2π � i f(Ri, φi)eimφiGn(Ri) (B8) where (Ri, φi) is the position of particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The reconstruction of the surface density field is then ˆΣdisc(R, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' t) = � m � n Cmn(t)eimφGn(R), (B9) where the coefficients Cmn(t) may have some time depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' APPENDIX C: MULTICHANNEL SINGULAR SPECTRAL ANALYSIS Singular Spectral Analysis (SSA) is a method of non- parametrically decomposing a time series into a sum of com- ponents that ideally capture different aspects of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Multichannel Singular Spectral Analysis (mSSA) extends SSA to include multiple series, such that one may identify co- herent signals between series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A full description and pedagog- ical examples may be found in Weinberg & Petersen (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this appendix, we briefly describe the important elements of SSA and describe the implementation used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C1 A conceptual introduction to mSSA In this section, we present a brief and intuitive introduction to mSSA through the conceptual relationship with principal component analysis (PCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' PCA characterises correlations within a data set by transforming to coordinates where most of the correlation is represented in a small number of di- mensions in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' mSSA itself is a generalisation of PCA: mSSA performs computations analogous to PCA on a grand trajectory matrix which is constructed to represent different variations in time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The extension of PCA to mul- tiple times allows for the decomposition of different series of samples of variables over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In our case, the variables are the individual BFE coefficients for each snapshot in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We can then simultaneously decompose struc- ture in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We refer the interested reader to Golyadina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (2001) for a more thorough description of SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 The goals of Principal Component Analysis The goal of PCA is the reduction of the data set’s dimen- sionality while retaining as much as possible of the variation present in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This is achieved by a linear transfor- mation or rotation to a new set of axes, the principal com- ponents (PCs), which are uncorrelated and ordered by their contribution to the total variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For a brief introduction, consider a data sample of N random variables with M chan- nels (or dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' PCA is mathematically equivalent to the eigen-analysis of the covariance matrix: C ∝ X⊺ · X (C1) where X is a M ×N matrix whose (i, j)th element is the data with zero-mean and unit variance for each of the M chan- nels: (xij − µi)/σi where µi, σ2 i is the mean and variance for channel i, respectively13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The sums over covariance between channels is effected by the matrix multiplication in equation (C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Then, the leading eigenvector or PC is the direction that maximises the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The next PC is the direction that maximises the variance, uncorrelated to the first, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Relationship of mSSA and Principal Component Analysis For our analysis, we have time series of N samples for each of M BFE coefficients that have been reduced to zero mean and unit variance: {ai,j : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' , M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='., N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each of the N samples corresponds to a phase-space snapshot in our simulation, providing a two-dimensional grid in coefficient channel (the i index) and time (the j index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A PCA anal- ysis of this matrix, as described above, identifies the sets of weighted coefficients (the principal components) which con- tain the highest proportion of the variance in each of the N temporal views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each of these sets represents a coherent spatial pattern that is present in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The mSSA algorithm adds the ability to simultaneously find dominant shapes in space that also evolve similarly in time14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For our case, this is done by constructing a new ma- trix where each row contains information from a sequence of L < N snapshots rather than a single one, effectively creat- ing a time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Each of K = N − L + 1 rows is of length M × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The ith row is a concatenation of the coefficients de- rived from snapshots i - i + L − 1, so each contains the same coefficients, but lagged by systematic amounts in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The resulting (M × L) × K matrix is known as the grand trajec- tory matrix (Ghil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Constructing grand covariance matrix and performing a PCA would now compare not one, 13 The covariance matrix is often normalised in PCA as C = 1 d X⊺ · X, where if the mean is determined from the data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=', the sample variance, we have d = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 14 SSA is a unique case where M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' That is, we only have one input channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this work, we advocate for use of mSSA specifically, but one could also apply the discussion here to SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) 18 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' but L snapshots at once to find repeating patterns at differ- ent lag times, effectively sliding a window of length L over the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The resulting PCs maximise the variance in the K = N − L + 1 overlapping views of the M time series simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For intuition, consider a set of coefficients that all have si- nusoidal variation with a single period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' At time lags that are multiples of the period, the variance will be large as the sig- nals reinforce each other coherently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' At incommensurate lags, the covariance will tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This allows mSSA to natu- rally find the coherent temporal signals in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For each oscillatory signal in our series, we will find a pair of eigen- vectors that represent the same frequency, just as in Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' A sinusoid was simply an example;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' the method will work any temporally coherent signal including exponential growth or decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' mSSA is purely non-parametric in this sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2 Details of mSSA implementation SSA is principal component analysis (PCA) of sequentially lagged L-length windows of a time series (where L is a user- specified window length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The key is that distinct features have different projections into L-lagged space and so PCA separates the different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' When working optimally on a dynamical system input, the different features will correspond to different dynamical phenomena, such as a galactic bar (as in Weinberg & Petersen 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The SSA procedure involves three main steps: embedding, singular value decomposition, and grouping/reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1 Embedding In this step of SSA one forms a matrix which represents the sequence of L-lagged windows of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This matrix is called the trajectory matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Consider an input time series ⃗s = {s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' sN} and window length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The trajectory matrix is formed as T = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 s1 s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' sN−L+1 s2 s3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' sN−L+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' sL sL+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' sN \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (C2) It is common to denote N − L + 1 as K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The anti-diagonals of T are equal by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Matrices with this property are called Hankel matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='2 Singular Value Decomposition: Principal Components and Singular Values After forming the trajectory matrix, the next step is to per- form a singular value decomposition (SVD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We begin by in- specting the dimensions of the trajectory matrix, T ∈ RL×K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To maximise computational efficiency, we construct the co- variance matrix such that it is min(K, L) × min(K, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Consider the SVD of the trajectory matrix T = UΛ1/2V⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (C3) The covariance matrix for L < K, C = TT⊺ = UΛU⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' When K < L, we can write the covariance as C = T⊺T = VΛV⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The choice in constructing the covariance matrix is to de- crease the computational complexity by choosing the rep- resentation with the smallest rank nullity15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We may then obtain the SSA principal components either directly from U ∈ RL×K, or as U = TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The SSA empirical orthogonal functions either come directly as V ∈ RL×K, or may be com- puted as V = (U⊺T)⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The singular values are given along the diagonal of Λ: that is, the ith singular value is σi = √λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We sort the singular values from highest to lowest and sort the columns of U and V accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Given the ith empirical orthogonal function Vi and a prin- cipal component Ui, we can form the matrix Ai = σiUiV ⊺ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Summed over all i, these matrices are the set of matrices which best approximate T and have orthogonal column/row spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this sense the SVD of C gives us a set of matrices which each carry independent information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We refer to the tuples (σi, Ui, Vi) as eigentriples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3 Grouping The various input coefficient series are often not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This, combined with sampling noise, may lead to informa- tion smearing between PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Of practical importance, there are often multiple eigenvectors and PCs that correspond to a particular dynamical signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The mSSA practitioner will need to group eigenvectors that describe the same or related sig- nals together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For example, even for a pure sinusoidal signal, the basic group will contain two eigenvectors, with identical eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The pair together describe the amplitude and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For typical features found in disc galaxies, such as arms and bars, the relevant groups often contain multiple eigen- vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' There are a variety of ways to motivate groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As a starting point, a group of PCs describing a single dy- namical features will often have similar eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Next, plotting the PCs over time by eye often reveals whether their evolution is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This can be quantified by (for example) taking a Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In particular, power spectra can clearly indicate which PCs are describing noise as they will have a broad frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The understanding of the noise can also be used to modify the window length to get better noise separation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' w-correlation matrices are a good grouping diagnostic (Weinberg & Petersen 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' These matrices quantify the correlation between PCs and can be used to guide group- ing by using the correlation as an indication of which PCs belong in groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Though we have listed two approaches to grouping, it is important to note that these alone do not es- tablish the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The grouping is determined by the content of the PCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We find grouping to be important primarily be- cause misgrouping can spread out the correlated signal and lead to an underestimate of a particular correlated degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' As an example, examination of PCs may indicate that two long duration trends are present in the same PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this case, the window length could be increased to separate those 15 The information content is the same in both, since the “wrong” choice simply increases the rank of the nullity which has no useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023) Near-equilibrium disc-halo dynamics 19 trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, it is also common that PCs may demon- strate the existence of multiple behaviours present in single PCs leading to a modification of the window length in the way described in the previous paragraph to achieve better separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The converse situation can also occur, where sin- gle processes are being split into many PCs, in which case the window length can be shortened to simplify grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The goal of grouping is a partition of the indices i into m distinct subsets such that each set of m subsets corresponds to a distinct feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The trajectory matrix reconstructed from select i indices is ˜T k = � i∈Im σiUiV ⊺ i (C4) where Im is a list of i indices in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' After segmenting the eigen-triples into groups, each ˜T k ideally corresponds to a distinct feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='4 Coefficient Reconstruction Given that the matrices ˜T k need not be (and almost certainly will not be) Hankel, we must convert them into the closest Hankel matrices that preserve the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The step of SSA in which this is done is referred to as reconstruction, resulting in the coefficient reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The final step is the reconstruction of the original series from the PC groups which is referred to as the reconstruc- tion stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In order for a trajectory matrix to unambiguously correspond to a time series it must be Hankel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' However, in general ˜T k need not be Hankel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' It is thus necessary to “Han- kelise” each ˜Tk by setting each anti-diagonal to the average value along the anti-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The “Hankelised” matrix is the best Hankel representation of the original matrix in terms of the Frobenius norm (see Weinberg & Petersen 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This antidiagonal averaging procedure results in reconstructed co- efficients for a set of pre-selected PCs, Im: ˜sk j = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 j j � n=1 U k n−j+1V k n if 1 ≤ j < L − 1, 1 L L � n=1 U k n−j+1V k n if L ≤ j ≤ N − L + 1 1 N − j + 1 N � n=N−L+1 U k n−j+1V k l if N − L + 2 ≤ j ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' = {˜s1,k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' ˜sN,k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (C5) The series ˜sk, where k is the group label (typically an integer), is the reconstructed coefficient series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='5 Power Spectra from Discrete Fourier Transforms We use the Discrete Fourier Transform (DFT) to estimate primary frequencies in reconstructed coefficients for the pur- poses of grouping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The DFT of a reconstructed coefficient is defined as ˜sk(ω) = F[˜sk(t)] = N−1 � j=0 e−i2πωtj/N ˜sk(tj) (C6) where C(tk) is a reconstructed coefficient time series of inter- est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' One may also compute the DFT of a PC directly for an alternate grouping strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Plotting the resulting frequency and DFT values results in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='6 Contrast measurement Using the reconstructed coefficients, one can construct the field of interest using a reduced version of either equation (B5) or equation (B9), depending on the component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The reduced version of each equation will sum over a selected set of indices corresponding to the input series to mSSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' For example, if one were to use disc coefficients Cm=0,n∈[0,5] as the input series ⃗s for the group indexed by k (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' equation C5), the surface density representation given by equation B9 would become ˜Σk disc(R, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' t) = � m=0 � n∈[0,5] ˜sk mn(t)eimφGn(R), (C7) where the notation ˜· indicates an approximated and trun- cated field representation (in this case surface density) from a select PC group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' To compute the contrast ∆Σ, we divide the selected field representation by the unperturbed m = 0, n = 0 representa- tion ˆΣdisc,00(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' t) = C00(t)G0(R), (C8) giving ∆Σ(R, φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' t) = ˜Σk disc ˆΣdisc,00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (C9) Analogous expressions exist for computing halo density con- trast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' In this work we consider only disc surface density con- trasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='7 Implementation of mSSA mSSA involves the exact same procedure as SSA, but with a grand-trajectory matrix (Ghil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' 2002) that is the con- catenation of the trajectory matrices for the individual series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' All other procedures are the same except for diagonal averag- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The choice of L is no longer symmetric around N 2 because only one of the dimensions is correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We form the grand trajectory matrix by concatenating tra- jectory matrices: H = � T0, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' , TM � ∈ RL×MK (C10) where M is the total number of different coefficient series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' The next step is to find the SVD of H such that we may write the grand-trajectory matrix as the sum of principal compo- nents, H = �r i=1 σiUiV ⊺ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' This could be computed directly but it is unnecessarily expensive and could be prohibitive for a large number of series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We therefore solve the smaller L×L eigenvalue problem instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' First note that H ∈ RL×MK and L ≤ N, thus HH⊺ ∈ RL×L is smaller than RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' If N is sufficiently modest (as is the case in this work with N = 599), we may find U by performing the eigendecomposition HH⊺ = UΛU⊺, (C11) which only requires the eigen-decomposition of the L × L matrix HH⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We can then find ΣV⊺ as U⊺H = U⊺(UΣV⊺) = ΣV⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' (C12) We then have everything required to write H = �L i UiσiV ⊺ i doing only the eigen-decomposition of an L × L matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We MNRAS 000, 000–000 (2023) 20 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' found that this optimisation, in the modest N limit, allowed the correlation of a much greater number of input series M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' After obtaining H = � i σiUiV ⊺ i , we do the same group- ing described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content='3 and obtain H = �m k=1 ˜Tk where ˜Tk = � i∈Ik σiUiV ⊺ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' We also must Hankelize each ˜Tk by applying the Hankelization algorithm from equation (C5) to each block of the grand trajectory matrix independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' Af- ter this procedure has been applied, each Hankelized PC group ideally corresponds to a different feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} +page_content=' MNRAS 000, 000–000 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE0T4oBgHgl3EQfUwDn/content/2301.02256v1.pdf'} diff --git a/VtE1T4oBgHgl3EQfIwPB/content/tmp_files/2301.02944v1.pdf.txt b/VtE1T4oBgHgl3EQfIwPB/content/tmp_files/2301.02944v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d73dd48f9711288af94c4c24c53ef27a47251d11 --- /dev/null +++ b/VtE1T4oBgHgl3EQfIwPB/content/tmp_files/2301.02944v1.pdf.txt @@ -0,0 +1,1226 @@ +Quantum Honest Byzantine Agreement as a +Distributed Quantum Algorithm +Marcus Edwards +July 23, 2020 +1 +arXiv:2301.02944v1 [quant-ph] 7 Jan 2023 + +Contents +1 +Quantum Honest Byzantine Agreement +3 +1.1 +Blockchain’s Resource Consumption and the Abundance of +Quantum Resources +. . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.2 +Coincidence-driven Consensus via Byzantine Agreement . . . . . +5 +1.2.1 +Distribution of Correlated Lists . . . . . . . . . . . . . . . +5 +1.2.2 +Consensus . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.3 +Casting to a Modular Quantum Neural Architecture . . . . . . . +8 +1.3.1 +Restricted Boltzmann Machines . . . . . . . . . . . . . . . +8 +1.3.2 +Associative Measuring Neurons . . . . . . . . . . . . . . . +10 +1.3.3 +Training Via Quantum Binding Commitment . . . . . . . +15 +1.3.4 +Commitment Schemes . . . . . . . . . . . . . . . . . . . . +15 +1.3.5 +Collapsing Hash Functions +. . . . . . . . . . . . . . . . . +18 +1.3.6 +A New Type of Machine Learning Algorithm . . . . . . . +19 +1.4 +Practical Implementation Limitations and Advantages . . . . . . +20 +1.4.1 +Practical Constraints Today . . . . . . . . . . . . . . . . . +20 +1.4.2 +Near-future Possibilities . . . . . . . . . . . . . . . . . . . +27 +2 +References +29 +2 + +1 +Quantum Honest Byzantine Agreement +Blockchain technology has three main components: +the network, consensus +algorithm and distributed data structure. Each of these brings with it particular +issues of scalability and efficiency. +By recasting the network and consensus +algorithm components of blockchain to a quantum algorithm, we show that +the efficiency and scalability of blockchain technology can be improved in the +near-term without requiring powerful quantum computers to be available. +1.1 +Blockchain’s Resource Consumption and the Abun- +dance of Quantum Resources +Crandall points out the abundance of resources in our world that could +be harnessed for information processing tasks in his book “Nanotechnology: +Molecular speculations on global abundance” [4]. +In light of the seemingly +endless resources around us, it is difficult to stomach the irresponsible use of +resources that we have encouraged through the design of our most prominent +blockchain systems including Bitcoin. +There are critical issues with the scaling properties and efficiency of these +blockchain technologies which require solutions if any significant distributed +ledgers are going to be possible to sustainably implement. The scaling properties +of the immutable distributed data structures used in blockchain networks have +been shown to cause demands on memory that are hard to justify. Blockchains +that are based on proof-of-work consensus schemes like Bitcoin also encourage +massively wasteful compute resource consumption. +Competition in Bitcoin’s +compute-intensive scheme coupled with the limitations of the blockchain data +structure implementation by Bitcoin also causes issues with throughput of the +system as a practical trading platform. The amount of transactions that can be +processed by Bitcoin is less than seven per second. This is far from the reported +3 + +47,000 per second achieved by VISA [5]. These issues have motivated some push +back against the spread of blockchain. China is seeking to stop Bitcoin mining +in the country, for example [6]. +From a business perspective, blockchain technology is not expected to be +viable for full adoption and practical use by mainstream banks for around +another ten years [7]. Even so, banks are beginning to implement prototypes +and blockchain applications of limited scale now. An IBM survey of 200 global +banks [8] showed that 65% of these banks intended to roll out blockchain-based +products between 2016 and 2019. +The majority of blockchain applications that are being developed do not have +solutions to the scalability and efficiency issues of their underlying cybersecurity +schemes. They are also not prepared to face the challenges of attackers equipped +with the quantum computers we expect to see developed within the next ten +years. +In the meantime, evidence is mounting that we will be capable of performing +complex computing tasks using the world’s smallest resources within the next +decade. +Subatomic particles with properties that allow for us to use them +in quantum computing applications include the spin- 1 +2 particles: +electrons, +protons, neutrons, neutrinos and quarks. These are especially convenient for +use in computing applications because of how the limitation of 2 observables +mimics the limitation of the 2 discrete levels used in digital logic and Boolean +algebra. +I explore the question of whether consensus networks like those in modern +blockchains might be possible to improve upon using more efficient hybrid +quantum-classical communication networks. +4 + +1.2 +Coincidence-driven Consensus via Byzantine Agree- +ment +Sun, Xin, et al. presented a voting protocol for blockchain in their 2019 paper +[1]. +Their protocol makes use of a method for ensuring consensus between +talliers from their previous work, called a Quantum Honest Success Byzantine +Agreement Protocol (QHBA) [2]. This protocol is used in their voting scheme +to identify dishonest ballot talliers. +Definition 1 (Honest Success Byzantine Agreement Protocol (HBA)) +An honest success Byzantine agreement protocol involves n agents. One of the +agents is the sender S, and holds an input value xs ∈ D, where D is a finite +domain. A protocol achieves honest success Byzantine agreement if the protocol +guarantees the following: +1. If the sender is honest, then all honest agents agree on the same output +value y = xs. +2. If the sender is dishonest, then either all honest receivers abort the +protocol, or all honest receivers decide on the same output value y ∈ D. +The protocol is p-resilient if the protocol works when less than a fraction of p +receivers are dishonest. +The QHBA is +m−2 +m -resilient. +m is the number of receivers, and is more +efficient than a classical HBA protocol when there are many dishonest receivers. +1.2.1 +Distribution of Correlated Lists +The first phase of the QHBA protocol is for correlated lists to be distributed +among the agents using quantum secure direct communication. +5 + +Let the sender be S = P1. Each agent Pi ∈ {P n +2 +1, ..., Pn} is tasked with +distributing a list of numbers Li +k to agent Pk ∈ {P1, ..., P n +2 } such that: +1. |Li +k| = l ∀ k ∈ {1, ..., n/2}, where l is a multiple of 6. +2. Li +1 ∈ {0, 1, 2}l. +l +3 numbers on Li +1 are 0. +l +3 are 1. +l +3 are 2. +3. Li +k ∈ {0, 1}l ∀ k ∈ {2, ..., n/2} +4. ∀ j ∈ {1, ..., l}, if Li +1[j] = 0, then Li +2[j] = ... = Li +n/2[j] = 0 +5. ∀ j ∈ {1, ..., l}, if Li +1[j] = 1, then Li +2[j] = ... = Li +n/2[j] = 1 +6. ∀ j ∈ {1, ..., l}, if Li +1[j] = 2, then ∀ k ∈ {2, ..., m} the probability that +Li +k[j] = 0 and that Li +k[j] = 1 are equal. +If the number of receivers that report non-compliant lists from a distributor +passes a threshold, then that distributor is classified as dishonest. +1.2.2 +Consensus +Assuming h > n +2 , the following procedure can be used to reach a consensus. +First, the sender S sends a binary number B1k and a list of numbers ID1k +to each receiver Pk. ID1k should indicate all the positions on L1 where B1k +appears to Pk. An honest sender will send the same list to all receivers. +Each Pk will compare the B1k and ID1k to their list Lk. If any honest Pk +finds information that is not consistent, he/she sends ⊥ to the other receivers. +Otherwise, he/she sends B1k and ID1k to the other receivers. +After all these messages have been received, each honest Pk checks the +following: +1. If there were more than two agents who sent binary numbers and lists +that were consistent with Lk but some had different binary numbers, Pk +outputs ⊥. +6 + +2. If more than two agents sent the same binary numbers and lists which +were consistent with Lk, these agents are considered to be honest. Pk +outputs the binary number provided by these honest agents. +3. If more than two agents sent the same binary numbers and lists which +were consistent with Lk, any other agents are considered dishonest. If all +of the dishonest agents sent ⊥ to Pk, then Pk sets vk to the binary value +provided by the honest agents. +4. In all other cases, Pk outputs ⊥. +Consensus is achieved if at least n +4 agents output the same bit value. +Suppose Pj were a dishonest receiver, and j ≥ 2. Pj would want to send a +binary number Bjk and list of numbers IDjk which was consistent with Lk.On +Lj, there are +l +2 appearances of Bjk. On L1 there are only +l +3 appearances of +Bjk. So, there are +l +6 positions of discord x, where L1[x] = 2. If Pj selects a +discord position x then with probability 1 +2, Lk[x] ̸= Bjk. Pj has to avoid all +discord positions in order to avoid being identified as dishonest. This has a +( 2 +3) +l +3 probability of success which is very small when l is large. This is rationale +behind the checks made by Pk listed above. +In a nutshell, the receivers don’t know which instances of the false bit in +their strings from the distributors are random, so they have to just echo what +they were given by the sender. The final output is simply an agreed-upon bit. +If the receivers should chose to transmit the false bit and say it should occur at +a random index, they are recognized as dishonest when Bjk, IDjk are checked +against the other receivers’ lists L. +7 + +1.3 +Casting to a Modular Quantum Neural Architecture +I suggest that the QHBA protocol essentially reduces consensus to coincidence. +The volume of coincidence is the input parameter which drives a receiver to echo +its input. A lack of coincidence results in no useful output from a receiver, as +per subsection 2.1.3. This is a similar mechanism to the learning mechanism in +cognitive modular neural architectures like Haikonen’s architecture for artificial +intelligence [3]. +1.3.1 +Restricted Boltzmann Machines +Furthermore, the path of information between the participants in the QHBA +protocol is suggestive of a fully-connected feed-forward neural network. +The RBM-like connectivity graph of the Honest Success +Byzantine Agreement +At first glance, this appears to be a Restricted Boltzmann Machine (RBM). +Restricted Boltzmann machines are an early machine learning neural network +structure created by Geoffrey Hinton. +A restricted Boltzmann machine is a +shallow network with two layers. One is ”hidden”, one is ”visible”. The input +layer’s nodes accept the system’s inputs, process these inputs, and pass their +8 + +outputs onto the nodes of the next layer. Each node is a McCulloch-Pitts neuron +and has an activation function, typically of the following form. +yj = T( +� +i +wjixi + b) +The input xi is multiplied by a weight wji and added to a bias b. +The +result is passed to an activation transfer function T, which produces a node’s +output. This output is an amplification or suppression of the strength of the +signal passing through it. +In a fully-connected RBM, each node of the first layer outputs to each node of +the second. Each combination of source and destination nodes ji has a different +weight. +The jth McCulloch-Pitts neuron in a network takes a set of weighted inputs +xi and a bias signal, and outputs a signal yj which transforms its inputs +according to a transfer function T. +A simple instance of a McCulloch-Pitts neuron is the perceptron, which is +a threshold binary classifier. Its output is always a 0 or 1. The perceptron will +output a value of 1 if the following condition is satisfied. +( +� +i +wixi + b) > 0 +This type of neuron then plays the role of summarizing its inputs by putting +them into a category. Artificial intelligence is most useful for the categorization +of data which has very dense informational data points which share a structural +pattern. A set of these neurons is typically arranged, and each neuron given the +responsibility to categorize a different subset of a ”training data set”. A neural +network’s output is often simply used to update a statistical model. This might +mean updating a regression or more advanced statistical model of the training +9 + +data. +As a thought experiment, let us consider the topology of the computer +network implied by the QHBA protocol as a neural network. In this paradigm, +the means of network communication between each protocol participant +corresponds to a weighted I/O channel between neurons. +The participants +themselves correspond to neurons. The training data set is simply the sender’s +list L1, and the output is the agreed-upon bit B1. +1.3.2 +Associative Measuring Neurons +If our network is to perform the QHBA protocol, the neurons must clearly be +modified. Rather than the receiver neurons simply outputting a binary classifier +when the perception threshold is reached, our version of honest receiver neurons +must output its input, Bjk, IDjk when the amount of agreement or coincidence +of its inputs’ values passes a threshold. This mechanism will actually suffice for +the distribution neurons as well, with one small tweak. Rather than having a +neuron simply output its input, we can have the neurons both measure and then +output their inputs. This will not change the function of the reveiver neurons, +but will allow the distributors to achieve the replacement of 2’s in the list L1 +provided to their inputs by the sender S with probabilistically distributed 1’s +and 0’s. +Definition 2 (Associative Measuring Neuron) An associative measuring +neuron will conditionally propagate a quantum state from its quantum input +to its output. Its output may be a classical channel or quantum channel that +will accept only basis states. Let an associative measuring neuron k have the +following output yk in terms of inputs x from n neurons, where |Xk > will +be a superposition of basis vectors weighted appropriately to represent their +multiplicities as inputs to the neuron, and the weight of |0⊗l > will represent the +10 + +neuron’s bias. Then the neuron achieves the non-linear effect of collapsing its +output state to an that of an input which is sufficiently present so as to overcome +the neuron’s bias. +yk = M|Xk > +(1) +The behaviour of an associative measuring neuron is then to output the most +recurring input with high likelihood, unless no input is repeated sufficiently +enough, in which case the measurement will yield |0⊗l > with high probability. +The input coincidence threshold is manifested by bk, which is the coefficient +on the |0⊗l > basis state and should be trainable as well as the weights w. A +simple way to achieve this functionality would be to implement an associative +measuring neuron using a series of conditionally applied quantum gates. +Associative Measuring Neuron Circuit +In this circuit, D is the Grover Diffusion Operator. +Grover Diffusion Operator +Each Uki is a circuit composed of controlled Pauli-Z phase flip gates. Each +of the Uki is a parameterized operator that takes a set of qubit indexes IDki, a +boolean value Bki and a weight wki. All of these parameters are classical. +11 + +Uki +Ukj +Ukl +Yk +H +D +D +DHH +20><0-I +H&If the boolean Bki is 1, then Uki is simply a multiply controlled CZ gate with +all qubits in IDki included as controllers in x and targets in y. For example, if +l = 6; a, c ∈ IDki and b, d, e, f, g /∈ IDki we would have the following. +Parameterized Oracle Example with Bki = 1 +If the boolean Bki is 0, then a Pauli-X gate is first applied to the input xi. +12 + +Yka +Z +Ykb +yke +Z +ykd +ykg +Lia +Tib +Lic +Lid- +Tie +Ti +LigYka +Z +ykb +Yke +Z +Ykd +ke +Ykg +Tia +Tib- +Tic +Lid +Lie- +Lif +TigParameterized Oracle Example with Bki = 0 +The weight wki dictates the number of times the operators Uki and D are +repeated. The larger the weight with respect to other weights for other inputs +to the same neuron, the more repeats. When there is only one weight and it +is 1, then the operator should not be applied at all. Otherwise, Uki and D +should be repeated a number of times proportional to the number of standard +deviations Ni the weight wki is above the mean weight ¯ +wk. A negative N should +be reflected as well, so we will always either subtract from or add to a default +number of repetitions. This default will exactly be the bias bk and will be a +learned parameter itself. +Ni = +wki − ¯ +wk +� +1 +l +�l +j=1(wkj − ¯ +wk)2 +(2) +The number of repetitions total will be ⌊bk + Ni⌋. +The function of the associative measuring neuron is comparable to a number +of competing Grover searches [12] performed on the same quantum state. The +algorithm is designed such that the effect of a search is proportional to the +multiplicity of its corresponding input list Li +k = xi, if that list matches the +paramters IDki and Bk. In the case that IDki and Bk correspond correctly with +xi, then the effect of Uki is to impose a negative phase on the bits corresponding +to Bk in yk, which effectively ”tags” that state for amplitude amplification. In +the case that xi does not correspond with IDki and Bk, the controlled Z gate is +not applied since not all of its controlling qubits are 1’s. Hence, the bad value +contributes nothing to the neuron’s final output state. +S should encode a 2 into L1 by simply applying a Hadamard gate to +13 + +the corresponding qubits in a qubit register with a size equal to the length +of L, preparing and sending this entire qubit register L to each distributor +individually. With each message, S sends the parameters IDki and Bk. The +effects of the associative measuring neuron’s operations will be to exactly +replace any 2’s in the list L1 provided to their inputs by the sender S with +probabilistically distributed 1’s and 0’s via measurement of the corresponding +single qubit states which will be |0>+|1> +√ +2 +or |0>−|1> +√ +2 +depending on whether Bk +is 1 or 0. The algorithm will leave the rest of the state unchanged and simply +measure it. +The receivers will also be associative measuring neurons and perform the +same process. However, it will be assumed that they are more likely to have +conflicting inputs, and that their weights will not all agree. Also, receivers will +receive their inputs x from distributors and other receivers, but the parameters +IDki and Bk will still be provided directly by S. The receivers should have a +final state that approximates the following. +|Xk >= bk|0⊗l > + +� +i +� +j̸=i +(< xi|xj > wkiwkj)|xi > +(3) +The weights and biases will be naturally normalized. +b2 +k + +� +i +� +j̸=i +δ(xi, xj)(wkiwkj)2 = 1 +(4) +While the machine begins with a nearly equal number of distributors and +receivers, the neurons which do not receive consistent data do not output +information and their input sources are considered unreliable. This decreases +14 + +the number of useful neurons in the network and may reduce the size of either +of the two layers. This is how the QHBA selects trusted paths of information +through the network. This is not dissimilar to the way that pathways between +neurons are created through learning in Haikonen’s cognitive modular neural +network. +1.3.3 +Training Via Quantum Binding Commitment +We can make use of an optimally secure mechanism called a Quantum Binding +Commitment to define the training data for the system. In our network, we +will want the consensus to result in agreement on a single bit. So, the measure +of fidelity used in our training is trivial: a single bit (a sender’s vote) that is +known initially only by the sender and the training code which can be a very +simple, visible, immutable and infinitely running script. +Quantum Bit Commitment Training +1.3.4 +Commitment Schemes +Computationally binding commitment schemes between two parties are +composed of two phases. +The Commitment Phase allows one party to send +15 + +Commitment +ML Algorithmthe other party some information c related to a message m which does not give +the receiver any information about m itself. However, the act of sending c binds +the sender to provide the message m in the second stage, the Open Phase. In the +Open Phase, the sender transmits m to the receiver and proves to the receiver +that m does indeed correspond to c by providing a signature that ”opens c to +m”. +In our system, a successful opening will occur after a consensus is reached +on a vote with well-tuned weights and biases. In the case that an opening is +unsuccessful, this is used as negative feedback to motivate the ML to adjust its +descent of the gradient. +A classical definition of a computationally binding is the following from +Unruh [21]. +Definition 3 (Classical-style binding) No algorithm A can output a +commitment c and two signatures s, s’ that open c to two different messages +m and m’. +Computationally binding commitment schemes have been studied and +defined in the quantum setting. Interestingly, when the algorithm A is allowed +to be a quantum polynomial time algorithm, this definition was shown to be +inadequate. +While definition 3 holds for a particular classical-style binding +commitment, Ambainis, Rosmanis, and Unruh showed that for this particular +binding a quantum polynomial time algorithm A employed by an adversary +could open c to any message that the adversary wished [22]. +Therefore Unruh was motivated to define a different type of binding that +was useful in the quantum case. The new binding property is demonstrated by +a pair of quantum games. +Let A, B be algorithms and S, M, U be quantum registers. +Vc is a +16 + +measurement which verifies that that U opens M. +Mok measures m in the +computational basis if ok = 1. +The first game Game1 consists of four steps: +(S, M, U, c) ← A(1γ) +ok ← Vc(M, U) +m ← Mok(M) +b ← B(1γ, S, M, U) +The second game Game2 omits the measurement in step three but is +otherwise the same: +(S, M, U, c) ← A(1γ) +ok ← Vc(M, U) +b ← B(1γ, S, M, U) +A commitment scheme is ”collapse-binding” iff for any quantum polynomial +time valid adversary, cAdv = |Pr[b = 1 : Game1] − Pr[b = 1 : Game2]| is +negligible. +This essentially expresses that if an adversary (A, B) provides a classical +commitment c, there must be only one message he/she can open c to. +A +outputs a superposition of messages M and a superposition of corresponding +opening signatures U. S is the adversary’s state. The assertion that |Pr[b = 1 : +17 + +Game1]−Pr[b = 1 : Game2]| is negligible limits the value of M to computational +basis vectors for collapse-binding commitments. No quantum polynomial time +algorithm B should be able to distinguish between the value of M whether M +is measured in the computational basis or not. +1.3.5 +Collapsing Hash Functions +We will have to choose a specific collapse-binding commitment scheme for our +system to use. However, the specific choice is relatively arbitrary as long as it +satisfies the collapse-binding property. +The games used to define the collapse-binding property of commitment +schemes can also be applied to classify hash functions that are collapsing. +Assume H is a one-to-one hash function. +Definition 4 (Collapsing hash function - informal) H is a collapsing +hash function iff no quantum polynomial time algorithm B can distinguish +between Game1 and Game2. +An adversary is valid if A outputs a classical +value c and a register M where H(m) = c. +This game based definition was clarified and made mathematical by Fehr in +2018 [23]. +Definition 5 (Collapsing hash function - formal) A function H X → Y +is ∈(q)-collapsing if +cAdv[H](q) := +sup +SMCU +δq(M, M|CU) ≤∈ (q) +for all q. The supremum is over all states SMCU = S H(M) CU with +18 + +complexity ≤ q. +The collapsing property of a hash function is a counterpart of collision +resistance. Unruh shows that the random quantum oracle is a collapsing hash +[21] and so some hash function based commitment schemes are collapsing in the +random oracle model. Unruh also showed that Merkle-Damgard hash functions +are collapsing if their underlying compression algorithms are, which implies that +SHA-2 is collapsing [24]. Czajkowski, et al. showed the same for Sponge hashes +with certain conditions [25]. Sponge hash construction underlies SHA-3. +1.3.6 +A New Type of Machine Learning Algorithm +Our approach does not exactly fit into the category of supervised machine +learning, since the idea here is not to train the neural network using a predefined +data set until it reaches a fidelity threshold, and then to use the machine +in production afterwards. Rather, our machine will be continuously training +in production to achieve the effect of learning and accounting for the shifting +behaviour of the participants in the network. +The algorithm will train both weights and biases. +As in any traditional +RBM, each bias will effect all of the inputs to its neuron, while the weights +will each be specific to channels between neurons. Hence, I expect that this +approach will mitigate the impact of both dishonest individuals and groups of +collaborating dishonest individuals. +Another interesting paradigm shift has taken place. ML usually variationally +combines well-defined, trivially simple and reliable elements (the neurons) to +model uncertain, complex and large data sets (the training data). In our case, +this is basically reversed. The training data is a single bit, for which we have an +expectation value at the onset. The neurons are the uncertain elements. While +we have defined the behaviour of the honest receiver neurons in the previous +19 + +subsection, we expect some participants to be dishonest and therefore deviate +from this neuron model. +The question that this algorithm is answering is fundamentally how to +orchestrate unreliable and complex elements of a fair consensus system such +that collaborative productivity is maximized. Since this new type of machine +learning algorithm seems to be a suitable solution, there is the suggestion that +a symbiosis of machine and human information processing can be useful for +maximizing productivity. +It would be interesting for future works to consider other applications where +such a ”symbiotic” information processing approach could be useful. +1.4 +Practical Implementation Limitations and Advantages +1.4.1 +Practical Constraints Today +It is an intended feature of the associative measuring neuron’s design that if +the size of input lists and outputs is l = 6, its behaviour can be realized today +using IBM’s commercially available Q System One or IBM Q 16 Melbourne +system which is free to use for research purposes. Some restrictions must be +applied to the neuron in order to ensure that it can be implemented using either +system, since the superconducting Transmon qubit networks of these systems +are not fully-connected. +In order to make the associative measuring neuron +compatible with the Melbourne, a sender simply must choose IDki and Bk that +specify a controlled Z operation that is possible to implement. Many control +configurations can be achieved using qubit swapping. +20 + +Y +13 +12 +11 +10 +9 +8IBM Q 16 Melbourne Connectivity Graph +The number of D and Uki operations applied, the less fidelity we will have +in the neuron’s outputs due to decoherence. +A minimum fidelity should be +chosen and used to select the range of possible values that will be taken by the +default repetition bias bk. This fidelity can be dynamically chosen based on +the calibration parameters of the Melbourne, for example, which fluctuate but +are available at a given time. The average T1 and T2 times for the IBM Q +System One are reportedly 74µs and 69µs respectively [14]. The Melbourne’s +decoherence times are similar but vary depending on the qubits involved, as +evidenced by the figure. IBM reports that the average decoherence times for the +Melbourne are T1 = 67.50µs and T2 = 22.40µs [16]. We only will consider the +Melbourne’s limitations thoroughly, since it is less advanced and more limited +than the IBM Q System One. +IBM Q 16 Melbourne Calibration Details July 17th 2019 +The gate times of the Melbourne are updated continuously and published +publicly [17], and the average amount of time required for a CX gate is around +350ns. +21 + +qubit +multi_qb_gate_error +T1 (us) +T2 (us) +Frequency (GHz) +readout error +gate_error +Q0 +73.32348273 +23.48828043 +5.100090141 +0.0215 +0.004031062 +Q1 +CX1 0:0.03, CX1 2:0.04 +63.23181621 +116.7289054 +5.238609742 +0.054 +0.012242205 +Q2 +CX2_3: 0.04 +46.13953307 +74.56571753 +5.032644087 +0.1864 +0.010450744 +Q3 +81.05055849 +74.78940464 +4.896205701 +0.047 +0.002494886 +Q4 +CX4 3:0.03, CX4 10:0.04 +55.43102145 +27.63898146 +5.028667392 +0.1226 +0.002551687 +Q5 +CX5 4:0.05,CX5 6:0.05,CX5 9:0.07 +27.79450766 +50.71953989 +5.06718735 +0.0568 +0.004714312 +Q6 +CX6 8:0.04 +56.16840169 +56.0630866 +4.923906934 +0.0478 +0.004816689 +Q7 +CX7 8:0.03 +32.50641909 +45.28966051 +4.974534967 +0.0598 +0.004438222 +Q8 +47.68062524 +71.45643335 +4.739563654 +0.0389 +0.004361702 +Q9 +CX9 8:0.04, CX9 10:0.05 +38.43726664 +79.71232612 +4.963421912 +0.0443 +0.006372041 +Q10 +56.99362705 +69.83941723 +4.945065458 +0.037 +0.003278348 +Q11 +CX11 3:0.05,CX11 10:0.05,CX11 12:0.06 +57.53451171 +71.43323367 +5.004981691 +0.0357 +0.0044898 +Q12 +CX12 2: 0.06 +78.13277541 +117.4664528 +4.760047973 +0.0918 +0.007732648 +Q13 +CX13_1:0.12, CX13_12:0.1 +21.39891833 +41.28178002 +4.968495889 +0.0498 +0.011006778IBM Q 16 Melbourne Gate Time Details August 7th 2019 +A CZ gate is realized in the IBM system using a CX and two single qubit +Hadamard gates. +Each Uki is a pair of controlled CCZ gates, and either 0 +or 6 X gates. +A CCZ gate can be realized using CNOT, T †, and T gates +via an optimal decomposition [20]. This requires six CX gates. The Grover +diffusion operator can be realized using Hadamard gates surrounding a multiply +controlled Z operation as well, as per [19]. +CCZ Gate Acheivable Using IBM Q +Generally, the Grover diffusion operator would involve a multiply controlled +Z gate with a number of controls equal to the size of the output register, minus +one. We can get away with simplifying the Grover diffusion operator for our +case by realizing that the operator will only ever be used to rotate the state +22 + +cX Gate +GFGateTime (ns) +CX1_0 +239 +CX1_2 +174 +CX2_3 +261 +CX4_3 +266 +CX5_4 +300 +CX5_6 +300 +CX7_8 +220 +CX9_8 +434 +CX9_10 +300 +CX11_10 +261 +CX11_12 +261 +CX13_12 +300 +CX13_1 +652 +CX12_2 +1043 +CX11_3 +286 +CX4_10 +261 +CX5_9 +348 +CX6_8 +348towards basis states with two non-zero qubit values. During each such rotation, +the diffusion operator can be realized by a single CZ gate which involves the +two qubits that correspond to the particular input xi’s corresponding indexes +ID. +This will rotate the high dimensional output state towards the target +basis state in the relevant degrees of freedom, and leave the other degrees of +freedom untouched. However, each input xi may adjust the overall output state +in different degrees of freedom, and together rotate the state in any arbitrary +direction. Using this approach, the Grover diffusion operator can be realized +using six single qubit gates and one CX gate. +The single qubit gates involved in the algorithm each have a time penalty +as well. These time penalties can be understood by decomposing each unitary +gates into its set of actual physical gates that are used to implement them in +IBM’s system. IBM’s computers support three types of single qubit gates, the +first two (u1, u2) are relevant for us: +u1(λ) = +� +�� +1 +0 +0 +eλi +� +�� +u2(φ, λ) = +1 +√ +2 +� +�� +1 +−eλi +eφi +e(φi+λi) +� +�� +Any single qubit gate which has the form given by u1 is implemented using +Frame Change (FC) operation, which does not physically take any time but +actually influences the frame of the following operation and takes no time in +and of itself. We can see that the T and T † gates do have the corresponding +form. +23 + +T = +� +�� +1 +0 +0 +e +π +4 i +� +�� = u1(π +4 ) +T † = +� +�� +1 +0 +0 +e− π +4 i +� +�� = u1(−π +4 ) +u1 Frame Change Physical Gate +The Hadamard gate is also used in our algorithm, and matches the form +of u2. +Any gate which has the form of u2 is implemented using a physical +Gaussian-Derivative (GD) pulse parameterized by two frame changes. A GD +pulse takes 60ns itself, and invokes a 10ns buffer time. +H = +1 +√ +2 +� +�� +1 +1 +1 +−1 +� +�� = u2(2π, 3π) +u2 Frame Change Physical Gate +The final type of gate that is relevant for our work is the CX gate, which +makes use of both FC and GD physical gates as well as Gaussian Flattop +(GF) pulses. We have already addressed the time requirements for CX gates +depending on the qubits involved. +24 + +U1 +FC +bm +(M) +(-入)U2 +FC +GD +FC +bm +(9, 入) +(-入) +(π/2,π/2) +(-Φ)CX Physical Gate +Understanding this, we may say that the time requirement for Uki is at most +equivalent to that of two CCZ gates and six X gates. +T(Uki) ˙= 2 · 0ns + 6 · 350ns = 2100ns +(5) +Similarly, the time requirement for our simplified Grover diffusion operator +is that of four Hadamard gates, two X gates and one CX gate. +T(D) ˙= 2 · 0ns + 4 · 70ns + 1 · 350ns = 640ns +(6) +The overall time cost of a repetition of DUki is then given by equation (7). +Trep = T(Uki) + T(D) = 2740ns +(7) +The time requirement for an associative measuring neuron’s operation in its +entirety will then be given by equation (8). +Tassoc = +� +i +⌊bk + Ni⌋ · 2740ns +(8) +25 + +FC +GD +GD +I +Control: wc +(乙/μ) +(/-") +(t,0) +GF +GF +CR: WT +(π/4,0) +(π/4, ) +- ++ +GD +Target: WT +(π/2,0) +T +- +-To ensure this operation completes within an acceptable window, we simply +enforce that Tassoc < T2. The most expensive associative measuring neuron +operation will involve +|P | +2 +inputs xi. +So, for example a system with ten +participants would yield a maximal Tassoc time of max(Tassoc| |P|). +max(Tassoc| |P|) = +|P | +2 +� +i=0 +⌊bk + Ni⌋ · 2740ns +In this scenario, Ni would be standard deviations of each |P | +2 points. To keep +an associative measuring neuron operation under the shortest time constraint, +which is T2 = 22.40µs on the Melbourne, we must limit either the number of +participants in the network |P|, or cap the number of standard deviations Ni at +some maximum range. It is more appealing for the machine learning algorithm +to take the number of participants as a parameter and adjust the range of the +maximum considered standard deviation. So, we can define a maximum range +max(Ni| |P|). +T2 = max(Tassoc| |P|) = +|P | +2 +� +i=0 +⌊bk + max(Ni)⌋ · 2740ns +22.40µs = +|P | +2 +� +i=0 +⌊bk + max(Ni)⌋ · 2740ns +22.40µs +2740ns = +|P | +2 +� +i=0 +⌊bk + max(Ni)⌋ +26 + +22.40µs +2740ns − +|P | +2 +� +i=0 +bk ˙= +|P | +2 +� +i=0 +max(Ni) +22.40µs +2740ns − +|P | +2 +� +i=0 +bk ˙= |P| +2 max(Ni) +max(Ni| |P|) ˙= +2 +|P| · 22.40µs +2740ns − +|P | +2 +� +i=0 +bk +In the worst case, � |P | +2 +i=0 bk → |P | +2 +since 0 ≤ bk ≤ 1. +max(Ni| |P|) ˙= +2 +|P| · 22.40µs +2740ns − |P| +2 +The system will become functionally useless when max(Ni| |P|) approaches +0. Therefore we can conclude that the system will be able to handle only 6 +participants if our quantum associative measuring neurons were used at each +node today. +Also, if this system were implemented today, each participant would not +have a local quantum computer to use for their associative measuring neuron +operations. Rather, they would need to delegate their quantum computations +to a central quantum computer. Today, the best option would be IBM’s system. +The amount of time spent in the queue waiting for each others’ operations to +complete would render the speedup from using Grover’s search pointless. +1.4.2 +Near-future Possibilities +Despite the conclusion that this system is not practical to implement today, +the work we did in the last subsection gives us a method for predicting how +27 + +useful the system will be in the future, when we have access to better quantum +computers. +IBM claims that they intend to eventually improve their coherence (T2) +times to 1-5 milliseconds, and suggest that they are exponentially approaching +this goal according to a relationship similar to Moore’s law for integrated +electronics [14]. Assuming this goal is reached within the next decade, which +is generally considered to be feasible with at least a non-zero possibility, our +scheme would be able to support roughly 85 participants in each vote. +The true randomness of the probabilistic outcomes from measuring the +states |0>+|1> +√ +2 +and |0>−|1> +√ +2 +is a valuable cryptographic asset when a quantum +associative measuring neuron is used for a security protocol due to the outcome +being truly random. +Also, the Grover’s search algorithm provides a known +quadratic speedup over equivalent classical methods, when the number of +applied operations is compared to the number of classical records checked for +the value searched for [13]. +However, it is important to point out that an associative measuring neuron +with a limited repetition capacity can be easily classically simulated. So, the +entire system described thus far could theoretically be replaced with a classical +equivalent. This would mean that we do not gain the security and efficiency +benefits of the quantum algorithms employed. However, it would mean that +scaling the system to support any arbitrarily large number of users would be +possible. +An optimal network scheme would incorporate both quantum and classical +elements to take advantage of as much quantum security and speedup as possible +with the resources available whilst also supporting an arbitrary number of users. +The +machine +learning +inspired +element +of +the +consensus +algorithm +implemented at any scale would be beyond the capabilities of any quantum +28 + +computing technology that exists today. However, it would be well within the +reach of modern classical technology. So, we will assume that it is purely classical +for the forseeable future. However, it would be interesting for a future work to +look into how quantum machine learning might increase the efficiency of this +component as well. +On the other hand, quantum-secure communication channels are already +being established and demonstrated in the world by companies like NXM [18]. +We posit that quantum networks will also be available for practical use in the +near-term, and we can expect to use these as a resource. The ability to perform +a Hadamard gate, transmit and measure the resulting state is already quite +feasible. So, we can assume that at least some of the channels used by the sender +of any vote can benefit from the pure randomness of the quantum approach for +encoding 2’s into the lists L1k. +Our system does not make assumptions on the number of participants who +will be interested in participating in any given vote. 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Bertoni, +Guido, +et al. +“On the Indifferentiability of the Sponge +Construction.” Advances in Cryptology – EUROCRYPT 2008 Lecture +Notes in Computer Science, 2008, pp. 181–197., doi:10.1007/978-3-540- +78967-3 11. +32 + diff --git a/XNFOT4oBgHgl3EQf8TTm/content/tmp_files/2301.12966v1.pdf.txt b/XNFOT4oBgHgl3EQf8TTm/content/tmp_files/2301.12966v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..147c1a22cc3ff3fb3197699efb83e4cd13d5ba17 --- /dev/null +++ b/XNFOT4oBgHgl3EQf8TTm/content/tmp_files/2301.12966v1.pdf.txt @@ -0,0 +1,1257 @@ +Based on Springer Nature LATEX template +Lyapunov Exponents for Temporal Networks +Annalisa Caligiuri1, Victor M. Egu´ıluz1, Leonardo di +Gaetano2, Tobias Galla1 and Lucas Lacasa1* +1Institute for Cross-Disciplinary Physics and Complex Systems (IFISC), +CSIC-UIB, Palma de Mallorca, Spain. +2Department of Network and Data Science, Central European University, +1100 Vienna, Austria. +*Corresponding author(s). E-mail(s): lucas@ifisc.uib-csic.es; +Abstract +By interpreting a temporal network as a trajectory of a latent graph +dynamical system, we introduce the concept of dynamical instabil- +ity of a temporal network, and construct a measure to estimate the +network Maximum Lyapunov Exponent (nMLE) of a temporal net- +work trajectory. Extending conventional algorithmic methods from +nonlinear time-series analysis to networks, we show how to quantify +sensitive dependence on initial conditions, and estimate the nMLE +directly from a single network trajectory. We validate our method +for a range of synthetic generative network models displaying low +and high dimensional chaos, and finally discuss potential applications. +Keywords: Lyapunov exponent, temporal networks, chaos, complex systems +1 Introduction +Temporal networks (TNs) [1–3] are graphs whose topology changes in time. +They are minimal mathematical models that encapsulate how the interaction +architecture of elements in a complex system changes dynamically. TNs have +been successfully used in a variety of areas ranging from epidemic spreading +[4] or air transport [6] to neuroscience [7] to cite a few, and it has been shown +that important dynamical processes running on networks (e.g. epidemics [4], +synchronization, search [5]) display qualitatively different emergent patterns +when the substrate is a TN, compared to the case of a static network. These +1 +arXiv:2301.12966v1 [physics.data-an] 30 Jan 2023 + +Based on Springer Nature LATEX template +2 +Lyapunov Exponents for Temporal Networks +effects are particularly relevant when the timescale of the dynamics running +on the graph is comparable to that of the intrinsic evolution of the network, +i.e., when there is no manifest separation of timescales. Relatively lesser +work has, however, considered the intrinsic dynamics of the network from a +principled point of view. Recently, a research programme has been proposed +[8] in which TNs are to be interpreted as the trajectories of a latent ‘graph +dynamical system’ (GDS). The GDS provides an explicit model for the time- +evolution of the network. Similar to a conventional dynamical system (whose +output is a time series of scalar or vector quantities), the output of a GDS is +a time series of networks, i.e. a TN. +The dynamics of TNs and GDS are indeed the objects of ongoing research. +For instance, in [8] the authors consider how to extend the autocorrelation +function of a signal to a graph-theoretical setting. They explore how TNs can +oscillate and how harmonic modes, as well as decaying linear temporal correla- +tions of various shapes, emerge. In a similar fashion, the memory of a temporal +network has been studied from different angles, including the concept of mem- +ory shape [9] as a multidimensional extension of memory (high order Markov +chain theory) in conventional time series. In this work, we further pursue the +abovementioned research framework programme, and consider the problem of +dynamical instability and chaos quantification in TNs. Interpreting temporal +networks as trajectories in graph space, we aim to generalise the concept +of Lyapunov exponents as quantifiers of the sensitivity to initial conditions. +Our objective is to define and measure Lyapunov exponents and sensitive +dependence to initial conditions of the network as a whole. Our approach is +therefore not to quantify chaos in the dynamics for example of every link, but +rather to quantify chaos for the collective dynamics of the whole network. +Since TNs in applications are frequently observed empirically, we focus our +implementation and inference of network Lyapunov exponents solely on the +observation of a single (long) TN trajectory, without the need to access the +underlying GDS (but of course, the framework is also applicable if the GDS +is explicitly accessible). Our algorithmic implementation can thus be seen as +a conceptual network generalization of the classical algorithms by Wolf and +Kantz [10–12, 14], originally proposed to quantify sensitive dependence on +initial conditions directly from empirically observed time series (see also [13] +for a similarly seminal work). +Importantly, any new method needs to be validated. In our case, this is +not trivial, since the notion of chaotic TNs is not common in existing litera- +ture. A second objective of this work is thus to propose synthetic generative +models of chaotic TNs, which can be used as templates to validate the meth- +ods we develop for the quantification of chaos in TNs. These methods, once +validated, can then be used in wider applications and further research. + +Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +3 +The rest of the paper is organised as follows. In Section 2 we introduce the +theoretical background to our work, and we set the notation. We derive the +network analog of the maximum Lyapunov exponent (MLE), and we outline +an algorithmic implementation to estimate quantities such as the spectrum +of local expansion rates, trajectory-averaged and volume-averaged expansion +rates, and the network maximum Lyapunov exponent (nMLE). In Section 3 +we consider the relatively simple case of random network dynamics as a first +example, and show how the method works in such scenario. Then, in Section 4 +we introduce a generative model of (low-dimensional) chaotic temporal net- +works. This model provides us with ‘ground-truth’ access to the nMLEs of the +network trajectories that the model generates. We show that the method we +propose to infer nMLEs from a trajectory of networks correctly reconstructs +this ‘true’ exponent. We also assess how the estimation of the nMLE is affected +when the chaotic network trajectory is polluted with certain amounts of noise, +and discuss at this point how to estimate negative nMLEs as well. In Section 5 +we introduce a different generative model of (high-dimensional) chaotic TNs. +We demonstrate that the generated TNs show sensitive dependence to initial +conditions, and that the nMLE varies as expected as a function of a network’s +coupling parameter. In Section 6 we finally conclude and discuss potential +applications of the method. +2 Theory and method +2.1 Lyapunov exponents for graph dynamical systems +In nonlinear time series analysis, the maximum Lyapunov exponent λMLE of +a dynamical system quantifies how two trajectories that are initially close +separate over time. More precisely, one imagines two copies of the system, +which are started from initial conditions at time t = 0 which are a distance d0 +apart. One then defines +λMLE = lim +t→∞ lim +d0→0 +1 +t ln dt +d0 +, +(1) +where dt is the distance between the two copies of the system at time t. +In practice, the distance d0 is often small but finite, and the limit d0 → 0 may +not be accessible. It then turns out that the long-time limit t → ∞ is not +accessible either as the growth of dt is bounded by the size of the attractor +of the system [16]. In such cases, the behaviour of dt usually undergoes a +cross-over (at a time which we label τ), between an exponentially expanding +phase (t < τ) to a saturated phase (t > τ). In the latter regime, dt fluctuates +around the attractor’s size. We will call τ the saturation time. +In the network setting, we assume there exists a (sometimes unknown) +graph dynamical system that determines the evolution of a graph over time. +We focus on discrete time. The GDS is then a map, which determines how + +Based on Springer Nature LATEX template +4 +Lyapunov Exponents for Temporal Networks +one network evolves in the next time step. For simplicity, we assume that +the set of vertices is fixed, and that the vertices are distinguishable from +one another and labelled. Thus, only the set of edges between these ver- +tices evolves in time. A trajectory of the GDS then consists of a sequence +of network snapshots. These trajectories define the TNs generated by the +system [1, 3]. Each trajectory is given by the sequence of adjacency matrices +S = (At)t≥0, At = {aij; t}; i, j = 1, 2, . . . , n where aij; t = 1 if the vertices i +and j are connected at time t, and zero otherwise (this symmetric choice is +for simple, undirected networks, but the method works essentially along the +same lines for non-simple and directed networks, perhaps except for different +normalization factors in the definition of the distance, see below). As such, +we are considering labelled, unweighted networks of n nodes. +It is not clear a priori if the specific choice of the distance used to quantify +the deviation between two originally close network trajectories is critical. We +conjecture that, as long as this distance is based on the full adjacency matrix +–not on a projection of it–, results should hold independent of the specific met- +ric. This is based on the fact that in dynamical systems, the MLE is invariant +under different choices of the underlying norm ||.|| [14, 16]. There exist many +graph distances [18]. For simplicity, we take an intuitive definition of such a +distance which is based on the amount of edge overlap between two networks: +given two networks with the same number of nodes n and adjacency matrices +A and B, +d(A, B) = 1 +2 +� +i,j +|aij − bij|, +(2) +where i and j take values from 1 to n1. In our setting, networks are simple +and unweighted. In the particular case where A and B have the same number +of edges, the distance defined in Eq. (2) is indeed a rewiring distance, i.e., it +is given by the number of unique rewirings needed to transform A into B, +and therefore d is a positive integer-valued function. One can then further +normalize d as appropriate such that it is defined in [0, 1], as we will show +later. Further details can be found in the Appendix, where we also introduce +alternative distances. +2.2 Inference of network Lyapunov exponents +2.2.1 Local expansion rates +We are interested in quantifying sensitive dependence on initial conditions (and +in particular, the network version of λMLE, which we here call λnMLE) when +the mechanics of the GDS is not known, and when we only have access to +a (single) discrete-time network trajectory S = (A0, A1, A2, . . . ) of adjacency +matrices. This is analogous to the case in which one would like to reconstruct +the MLE of a conventional dynamical system from a single time series. The +1The pre-factor is used to have a normalized distance when networks are simple, undirected +and have the same number of links, otherwise a different normalization is needed. + +Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +5 +standard approach consists in using Wolf’s or Kantz’s algorithms [10–12, 14]. +The central idea is here to look for recurrences in the orbit, finding points in +the time series which might be temporally separated but which are close in +phase space. One then monitors the deviation of those points over time. Here, +we extend this approach to the case of a time series of networks, i.e. a TN. +We start by fixing an element At from S, where t is an arbitrary point in +time. This adjacency matrix will be the ‘initial condition’ for our analysis. To +extend Wolf’s algorithm we then proceed to look in S for another element At′ +at a different time t′, such that d(At, At′) < ϵ, where ϵ is a small threshold +chosen before the analysis begins2. This recurrence in phase space allows us +to use a single trajectory to explore how two close networks separate over +time. We then set d0 := d(At, At′) as the initial distance. We then proceed to +measure how distance evolves over time as we separately track the evolution +of At+k and At′+k in S, where k = 1, 2, . . . . We write dk = ||At+k − At′+k||. +Without loss of generality, we can always write the successive distances in +terms of a sequence of local expansion rates ℓ1, ℓ2, . . . , +dk = dk−1 exp(ℓk). +(3) +Each of the ℓk can be positive (local expansion), negative (local contraction), +or zero. Equation (3) generally models the case where two initially close tra- +jectories (d0 < ϵ) deviate from each other over time. The ℓk can depend on k, +since the expansion rates can vary as the trajectories pass through different +parts of the attractor [14]. We then define a trajectory-averaged expansion rate +ℓ as follows +ℓ = 1 +τ +τ +� +k=1 +ℓk = 1 +τ +τ +� +k=1 +ln dk +dk−1 +, +(4) +where τ is the saturation time defined earlier. Since we are considering a fixed +trajectory (not an ensemble of trajectories), we thus have +ℓ = 1 +τ ln dτ +d0 +. +(5) +Provided the GDS is ergodic (i.e., that a single and long enough orbit ade- +quately visits the whole graph phase space), ℓ converges to the network +Maximum Lyapunov Exponent λnMLE in the limit of large τ, independent of +the choice of the initial adjacency matrix At. However, in practice, τ will be +finite, and thus we cannot readily assume that ℓ fully describes the long-term +behaviour, or that it is independent of the initial condition At. It is thus inter- +preted as a local Lyapunov exponent [15], and an average of this quantity over +different initial conditions At will be required, as discussed further below. +2In practice, At′ is selected as the closest network from At within the ball of radius ϵ. + +Based on Springer Nature LATEX template +6 +Lyapunov Exponents for Temporal Networks +Fig. 1 Illustration of a temporal network S = (G0, G1, G2, ...) as a trajectory of a latent +graph dynamical system (GDS) and the methodology to compute its network Maximum +Lyapunov exponent (Kantz version) λK +nMLE. Each element of the trajectory is a network, +i.e. a snapshot of the temporal network. λK +nMLE is estimated directly from S (i.e., without +accessing the GDS directly) by looking at recurrences in S and quantifying the average +expansion around different network snapshots. In this illustration, a ball of radius ϵ around +an arbitrary G0 is fixed, and four recurrences are found where d(G0, Gr) < ϵ. The initial +distance d(G0, Gr) is averaged over the four recurrences and the average distance after one +time step d(G1, Gr+1) is computed. λK +nMLE is computed by averaging over time, volume and +different initial conditions G0 (see the text for details). +2.2.2 Wolf and Kantz methods of estimating maximum +Lyapunov exponents for temporal networks +The time it takes for two trajectories to reach a distance of the order of the +attractor’s size depends on how close these two trajectories were initially. In +other words, the saturation time τ depends on d0, and therefore on the choice + +Go +G1 +S= (Go,G1, G2, ..., Grt, Gr+1, ..., Gr2, Gr+1, ...) +Gr2+1 +Gr3 +Gra +Go +Gr2 +G1Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +7 +of the threshold ϵ. The limits d0 → 0 and τ → ∞ are thus related to one +another. Conceptually, we would like the trajectories to be as close as possible +initially, so that we can monitor the expansion rate for long times, allowing +us to obtain a global MLE as opposed to a local Lyapunov exponent. To do +this, we need to track the expansion for sufficiently long, but we also need to +avoid the regime in which the distance is limited by the characteristic size of +the attractor. +Generalised Wolf approach to measuring the nMLE. We now construct +the generalisation of Wolf’s approach, which will yield an estimation of the +nMLE that we call λW +nMLE. The aim is to compute ⟨ℓ⟩ = ⟨ 1 +τ ln dτ +d0 ⟩, where the +average is over choices of At and At′. In practice one considers a set of w +initial choices of At, which we index i = 1, . . . , w, and for each of these choices, +one additional point At′ on the trajectory such that d(At, At′) < ϵ. One then +obtains +λW +nMLE = 1 +w +w +� +i=1 +ℓ(i), +(6) +where ℓ(i) is the trajectory-averaged expansion rate ℓ computed for the i-th +initial condition, obtained from Eq. (4). +Algorithmically, this approach has the advantage that we do not need to fix +the choice of τ, we are able to flexibly adjust τ for each initial condition, +according to the specific d0 we are able to find in S. On the other hand, this +approach is point-wise, in the sense that for each choice of At one only con- +siders a single At′ nearby. As a consequence, this method does not necessarily +capture the average expansion rate around each initial condition At. +Generalised Kantz approach to measuring the NMLE. In order to cal- +culate such an average expansion rate (for a given choice of At) one would, for +a fixed At, have to average over the expansion rates for choices of At′ in an +ϵ-ball about At. This volume-averaging is the basis of Kantz’s generalization +[11, 12] of Wolf’s algorithm [10], see Fig.1 for an illustration. For a given ini- +tial condition At Kantz’ method provides a trajectory and volume averaged +expansion rate ⟨ 1 +τ ln dτ +d0 ⟩volume. We will write Λ for the volume-averaged expan- +sion rate. For fixed At, this could algorithmically be obtained as follows. One +chooses N different At′ from the trajectory S, all within distance ϵ from At. +We label these j = 1, . . . , N. For each At′ one then computes ℓ(j) via Eq. (4). +Then one sets +Λ(At) = 1 +N +N +� +j=1 +ℓ(j), +(7) +where N is the number of initial conditions inside a ball of radius ϵ and cen- +tered at At that we have found in the sequence S. +In practice, Kantz algorithm proceeds slightly differently. Instead of first com- +puting the ℓ(j), and then averaging the expansion rates, the average over the +At′ is instead computed at the level of distances. That is to say, one makes N + +Based on Springer Nature LATEX template +8 +Lyapunov Exponents for Temporal Networks +choices of At′ as described above, and then obtains dk(j) for each j = 1, . . . , N +and k = 0, 1, 2, . . . (k runs up to the relevant cut-off time). One then sets +Λ(At) = 1 +τ ln +N −1 �N +j=1 dτ(j) +N −1 �N +j=1 d0(j) +, +(8) +where τ is a priori fixed for all j. The numerator in the logarithm represents +the volume average (over choices of At′ in a ball about At) of dτ, and the +denominator is the volume average of d0. +We stress that Eqs. (7) and (8) are mathematically different and do not +necessarily lead to the same results. While Eq. (7) allows adapting the precise +value τ (which depends on d0) for each trajectory, Eq. (8) instead requires +us to use a uniform τ across all trajectories. The latter expression looks at +the expansion in time of an initially small volume centered on At, and is +closer in spirit to capturing the underlying nonlinear dynamics than Eqs. (7). +Accordingly, Eq. (8) is the basis of the Kantz’ method. +The quantity Λ in Eq. (8) is still a local quantity, in the sense that it +was computed for a phase space volume around a fixed choice of At. In prin- +ciple, the local volume-averaged expansion rate could vary across different +regions in phase space. To capture the global long-term behaviour we there- +fore additionally average over choices of At, and then finally obtain the global +volume-averaged network maximum Lyapunov exponent +λK +nMLE = +�1 +τ ln ⟨dτ⟩volume +⟨d0⟩volume +� +At +, +(9) +where we have written ⟨· · · ⟩At, for the average over initial conditions At. In +practice this is carried out by averaging over a set of w choices of At, i.e., +λK +nMLE = 1 +w +w +� +j=1 +Λ(A(j) +t ) +(10) +where Λ(A(j) +t ) stands for the expression in Eq. (8) for the fixed choice +At = A(j) +t . +We note that the value of the saturation time τ or the radius of the ball +ϵ will have to be selected after some numerical exploration. Indeed, a better +estimate of the Lyapunov exponent is obtained when the cut-off time τ is +large. This, in turn, is the case when the initial distance between At and At′ is +small, hence favouring choosing a relatively small value of ϵ. However, a small +value of ϵ complicates the task of finding points At′ that are at most a distance +ϵ away from At on the given trajectory. In practice, a trade-off is to be struck. + +Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +9 +To summarise, from a given TN trajectory (i.e. a sequence of network +snapshots) we first measure the local expansion rates {ℓk}τ +k=1 via Eq. (3) +for a fixed choice of At and At′. The set of ℓk obtained in this way provide +information on the fluctuations of the local expansion rate (for fixed At and +At′), and its trajectory-average ℓ. We can then proceed along two alternative +routes. In the first approach (i) we average ℓ over different choices for the +initial condition At [Eq. (6)], and obtain Wolf’s approximation to the nMLE +λW +nMLE. Alternatively, (ii) we can initially perform, for each initial condition +At, an average over At′. This is done by computing the local expansion of a +volume ⟨dk⟩volume and then averaging this over time [Eq. (8)]. This is then +repeated for different choices of At, and once hence obtains a distribution +P(Λ) describing the fluctuations across different points in the network phase +space. Its mean provides Kantz’s approximation to the nMLE λK +nMLE. +In the following sections, we present a validation of this method for random, +low-dimensional and high-dimensional chaotic temporal networks. +3 The white-noise equivalent of a temporal +network: independent and identically +distributed random graphs +Before addressing the case of chaotic dynamics, we briefly discuss the case of +random network trajectories, with no correlations in time. One then expects +no systematic expansion or compression in time, and the resulting Lyapunov +exponent should hence vanish3. We here seek to verify that this is indeed the +case for the procedures we have introduced to estimate the Lyapunov expo- +nents of TNs. Studying this is of interest, among other reasons, because an +empirically obtained time series may appear random. It is then important to +be able to decide if the trajectory is consistent with an uncorrelated random +trajectory in network space, or with a deterministic chaotic model. +Here we study the simple case where S is an independently drawn sequence of +Erd¨os-R´enyi graphs with n nodes and in which the probability that any two +nodes are connected is p. This is an analog to white noise for TNs, i.e., a situ- +ation in which the TN displays delta-distributed autocorrelation function [8]. +At odds with a deterministic GDS, the distances between different points on a +network trajectory are now random variables. More precisely, since all elements +of S are the adjacency matrices of Erd¨os-R´enyi graphs, the elements of these +matrices are Bernoulli variables, taking values zero (with probability 1 − p) or +one (with probability p). For independent adjacency matrices A and B, the +possible values of |aij − bij| are then zero with probability p2 + (1 − p)2, and +3By construction, d1 > d0 as we force d0 < ϵ, so in order to avoid spurious expansions at k = 1, +in this section we don’t take into account d0 in the estimation of finite Lyapunov exponents, i.e. +our starting time is k = 1. + +Based on Springer Nature LATEX template +10 +Lyapunov Exponents for Temporal Networks +one with probability 2p(1 − p). Thus we have +d(A, B) ∼ Binomial[n(n − 1)/2, 2p(1 − p)]. +(11) +Eq. (11) is numerically verified in panel (a) of Fig. 2. +Fig. 2 +Panel (a): Probability distribution of the distance between consecutive networks in +an i.i.d. sequence of |S| = 105 Erdos-Renyi graphs with n = 100 nodes and p = 0.01 (black +line). The plot is in semi-log, where a Gaussian shape appears as an inverted parabola, so +we can better appreciate the tails. Blue solid line is Eq. (11). Panel (b): P(ℓ) (black) and +P(Λ) (red) associated with a sequence of |S| = 5·104 i.i.d. ER networks (n = 100, p = 0.01), +for τ = 1 and w = O(104) initial conditions. The mean of both distributions (nMLE) is +essentially zero for both methods, but the dispersion around the mean is larger in Wolf’s +approach. The solid blue line is the theoretical prediction, i.e. a Gaussian distribution with +mean 0 and variance as in Eq. (14) with τ = 1. The inset of panel (b) shows how the variance +of ℓ shrinks as τ increases (resulting in a much lower uncertainty around a zero nMLE): dots +are different numerical simulations with |S| = 104 and w = 103, for different τ; the blue line +is Eq. (14). +The quantity ℓ in Eq. (5) is given by ℓ = 1 +τ (ln dτ − ln d0). As we have just +established, d0 and dτ are independent binomial random variables following +the distribution in Eq. (11). For large networks (n ≫ 1) this distribution can +be approximated as a Gaussian, with mean µ = qn(n − 1)/2 and variance +σ2 = q(1 − q)n(n − 1)/2, where we have written q ≡ 2p(1 − p). Writing +d0 = µ + σz0, with z0 a standard Gaussian random variable, we have +ln d0 ≈ ln +� +µ(1 + σ +µz0) +� += ln µ + σ +µz0 − 1 +2 +σ2 +µ2 z2 +0 + . . . , +(12) +after an expansion in powers of σ/µ, where the latter quantity is of order +O(1/n). The same expansion can be carried out for dτ, and we therefore find +ℓ = 1 +τ +�σ +µ(zτ − z0) + 1 +2 +σ2 +µ2 (z2 +0 − z2 +τ) +� ++ . . . . +(13) + +(a) +10-2 +P + simulation +10 +— theory +10-5 +60 +80 +100 +120 +140 +dsimulation +(b) +10-2 +Eq. (14) +4 + P() +- theory (Wolf) +3 +L P(Λ) +P(0), P(Λ) +1 +10 +1 +2 +0 +-0.5 +0 +0.5 +l, ^Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +11 +We note that the second term in the bracket is of sub-leading order in 1/n. +Hence, ℓ is to lowest order in 1/n approximately Gaussian, with mean zero +and variance +Var(ℓ) = 2 +τ 2 +σ2 +µ2 = 1 +τ 2 +4(1 − q) +qn(n − 1) +(14) +This theory has been numerically verified, and in panel (b) of Fig.2 we plot +P(ℓ) both for τ = 1 (outer panel) and Eq.14 for increasing values of τ (inset +panel). +The case of Λ (Kantz-version) should intuitively converge even faster than +ℓ (Wolf-version) since in this case we are carrying out two averages instead +of just one, i.e. P(Λ) should have a smaller variance than P(ℓ), for a given +τ. This is confirmed in panel (b) of Fig. 2, where we also observe that both +methods yield the same (correct) estimation of the nMLE, which in this case +is approximately zero (both estimates are of the order of 10−6). Note that the +main panels of Fig. 2) are for the case τ = 1, so it is a worst-case scenario: +as τ increases Var(ℓ) shrinks [Eq. (14)] and the uncertainty around the null +shrinks accordingly [see inset of Fig. 2(b)]. +We were not able to find a closed-form solution for P(Λ) as averages inside +the ϵ-ball are random variables whose distribution explicitly depends on the +specific initial condition At: this calculation is left as an open problem. In +anycase, we conclude that an i.i.d. temporal network has a null MLE Lyapunov +exponent and the methodology (in both variants) correctly estimates it. +4 Low-dimensional chaotic networks +4.1 Network generation: the dictionary trick +To be able to validate the method in the context of chaotic dynamics, we +ideally need to have access to chaotic network trajectories with a ground true +nMLE. This is difficult as a general theory of chaotic GDS is not yet accessible. +To circumvent this drawback, in this section we develop a method to construct +(low-dimensional) chaotic network trajectories by symbolising in graph space- +time series from low-dimensional chaotic maps. The method of graph-space +symbolisation was first proposed as a so-called ‘dictionary trick’ in [8] and +consists of the following steps: +• We construct a network dictionary D. This is a set of networks that allows us +to map a real-valued scalar x ∈ [0, 1]4 into a network, such that the distance +between two scalars is preserved in graph space. The set D is therefore +ordered and equipped with a metric, such that the distance between two real- +valued scalars |x−x′| is preserved in the graph symbols. More concretely, the +dictionary of networks D = (G1, G2, ..., GL) such that d(Gp, Gq) ∝ |p − q| +(one can subsequently normalize d according to the length of the dictionary, +such that we have d ∈ [0, 1]). +4We choose the interval [0, 1] without loss of generality. + +Based on Springer Nature LATEX template +12 +Lyapunov Exponents for Temporal Networks +• Once such dictionary is built, any one-dimensional time series can be +mapped into a sequence of networks. In particular, we can map chaotic +time series with well-known MLEs into network trajectories, from which an +independent estimate of the nMLE can be obtained. +Algorithmically, the dictionary is generated sequentially with G1 ∼ ER(p) +(an Erd¨os-R´enyi graph with parameter p) and then iteratively constructing +Gk+1 from Gk by rewiring a link that (i) has not been rewired in any previous +iteration of the algorithm, (ii) into a place that did not have a link in any pre- +vious iteration of the algorithm. It is easy to see that such algorithm ensures +that D provides a partition of [0, 1] of the form [0, 1] = ∪L−1 +k=0 [k/L, (k + 1)/L], +where L is the number of networks in the dictionary. The dictionary is thus +metrical, in the sense that the rewiring distance between any two elements +in the dictionary is (for a sufficiently large refinement L) arbitrarily close to +the associated real-valued scalars in the original interval. Once the dictionary +is established, we can then generate synthetic temporal network trajectories +as symbolizations of unit interval dynamics by matching points in the subin- +terval [k/L, (k + 1)/L] to the symbol Gk+1. The resulting temporal network +S inherits, by construction, the properties of the scalar time series, and in +particular can be used to generate chaotic TNs. +Fig. 3 Panel (a): Semi-log plot of the distance dk as a function of iteration index k, for +two initially close network trajectories sampled from S. We can appreciate an initial expo- +nentially expanding phase, followed by a saturation phase, although the local expansion +rate strongly fluctuates. Panel (b) Volume-averaged distance ⟨dk⟩volume as a function of +time k, for N = 17 initial graph conditions inside a volume centered at an initial graph of +n = 500 nodes and m = 2000 edges. Network dynamics evolve according to a logistic map +as described in the text, whose true Lyapunov exponent is ln 2 ≈ 0.693. We can see how the +volume enclosing the graphs on average expands exponentially fast –with an exponent close +to ln 2, as expected– until it reaches the attractor size, what happens at the saturation time +τ ≈ 10. + +In2·k +n = 500 +0.1 +m = 2000 +0.01 +(a) +0 +10 +20 +30 +kn = 500 +m = 2000 +N = 17 +t (saturation time) +0.01 +In2·k +(b) +0 +5 +10 +15 +20 +25 +30 +kSpringer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +13 +4.2 Results for the logistic map +As a first validation, we consider the fully chaotic logistic map +xt+1 = 4xt(1 − xt), +xt ∈ [0, 1], +(15) +that generates chaotic trajectories with λMLE = ln 2 ≈ 0.693. Using the dic- +tionary trick, from a signal extracted from Eq. (15) we generate a temporal +network trajectory S of |S| = 3000 network snapshots. In this validation, +networks have n = 500 nodes and m = 2000 edges. +Fig. 4 Panel (a): Approximation to λW +nMLE following Wolf’s approach (see text), computed +by averaging ℓ over w randomly sampled initial conditions [Eq.6], as a function of w. We +can see that the exponent converges to the ground true exponent ln 2 as w increases. Inset +in (a): Probability distribution P(ℓ), sampled by estimating ℓ for w = 500 different initial +graph conditions sampled randomly from S. The mean of this empirical distribution is +λW +nMLE = ⟨Λ⟩At ≈ 0.685, very close to the true exponent ln 2 ≈ 0.693. Panel (b): Same as +panel (a), but using Kantz’s approach (see the text), where we compute the volume and +trajectory averaged expansion rate Λ for w initial conditions. Convergence properties are +similar in both cases. +For illustration, in panel (a) of Fig. 3 we plot in semi-log scales the +(properly normalized) distance dk as a function of the iteration index k, for +two initially close network trajectories sampled from S. We can see an initial +exponentially expanding phase (whose exponent is an estimation of ℓ) followed +by a saturation, although the distance function shows strong fluctuations. To +cope with these, in panel (b) of the same figure we plot the volume-averaged +expansion ⟨dk⟩volume vs k for a ball of radius ϵ = 0.005 centered at a specific +initial graph from S. We can now clearly see the initial exponential phase +followed by a cross-over to a saturation phase. The cross-over marks the +saturation time τ where the distance reaches the attractor size. Note that the +slope of the exponential expansion (i.e. the estimate of Λ) is close to ln 2, the +true MLE. + +1.2 +(a) +2 +W +0.685 +1.0 +P1 +0 +0.8 +0 +0.5 +1.0 +1.5 +In 2 +0.6 +0 +100 +200 +300 +400 +500 +w (# initial conditions)1.1 +3 +(b) +XNMLE = 0.685 +1.0 +2 +P(Λ) +0.9 +1 +0 +0.8 +-0.5 +0 +0.5 +1.0 +V +- In 2 +0.7 +0 +100 +200 +300 +400 +500 +w (# initial conditions)Based on Springer Nature LATEX template +14 +Lyapunov Exponents for Temporal Networks +Figure 4 shows the estimated of the nMLE obtained both using Wolf’s +approach [panel (a)] and Kantz’s approach [panel (b)]. These are from aver- +aging ℓ (Wolf) and Λ (Kantz) over w = 500 initial graph conditions sampled +from S. In both cases, the average quickly stabilises for w ≈ 100, and we +obtain estimates λW +nMLE ≈ λK +nMLE ≈ 0.685, very close to the ground true +λMLE = ln 2 ≈ 0.693. +Fig. 5 Volume-averaged distance ⟨dk⟩volume as a function of time k, for a network dynamics +evolving according to a chaotic logistic map xt+1 = 4xt(1 − xt), polluted with extrinsic +Gaussian noise N(0, σ2) as described in the text, for four different noise intensities σ = +0, 10−3, 10−2, 10−1. The exponential expansion phase –which systematically suggests the +same exponent ln 2, as expected– is gradually erased as the noise intensity increases. +4.3 Noisy chaotic networks +To explore how noise contamination can complicate the estimation of the +nMLE, we proceed to generate a temporal network S from Eq. (15) by using +the dictionary trick, where before the network mapping, the original chaotic +signal is contaminated by a certain amount of white Gaussian noise N(0, σ2)5. +As we did in Section 3, we remove potential algorithmic biases by discarding +⟨d0⟩ for the computation of Λ. Results are summarised in Fig. 5. The main +observation is that noise pollution tends to reduce the extent of the exponen- +tial phase (i.e., the saturation time τ decreases). For small amounts of noise, +this phase is still observable, and the estimated nMLE continues to be con- +sistent with that of the noise-free case. When the noise intensity is above a +certain threshold, noise effectively hides the chaotic signal, and the exponential +phase can no longer be identified, resulting in an apparent vanishing nMLE. +These results are consistent with intuition and with the typical phenomenology +observed in noisy chaotic time series [10], [11]. +5Note that we discard realizations of the noise that take the scalar variable outside the unit +interval. + +10 +10° +=0 +0=10-3 +0=10-2 +0=10-1 +exp(In2 k) +10-4 +2 +4 +6 +8 +10 +12 +14 +kSpringer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +15 +4.4 Results for the parametric logistic map +Here we consider the logistic map xt+1 = rxt(1 − xt). For each value of the +parameter r > r∞, using the dictionary trick we generate a long sequence of +networks Sr with the desired chaoticity properties, and proceed to estimate +the network Lyapunov exponent using the method detailed in Section 2. In +panel (a) of Fig.6 we plot λW +nMLE vs λMLE of the map, for a range of values of +the parameter r. The agreement is excellent in the region of parameters where +the temporal network is chaotic. +Fig. 6 +Panel (a): Scatterplot of λW +nMLE, estimated from a temporal network Sr generated +via the dictionary trick (see the text) from a logistic map xt+1 = rxt(1 − xt) for a range +of values of r, vs the ground true λMLE. The solid line is the diagonal of perfect agreement +y = x, highlighting the good agreement found in the chaotic region. The legend states +the goodness of fit metric R2 of the fit of dk to an exponential function. The method is +unable to capture negative Lyapunov exponents (observe that in those cases the R2 of the +exponential fit is very bad), but these cases can easily be identified as periodic orbits using +the autocorrelation function [8], see text for details. Panel (b): Estimate of the negative +nMLE using two initially close temporal networks generated via the dictionary trick from the +logistic map at r = 3.4 (period-2 orbit), where one initial condition belongs to the period-2 +attractor and the other is outside the attractor (see the text for details). +4.5 A note on negative Lyapunov exponents +The classical approach to estimate the MLE from a single trajectory displayed +by Wolf and Kantz algorithms –based on recurrences of the trajectory– is, by +construction, unable to capture negative MLEs. The reason is straightforward: +once in the periodic attractor, the trajectory sequentially visits each element +of the periodic orbit, and thus we won’t find recurrences that are close but +away from the initial condition of interest. Accordingly, our method to esti- +mate nMLE cannot work in that case for the same reasons, as confirmed in +Fig. 6(a). This drawback can be solved using two alternative approaches. +First, it is well known that a periodic time series has an autocorrelation +function that peaks at the period of the time series. Interestingly, a recent +work [8] has operationalised a way to estimate the autocorrelation function + +10-2 +r = 3.4 +入MLE = -0.137 +exp(-0.143k) +10-3 +(b) +0 +5 +10 +15 +20 +25 +k0.8 +R? +< 0.2 +0.6 +E (0.2,0.7) +E +[0.7,0.8] +0.4 +E [0.8,0.9] +> 0.9 +0.2 +(a) ++ +0 +++ +-0.5 +0 +0.5 +ΛMLEBased on Springer Nature LATEX template +16 +Lyapunov Exponents for Temporal Networks +of temporal networks, whereby temporal networks that display periodicity +are well characterised by a network version of the autocorrelation function. +Accordingly, from a practical point of view, before attempting to estimate the +nMLE of a given temporal network, it is sensible to apply the procedure of [8] +and exclude that the temporal network is periodic –which would typically6 +mean a negative nMLE–. Once this test is done, it is sensible to conduct the +nMLE analysis presented in this paper. +Second, it is indeed possible to estimate negative nMLEs if one has access +to the latent graph dynamical system (GDS), as in this case one does not need +to undergo a Wolf/Kantz approach and one can generate through the GDS +temporal networks from close initial graph conditions. To illustrate this, in +panel (b) of Fig. 6 we plot the graph distance of two initially close networks +evolving according to the logistic map for a value of the map’s parameter for +which the orbit is periodic (the TNs are again generated via the dictionary +trick). One initial condition is set at one of the orbit elements, whereas the +other initial condition is a network close in graph space (but outside the peri- +odic attractor). As we can see, there is an exponential shrinking of the initial +distance, and the slope gives an estimate of the nMLE, which in this case is +negative and in good agreement with the theoretical result. +5 High-dimensional chaotic networks +We now consider the case of high-dimensional chaotic dynamics for temporal +networks. We first introduce a generative model, based on coupled Map Lat- +tices (CML). These are high-dimensional dynamical systems with discrete time +and continuous state variables, widely used to model complex spatio-temporal +dynamics [20] in disparate contexts such as turbulence [21, 25], financial mar- +kets [26], biological systems [27] or quantum field theories [28]. +Globally Coupled Maps (GCMs) [29] are a mean-field version of CMLS, where +the diffusive coupling between the entities in a CML is replaced with an all- +to-all coupling, mimicking the effect of a mean-field. We consider a globally +coupled map of m entities, of the form +xi(t + 1) = (1 − α)F[xi(t)] + α +m +m +� +j=1 +F[xj(t)], i = 1, 2, . . . , m, +(16) +where F(x) = 4x(1 − x), x ∈ [0, 1], where α ∈ [0, 1] is the strength of the +mean-field coupling. In the uncoupled case α = 0, the system is composed of m +independent fully chaotic dynamics. Its attractor is thus high-dimensional and, +since there are m Lyapunov exponents all equal to ln 2, we have λMLE = ln 2. +6Some pathological cases exist for which we can have seemingly periodic behavior but not a +negative MLE, e.g. when we have a disconnected attractor composed by a number of bands and +a trajectory that periodically visits the different chaotic bands + +Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +17 +At the other extreme, for complete coupling α = 1, the system is fully synchro- +nized (i.e., for any time t we have xi(t) = xj(t) for all i, j), and the dynamics +is reduced to the one-dimensional dynamics, again with λMLE = ln 2. We add +that complete synchronization is in fact known to occur for α > 1/2 [29]. +For intermediate coupling the system shows a number of different macroscopic +phases [29]. Among these one finds high-dimensional chaos for weak coupling +α < 0.2. This is the so-called ‘turbulent state’. Interestingly, for CMLs with +diffusive coupling, a scaling law has been established [22] +λMLE = log 2 − βα1/p, +(17) +where p indicates the type of nonlinearity of F(x), i.e. p = 2 for the logistic +map, p = 1 for tent maps, etc. Results for GCM are less clear. However, when +the mean-field coupling can be considered ‘thermalized’ (i.e., independent of x) +[23, 24] then Eq. (17) holds for β = 1. However such thermalization is known +to be true only for tent maps (p = 1) and not logistic maps. +Here we consider the range α ∈ [0, 0.2], i.e., the turbulent state of the +GCM. We interpret the collection {xi}m +i=1 as the (weighted) edge set of a fully +connected undirected network backbone of n nodes and m = n(n−1)/2 edges. +Once the time series of each edge {xi(t)}T +t=1 has been computed from Eq. (16), +we proceed to binarise each edge activity by using a two-symbol generating +partition as follows: values xi(t) < 1/2 are mapped into the symbol 0, and +xi(t) ≥ 1/2 onto the symbol 1[30]. Note that the use of a generating partition +ensures that the symbolised (binary) series preserves the chaotic properties +of the original signal [31–33]. Finally, we convert the (binary) evolution of +the edges into a time-dependent adjacency matrix, thereby constructing a +temporal network S. For values of α in the weak-coupling regime, we expect +the temporal network to display sensitive dependence on initial conditions. +In practice, the Wolf/Kantz methods of inferring the larges Lyapunov +exponent proposed in the paper would require a very long sequence S for close +enough recurrences to be observable in a system with large n. However, here +we have access to the actual underlying GDS via Eq. (16). Given that the goal +of this section is to show evidence that high-dimensional chaotic networks can +be generated and their nMLE be estimated, we can use the GDS to generate +the temporal network for any required initial condition. Accordingly, for a +given initial condition {xi(0)}m +i=1, we construct a perturbed copy {x′ +i(0)}m +i=1 +(where |x′ +i(0) − xi(0)| < ϵ for some small choice of ϵ), generate temporal +networks for both of these initial conditions, and track the network distance +between the copies over time. We do this for 100 replicas to extract a volume- +averaged distance, and then for 50 different initial conditions. Observe that, +at odds with the model developed in the previous section, here the number of +edges in each network snapshot is not fixed, and thus the network phase space +is substantially larger. Similarly, the normalization factor of the distance +function is now simply the total number of possible edges, n(n − 1)/2. + +Based on Springer Nature LATEX template +18 +Lyapunov Exponents for Temporal Networks +Fig. 7 Panel (a): Semi-log plot of the volume-averaged distance ⟨dk⟩volume as a function +of the time step k, for a temporal network extracted from the GCM model with coupling +constant α = 0, 0.05, 0.1. We observe an exponential phase, with different exponents for +each value of the coupling constant. The solid lines are the best exponential fits. Panel (b): +Estimate of the network maximum Lyapunov exponent λnMLE vs the coupling constant α +for temporal networks generated from a GCM of logistic maps (blue circles) and tent maps +(black squares). For each α, a total of 50 initial conditions were considered, and a ball of 100 +points for each initial condition was used. Error bars are standard deviation from the average +over 50 different temporal network realizations. Blue lines report the theoretical predictions +for logistic and tent CMLs [Eq. (17)], whereas the black line reports the theoretical prediction +for GCM with thermalised mean-field, applicable for tent GCMs only. +Results for a network of n = 100 nodes are shown in Fig. 7. In panel (a) we +plot ⟨dk⟩volume vs time k, for three different coupling constants α = 0, 0.05, 0.1 +in the weak coupling regime. In every case we find a clear exponential phase. +The exponent in the uncoupled phase α = 0 is indeed equal to ln 2, as expected, +further validating the method. For increasing values of the coupling, interest- +ingly, the nMLE seems to decrease and, as a byproduct, the saturation time τ +increases. In Fig. 7(b) we plot, as blue dots, the estimated λnMLE as a function +of the coupling α ∈ [0, 0.2], indeed showing a clear decrease. Such decrease +might be induced by the fact that the m degrees of freedom are now coupled +in some nontrivial way. Blue lines correspond to the theoretical predictions for +logistic and tent CMLs obtained from Eq. (17). For completeness, we repeated +the same analysis for network GCMs constructed from tent maps where λMLE +is explicitly known (black line): F(x) = 1 − 2|x|, with x ∈ [−1, 1] and a sym- +bolisation partition with x < 0 mapped to the symbol 0, and x ≥ 0 mapped +to 1. Results for this case are plotted as black squares in Fig. 7(b) +We conclude that (i) the TN thereby generated exhibits high-dimensional chaos +and its nMLE, reconstructed with the methods we have developed, shows the +expected behaviour, and (ii) this validation shows that the method works with +TNs where not only the position but also the total number of edges itself +fluctuates over time. + +(a) +10 +10- +2 +α=0 +exp(0.68 k) (R² = 0.99) +α = 0.05 +exp(0.56 k) (R² = 0.99) +α = 0.1 +10-3 +exp(0.44 k) (R2 = 0.99) +0 +5 +10 +15 +k0.7 +(b) +0.6 +0.5 +入nMLE +E +0.4 +logistic map network + log2 -βVa +0.3 +.... log2 - βa + tent map network +0.2 + log2 + log(1-a) +0 +0.1 +0.2 +aSpringer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +19 +6 Discussion +In this work, we propose to look at temporal networks as trajectories of a +latent Graph Dynamical System (GDS). This interpretation naturally leads +us to explore whether these trajectories can show sensitive dependence on +initial conditions, a fingerprint of chaotic behaviour. We have proposed a +method to quantify this, and defined and computed the network Maximum +Lyapunov Exponent (nMLE) for temporal network. Since the latent GDS +is rarely available in practice, our algorithm exploits the recurrences of the +temporal network in graph space. It generalizes the classical approaches of +Wolf and Kantz to networks. We have validated the method by generating +different synthetic GDS with known ground-truth nMLE. +Conceptually speaking, quantifying chaos in the trajectory of structured +objects (in our case, mathematical graphs) is somewhat close in spirit to quan- +tifying the dynamical stability of (lattice) spin systems. Thus our approach +shares some similarities with the damage-spreading [34] and self-overlap meth- +ods [35] in statistical physics, and their applications to cellular automata [36] +and random Boolean systems [37]. +Observe that we have focused on exponential expansion on nearby con- +ditions –i.e., sensitive dependence–, since one of the goals of the paper is to +conceptually postulate the existence of chaotic networks and to potentially +operationalise a way to measure this deterministic fingerprint in observed TNs, +without needs to having access to the underlying GDS. However, our approach +can be straightforwardly extended to non-exponential divergence, e.g. alge- +braic or otherwise, simply by suitably modifying the definition of expansion +rates, thus yielding a way to quantify other types of dynamical instability. +The rationale of this work is to consider graphs evolving over time as whole +–yet not punctual– objects [8], and thus consider its evolution in graph space. +It is however true that this approach might have a limitation for (large) real- +world temporal networks, as it is often difficult to observe recurrences in high- +dimensional space. A possible solution is to extract suitable scalar variables +from the network, analyse sensitive dependence on initial conditions in each +of them, and extract a consensus. We leave this approach for future work. +Observe that throughout this work we have considered labelled networks. +This choice was used for, convenience, illustration, and because TNs are usu- +ally labelled, but we expect that a similar approach is possible for unlabelled +TNs, i.e. graphs that evolve over time according to a certain graph dynamics. +In this latter case, each network snapshot is no longer uniquely represented +by a single adjacency matrix, in the sense that permutations of the rows and +columns of the matrix lead to an equally valid description. It is then clear +that one needs to use graph distances showing invariance under permutation +of rows and columns in the adjacency matrices [38]. This could be, for exam- +ple, distances based on the network spectrum, or graph kernels [39]. We leave +this interesting extension as a question for further research, as well as the + +Based on Springer Nature LATEX template +20 +Lyapunov Exponents for Temporal Networks +quantification of the full Lyapunov spectrum beyond the maximum one. +Finally we would like to add that the fact that the method does not rely +on knowing the GDS and instead directly estimates the nMLE from tempo- +ral network trajectories enables the investigation of these matters in empirical +temporal networks. We foresee a range of potentially interesting applications +in physical, biological, economic and social sciences –as indeed temporal net- +works pervade these disciplines–. This approach is specially appealing in those +systems where we don’t have access to the ‘equations of motion’ but it is sen- +sible to expect some underlying deterministic dynamics, i.e. physical systems, +but the approach is also extensible to systems with socially or biologically- +mediated interactions, for instance: do flocks of birds [40–42] or crowd behavior +[43], adequately modelled as temporal proximity networks, show chaos? +Appendix: Graph distances +Consider two adjacency matrices A, and B, each with binary entries (0 or +1), describing two simple unweighted graphs with n nodes. The so-called edit +distance [18] is a matrix distance defined as +d(A, B) = +n +� +i,j=1 +|aij − bij|. +(18) +The object d(A, B) counts the number of entries that are different in A and B. +For simple undirected graphs (symmetric adjacency matrices), we need +to account for the fact that the number of edges is only half the number of +positive entries of the adjacency matrix, and therefore d(A, B)/2 measures +the number of edges that exist in one graph but not on the other. We have +d(A, B)/2 = 0 if and only if A = B. It is also easy to see that d(A, B)/2 only +takes integer values for symmetric adjacency matrices A and B. If A ̸= B, then +1 ≤ d(A, B)/2 ≤ n(n − 1)/2. We have d(A, B)/2 = 1 when the two graphs are +identical except for one edge, which is present in one graph and absent in the +other. +One can directly use this unnormalized distance (as we do in Section 3) or +subsequently normalize d(A, B) using different strategies, e.g. one can divide +it over n(n − 1)/2 (as we do in Section 5), or just divide over the maximum +possible distance, if further restrictions are imposed between A and B (as we +do in Section 4). +If we further impose that both graphs have the same number of edges, then +the lower bound cannot be attained and 2 ≤ d(A, B)/2 when A ̸= B. This +lower bound is reached when we only need a single edge rewiring to get from + +Springer Nature 2021 LATEX template +Lyapunov Exponents for Temporal Networks +21 +the first graph to the second. One can thus define the rewiring distance +d(A, B) = 1 +4 +n +� +i,j=1 +|aij − bij|. +(19) +applicable for simple graphs (i.e. no self-links). This quanity measures the +total number of rewirings needed to transform A into B when the associated +graphs are simple, unweighted, undirected (symmetric adjacency matrices) +and have the same number of nodes and edges. +The rewiring distance above is based on the concept of non-overlapping +edges, i.e., edges that are present in one graph, but not in the other. Thus, +the edit and rewiring distances are based on |aij − bij| for the different edges, +and hence assign the same importance to the presence or absence of an edge. +One can instead construct measures of distance based on the number of links +that are present in both networks. If the edge ij is present in both graphs +then aijbij = 1, while this product is zero otherwise. One can prove that the +following function is a distance [19]: +d(A, B) = 1 − +1 +2|E| +n +� +i,j=1 +aijbij. +(20) +We replicated the analysis in Sec. +4 for the distance defined above, and +results (resulting nMLE) coincide. +Acknowledgments We thank Federico Battiston for helpful comments on +initial phases of this research, and Emilio Hern´andez-Fern´andez, Sandro Mel- +oni, Lluis Arola-Fern´andez, Ernesto Estrada, Massimiliano Zanin, Diego Paz´o +and Juan Manuel L´opez for helpful discussions around several aspects of the +work. AC acknowledges funding by the Maria de Maeztu Programme (MDM- +2017-0711) and the AEI under the FPI programme. LL acknowledges funding +from project DYNDEEP (EUR2021-122007), and LL and VME acknowledge +funding from project MISLAND (PID2020-114324GB-C22), both projects +funded by the Spanish Ministry of Science and Innovation. 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Traffic and related self-driven many-particle systems. +Reviews of modern physics 73, 4 (2001): 1067. + diff --git a/XNFOT4oBgHgl3EQf8TTm/content/tmp_files/load_file.txt b/XNFOT4oBgHgl3EQf8TTm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..74150910523a8f90ee8a4fa18c0b2023c2a8f3fc --- /dev/null +++ b/XNFOT4oBgHgl3EQf8TTm/content/tmp_files/load_file.txt @@ -0,0 +1,833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf,len=832 +page_content='Based on Springer Nature LATEX template Lyapunov Exponents for Temporal Networks Annalisa Caligiuri1, Victor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Egu´ıluz1, Leonardo di Gaetano2, Tobias Galla1 and Lucas Lacasa1* 1Institute for Cross-Disciplinary Physics and Complex Systems (IFISC), CSIC-UIB, Palma de Mallorca, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2Department of Network and Data Science, Central European University, 1100 Vienna, Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' E-mail(s): lucas@ifisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='uib-csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Abstract By interpreting a temporal network as a trajectory of a latent graph dynamical system, we introduce the concept of dynamical instabil- ity of a temporal network, and construct a measure to estimate the network Maximum Lyapunov Exponent (nMLE) of a temporal net- work trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Extending conventional algorithmic methods from nonlinear time-series analysis to networks, we show how to quantify sensitive dependence on initial conditions, and estimate the nMLE directly from a single network trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We validate our method for a range of synthetic generative network models displaying low and high dimensional chaos, and finally discuss potential applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Keywords: Lyapunov exponent, temporal networks, chaos, complex systems 1 Introduction Temporal networks (TNs) [1–3] are graphs whose topology changes in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' They are minimal mathematical models that encapsulate how the interaction architecture of elements in a complex system changes dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' TNs have been successfully used in a variety of areas ranging from epidemic spreading [4] or air transport [6] to neuroscience [7] to cite a few, and it has been shown that important dynamical processes running on networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' epidemics [4], synchronization, search [5]) display qualitatively different emergent patterns when the substrate is a TN, compared to the case of a static network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' These 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='12966v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='data-an] 30 Jan 2023 Based on Springer Nature LATEX template 2 Lyapunov Exponents for Temporal Networks effects are particularly relevant when the timescale of the dynamics running on the graph is comparable to that of the intrinsic evolution of the network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', when there is no manifest separation of timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Relatively lesser work has, however, considered the intrinsic dynamics of the network from a principled point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Recently, a research programme has been proposed [8] in which TNs are to be interpreted as the trajectories of a latent ‘graph dynamical system’ (GDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The GDS provides an explicit model for the time- evolution of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Similar to a conventional dynamical system (whose output is a time series of scalar or vector quantities), the output of a GDS is a time series of networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' a TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The dynamics of TNs and GDS are indeed the objects of ongoing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For instance, in [8] the authors consider how to extend the autocorrelation function of a signal to a graph-theoretical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' They explore how TNs can oscillate and how harmonic modes, as well as decaying linear temporal correla- tions of various shapes, emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In a similar fashion, the memory of a temporal network has been studied from different angles, including the concept of mem- ory shape [9] as a multidimensional extension of memory (high order Markov chain theory) in conventional time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In this work, we further pursue the abovementioned research framework programme, and consider the problem of dynamical instability and chaos quantification in TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Interpreting temporal networks as trajectories in graph space, we aim to generalise the concept of Lyapunov exponents as quantifiers of the sensitivity to initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Our objective is to define and measure Lyapunov exponents and sensitive dependence to initial conditions of the network as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Our approach is therefore not to quantify chaos in the dynamics for example of every link, but rather to quantify chaos for the collective dynamics of the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Since TNs in applications are frequently observed empirically, we focus our implementation and inference of network Lyapunov exponents solely on the observation of a single (long) TN trajectory, without the need to access the underlying GDS (but of course, the framework is also applicable if the GDS is explicitly accessible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Our algorithmic implementation can thus be seen as a conceptual network generalization of the classical algorithms by Wolf and Kantz [10–12, 14], originally proposed to quantify sensitive dependence on initial conditions directly from empirically observed time series (see also [13] for a similarly seminal work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Importantly, any new method needs to be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In our case, this is not trivial, since the notion of chaotic TNs is not common in existing litera- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' A second objective of this work is thus to propose synthetic generative models of chaotic TNs, which can be used as templates to validate the meth- ods we develop for the quantification of chaos in TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' These methods, once validated, can then be used in wider applications and further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 3 The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In Section 2 we introduce the theoretical background to our work, and we set the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We derive the network analog of the maximum Lyapunov exponent (MLE), and we outline an algorithmic implementation to estimate quantities such as the spectrum of local expansion rates, trajectory-averaged and volume-averaged expansion rates, and the network maximum Lyapunov exponent (nMLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In Section 3 we consider the relatively simple case of random network dynamics as a first example, and show how the method works in such scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Then, in Section 4 we introduce a generative model of (low-dimensional) chaotic temporal net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This model provides us with ‘ground-truth’ access to the nMLEs of the network trajectories that the model generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We show that the method we propose to infer nMLEs from a trajectory of networks correctly reconstructs this ‘true’ exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We also assess how the estimation of the nMLE is affected when the chaotic network trajectory is polluted with certain amounts of noise, and discuss at this point how to estimate negative nMLEs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In Section 5 we introduce a different generative model of (high-dimensional) chaotic TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We demonstrate that the generated TNs show sensitive dependence to initial conditions, and that the nMLE varies as expected as a function of a network’s coupling parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In Section 6 we finally conclude and discuss potential applications of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2 Theory and method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 Lyapunov exponents for graph dynamical systems In nonlinear time series analysis, the maximum Lyapunov exponent λMLE of a dynamical system quantifies how two trajectories that are initially close separate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' More precisely, one imagines two copies of the system, which are started from initial conditions at time t = 0 which are a distance d0 apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One then defines λMLE = lim t→∞ lim d0→0 1 t ln dt d0 , (1) where dt is the distance between the two copies of the system at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In practice, the distance d0 is often small but finite, and the limit d0 → 0 may not be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It then turns out that the long-time limit t → ∞ is not accessible either as the growth of dt is bounded by the size of the attractor of the system [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In such cases, the behaviour of dt usually undergoes a cross-over (at a time which we label τ), between an exponentially expanding phase (t < τ) to a saturated phase (t > τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In the latter regime, dt fluctuates around the attractor’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We will call τ the saturation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In the network setting, we assume there exists a (sometimes unknown) graph dynamical system that determines the evolution of a graph over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We focus on discrete time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The GDS is then a map, which determines how Based on Springer Nature LATEX template 4 Lyapunov Exponents for Temporal Networks one network evolves in the next time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For simplicity, we assume that the set of vertices is fixed, and that the vertices are distinguishable from one another and labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Thus, only the set of edges between these ver- tices evolves in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' A trajectory of the GDS then consists of a sequence of network snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' These trajectories define the TNs generated by the system [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Each trajectory is given by the sequence of adjacency matrices S = (At)t≥0, At = {aij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' t};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' i, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , n where aij;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' t = 1 if the vertices i and j are connected at time t, and zero otherwise (this symmetric choice is for simple, undirected networks, but the method works essentially along the same lines for non-simple and directed networks, perhaps except for different normalization factors in the definition of the distance, see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' As such, we are considering labelled, unweighted networks of n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is not clear a priori if the specific choice of the distance used to quantify the deviation between two originally close network trajectories is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We conjecture that, as long as this distance is based on the full adjacency matrix –not on a projection of it–, results should hold independent of the specific met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is based on the fact that in dynamical systems, the MLE is invariant under different choices of the underlying norm ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='|| [14, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' There exist many graph distances [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For simplicity, we take an intuitive definition of such a distance which is based on the amount of edge overlap between two networks: given two networks with the same number of nodes n and adjacency matrices A and B, d(A, B) = 1 2 � i,j |aij − bij|, (2) where i and j take values from 1 to n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In our setting, networks are simple and unweighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In the particular case where A and B have the same number of edges, the distance defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (2) is indeed a rewiring distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', it is given by the number of unique rewirings needed to transform A into B, and therefore d is a positive integer-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One can then further normalize d as appropriate such that it is defined in [0, 1], as we will show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Further details can be found in the Appendix, where we also introduce alternative distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 Inference of network Lyapunov exponents 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 Local expansion rates We are interested in quantifying sensitive dependence on initial conditions (and in particular, the network version of λMLE, which we here call λnMLE) when the mechanics of the GDS is not known, and when we only have access to a (single) discrete-time network trajectory S = (A0, A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' ) of adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is analogous to the case in which one would like to reconstruct the MLE of a conventional dynamical system from a single time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The 1The pre-factor is used to have a normalized distance when networks are simple, undirected and have the same number of links, otherwise a different normalization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 5 standard approach consists in using Wolf’s or Kantz’s algorithms [10–12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The central idea is here to look for recurrences in the orbit, finding points in the time series which might be temporally separated but which are close in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One then monitors the deviation of those points over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Here, we extend this approach to the case of a time series of networks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' a TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We start by fixing an element At from S, where t is an arbitrary point in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This adjacency matrix will be the ‘initial condition’ for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' To extend Wolf’s algorithm we then proceed to look in S for another element At′ at a different time t′, such that d(At, At′) < ϵ, where ϵ is a small threshold chosen before the analysis begins2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This recurrence in phase space allows us to use a single trajectory to explore how two close networks separate over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We then set d0 := d(At, At′) as the initial distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We then proceed to measure how distance evolves over time as we separately track the evolution of At+k and At′+k in S, where k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We write dk = ||At+k − At′+k||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Without loss of generality, we can always write the successive distances in terms of a sequence of local expansion rates ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , dk = dk−1 exp(ℓk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (3) Each of the ℓk can be positive (local expansion), negative (local contraction), or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Equation (3) generally models the case where two initially close tra- jectories (d0 < ϵ) deviate from each other over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The ℓk can depend on k, since the expansion rates can vary as the trajectories pass through different parts of the attractor [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We then define a trajectory-averaged expansion rate ℓ as follows ℓ = 1 τ τ � k=1 ℓk = 1 τ τ � k=1 ln dk dk−1 , (4) where τ is the saturation time defined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Since we are considering a fixed trajectory (not an ensemble of trajectories), we thus have ℓ = 1 τ ln dτ d0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (5) Provided the GDS is ergodic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', that a single and long enough orbit ade- quately visits the whole graph phase space), ℓ converges to the network Maximum Lyapunov Exponent λnMLE in the limit of large τ, independent of the choice of the initial adjacency matrix At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' However, in practice, τ will be finite, and thus we cannot readily assume that ℓ fully describes the long-term behaviour, or that it is independent of the initial condition At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is thus inter- preted as a local Lyapunov exponent [15], and an average of this quantity over different initial conditions At will be required, as discussed further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2In practice, At′ is selected as the closest network from At within the ball of radius ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Based on Springer Nature LATEX template 6 Lyapunov Exponents for Temporal Networks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 1 Illustration of a temporal network S = (G0, G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=') as a trajectory of a latent graph dynamical system (GDS) and the methodology to compute its network Maximum Lyapunov exponent (Kantz version) λK nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Each element of the trajectory is a network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' a snapshot of the temporal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' λK nMLE is estimated directly from S (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', without accessing the GDS directly) by looking at recurrences in S and quantifying the average expansion around different network snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In this illustration, a ball of radius ϵ around an arbitrary G0 is fixed, and four recurrences are found where d(G0, Gr) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The initial distance d(G0, Gr) is averaged over the four recurrences and the average distance after one time step d(G1, Gr+1) is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' λK nMLE is computed by averaging over time, volume and different initial conditions G0 (see the text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 Wolf and Kantz methods of estimating maximum Lyapunov exponents for temporal networks The time it takes for two trajectories to reach a distance of the order of the attractor’s size depends on how close these two trajectories were initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In other words, the saturation time τ depends on d0, and therefore on the choice Go G1 S= (Go,G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', Grt, Gr+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', Gr2, Gr+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=') Gr2+1 Gr3 Gra Go Gr2 G1Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 7 of the threshold ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The limits d0 → 0 and τ → ∞ are thus related to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Conceptually, we would like the trajectories to be as close as possible initially, so that we can monitor the expansion rate for long times, allowing us to obtain a global MLE as opposed to a local Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' To do this, we need to track the expansion for sufficiently long, but we also need to avoid the regime in which the distance is limited by the characteristic size of the attractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Generalised Wolf approach to measuring the nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We now construct the generalisation of Wolf’s approach, which will yield an estimation of the nMLE that we call λW nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The aim is to compute ⟨ℓ⟩ = ⟨ 1 τ ln dτ d0 ⟩, where the average is over choices of At and At′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In practice one considers a set of w initial choices of At, which we index i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , w, and for each of these choices, one additional point At′ on the trajectory such that d(At, At′) < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One then obtains λW nMLE = 1 w w � i=1 ℓ(i), (6) where ℓ(i) is the trajectory-averaged expansion rate ℓ computed for the i-th initial condition, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Algorithmically, this approach has the advantage that we do not need to fix the choice of τ, we are able to flexibly adjust τ for each initial condition, according to the specific d0 we are able to find in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' On the other hand, this approach is point-wise, in the sense that for each choice of At one only con- siders a single At′ nearby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' As a consequence, this method does not necessarily capture the average expansion rate around each initial condition At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Generalised Kantz approach to measuring the NMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In order to cal- culate such an average expansion rate (for a given choice of At) one would, for a fixed At, have to average over the expansion rates for choices of At′ in an ϵ-ball about At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This volume-averaging is the basis of Kantz’s generalization [11, 12] of Wolf’s algorithm [10], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For a given ini- tial condition At Kantz’ method provides a trajectory and volume averaged expansion rate ⟨ 1 τ ln dτ d0 ⟩volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We will write Λ for the volume-averaged expan- sion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For fixed At, this could algorithmically be obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One chooses N different At′ from the trajectory S, all within distance ϵ from At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We label these j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For each At′ one then computes ℓ(j) via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Then one sets Λ(At) = 1 N N � j=1 ℓ(j), (7) where N is the number of initial conditions inside a ball of radius ϵ and cen- tered at At that we have found in the sequence S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In practice, Kantz algorithm proceeds slightly differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Instead of first com- puting the ℓ(j), and then averaging the expansion rates, the average over the At′ is instead computed at the level of distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' That is to say, one makes N Based on Springer Nature LATEX template 8 Lyapunov Exponents for Temporal Networks choices of At′ as described above, and then obtains dk(j) for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , N and k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (k runs up to the relevant cut-off time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One then sets Λ(At) = 1 τ ln N −1 �N j=1 dτ(j) N −1 �N j=1 d0(j) , (8) where τ is a priori fixed for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The numerator in the logarithm represents the volume average (over choices of At′ in a ball about At) of dτ, and the denominator is the volume average of d0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We stress that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (7) and (8) are mathematically different and do not necessarily lead to the same results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (7) allows adapting the precise value τ (which depends on d0) for each trajectory, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (8) instead requires us to use a uniform τ across all trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The latter expression looks at the expansion in time of an initially small volume centered on At, and is closer in spirit to capturing the underlying nonlinear dynamics than Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Accordingly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (8) is the basis of the Kantz’ method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The quantity Λ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (8) is still a local quantity, in the sense that it was computed for a phase space volume around a fixed choice of At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In prin- ciple, the local volume-averaged expansion rate could vary across different regions in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' To capture the global long-term behaviour we there- fore additionally average over choices of At, and then finally obtain the global volume-averaged network maximum Lyapunov exponent λK nMLE = �1 τ ln ⟨dτ⟩volume ⟨d0⟩volume � At , (9) where we have written ⟨· · · ⟩At, for the average over initial conditions At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In practice this is carried out by averaging over a set of w choices of At, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', λK nMLE = 1 w w � j=1 Λ(A(j) t ) (10) where Λ(A(j) t ) stands for the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (8) for the fixed choice At = A(j) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We note that the value of the saturation time τ or the radius of the ball ϵ will have to be selected after some numerical exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Indeed, a better estimate of the Lyapunov exponent is obtained when the cut-off time τ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This, in turn, is the case when the initial distance between At and At′ is small, hence favouring choosing a relatively small value of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' However, a small value of ϵ complicates the task of finding points At′ that are at most a distance ϵ away from At on the given trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In practice, a trade-off is to be struck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 9 To summarise, from a given TN trajectory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' a sequence of network snapshots) we first measure the local expansion rates {ℓk}τ k=1 via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (3) for a fixed choice of At and At′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The set of ℓk obtained in this way provide information on the fluctuations of the local expansion rate (for fixed At and At′), and its trajectory-average ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We can then proceed along two alternative routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In the first approach (i) we average ℓ over different choices for the initial condition At [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (6)], and obtain Wolf’s approximation to the nMLE λW nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Alternatively, (ii) we can initially perform, for each initial condition At, an average over At′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is done by computing the local expansion of a volume ⟨dk⟩volume and then averaging this over time [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is then repeated for different choices of At, and once hence obtains a distribution P(Λ) describing the fluctuations across different points in the network phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Its mean provides Kantz’s approximation to the nMLE λK nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In the following sections, we present a validation of this method for random, low-dimensional and high-dimensional chaotic temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 3 The white-noise equivalent of a temporal network: independent and identically distributed random graphs Before addressing the case of chaotic dynamics, we briefly discuss the case of random network trajectories, with no correlations in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One then expects no systematic expansion or compression in time, and the resulting Lyapunov exponent should hence vanish3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We here seek to verify that this is indeed the case for the procedures we have introduced to estimate the Lyapunov expo- nents of TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Studying this is of interest, among other reasons, because an empirically obtained time series may appear random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is then important to be able to decide if the trajectory is consistent with an uncorrelated random trajectory in network space, or with a deterministic chaotic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Here we study the simple case where S is an independently drawn sequence of Erd¨os-R´enyi graphs with n nodes and in which the probability that any two nodes are connected is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is an analog to white noise for TNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', a situ- ation in which the TN displays delta-distributed autocorrelation function [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' At odds with a deterministic GDS, the distances between different points on a network trajectory are now random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' More precisely, since all elements of S are the adjacency matrices of Erd¨os-R´enyi graphs, the elements of these matrices are Bernoulli variables, taking values zero (with probability 1 − p) or one (with probability p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For independent adjacency matrices A and B, the possible values of |aij − bij| are then zero with probability p2 + (1 − p)2, and 3By construction, d1 > d0 as we force d0 < ϵ, so in order to avoid spurious expansions at k = 1, in this section we don’t take into account d0 in the estimation of finite Lyapunov exponents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' our starting time is k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Based on Springer Nature LATEX template 10 Lyapunov Exponents for Temporal Networks one with probability 2p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Thus we have d(A, B) ∼ Binomial[n(n − 1)/2, 2p(1 − p)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (11) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (11) is numerically verified in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2 Panel (a): Probability distribution of the distance between consecutive networks in an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' sequence of |S| = 105 Erdos-Renyi graphs with n = 100 nodes and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='01 (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The plot is in semi-log, where a Gaussian shape appears as an inverted parabola, so we can better appreciate the tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Blue solid line is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Panel (b): P(ℓ) (black) and P(Λ) (red) associated with a sequence of |S| = 5·104 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' ER networks (n = 100, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='01), for τ = 1 and w = O(104) initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The mean of both distributions (nMLE) is essentially zero for both methods, but the dispersion around the mean is larger in Wolf’s approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The solid blue line is the theoretical prediction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' a Gaussian distribution with mean 0 and variance as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (14) with τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The inset of panel (b) shows how the variance of ℓ shrinks as τ increases (resulting in a much lower uncertainty around a zero nMLE): dots are different numerical simulations with |S| = 104 and w = 103, for different τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' the blue line is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The quantity ℓ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (5) is given by ℓ = 1 τ (ln dτ − ln d0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' As we have just established, d0 and dτ are independent binomial random variables following the distribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For large networks (n ≫ 1) this distribution can be approximated as a Gaussian, with mean µ = qn(n − 1)/2 and variance σ2 = q(1 − q)n(n − 1)/2, where we have written q ≡ 2p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Writing d0 = µ + σz0, with z0 a standard Gaussian random variable, we have ln d0 ≈ ln � µ(1 + σ µz0) � = ln µ + σ µz0 − 1 2 σ2 µ2 z2 0 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , (12) after an expansion in powers of σ/µ, where the latter quantity is of order O(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The same expansion can be carried out for dτ, and we therefore find ℓ = 1 τ �σ µ(zτ − z0) + 1 2 σ2 µ2 (z2 0 − z2 τ) � + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (13) (a) 10-2 P simulation 10 — theory 10-5 60 80 100 120 140 dsimulation (b) 10-2 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (14) 4 P() theory (Wolf) 3 L P(Λ) P(0), P(Λ) 1 10 1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 l, ^Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 11 We note that the second term in the bracket is of sub-leading order in 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Hence, ℓ is to lowest order in 1/n approximately Gaussian, with mean zero and variance Var(ℓ) = 2 τ 2 σ2 µ2 = 1 τ 2 4(1 − q) qn(n − 1) (14) This theory has been numerically verified, and in panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 we plot P(ℓ) both for τ = 1 (outer panel) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='14 for increasing values of τ (inset panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The case of Λ (Kantz-version) should intuitively converge even faster than ℓ (Wolf-version) since in this case we are carrying out two averages instead of just one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' P(Λ) should have a smaller variance than P(ℓ), for a given τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is confirmed in panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2, where we also observe that both methods yield the same (correct) estimation of the nMLE, which in this case is approximately zero (both estimates are of the order of 10−6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Note that the main panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2) are for the case τ = 1, so it is a worst-case scenario: as τ increases Var(ℓ) shrinks [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (14)] and the uncertainty around the null shrinks accordingly [see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We were not able to find a closed-form solution for P(Λ) as averages inside the ϵ-ball are random variables whose distribution explicitly depends on the specific initial condition At: this calculation is left as an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In anycase, we conclude that an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' temporal network has a null MLE Lyapunov exponent and the methodology (in both variants) correctly estimates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 4 Low-dimensional chaotic networks 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 Network generation: the dictionary trick To be able to validate the method in the context of chaotic dynamics, we ideally need to have access to chaotic network trajectories with a ground true nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is difficult as a general theory of chaotic GDS is not yet accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' To circumvent this drawback, in this section we develop a method to construct (low-dimensional) chaotic network trajectories by symbolising in graph space- time series from low-dimensional chaotic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The method of graph-space symbolisation was first proposed as a so-called ‘dictionary trick’ in [8] and consists of the following steps: We construct a network dictionary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is a set of networks that allows us to map a real-valued scalar x ∈ [0, 1]4 into a network, such that the distance between two scalars is preserved in graph space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The set D is therefore ordered and equipped with a metric, such that the distance between two real- valued scalars |x−x′| is preserved in the graph symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' More concretely, the dictionary of networks D = (G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', GL) such that d(Gp, Gq) ∝ |p − q| (one can subsequently normalize d according to the length of the dictionary, such that we have d ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 4We choose the interval [0, 1] without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Based on Springer Nature LATEX template 12 Lyapunov Exponents for Temporal Networks Once such dictionary is built, any one-dimensional time series can be mapped into a sequence of networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In particular, we can map chaotic time series with well-known MLEs into network trajectories, from which an independent estimate of the nMLE can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Algorithmically, the dictionary is generated sequentially with G1 ∼ ER(p) (an Erd¨os-R´enyi graph with parameter p) and then iteratively constructing Gk+1 from Gk by rewiring a link that (i) has not been rewired in any previous iteration of the algorithm, (ii) into a place that did not have a link in any pre- vious iteration of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is easy to see that such algorithm ensures that D provides a partition of [0, 1] of the form [0, 1] = ∪L−1 k=0 [k/L, (k + 1)/L], where L is the number of networks in the dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The dictionary is thus metrical, in the sense that the rewiring distance between any two elements in the dictionary is (for a sufficiently large refinement L) arbitrarily close to the associated real-valued scalars in the original interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Once the dictionary is established, we can then generate synthetic temporal network trajectories as symbolizations of unit interval dynamics by matching points in the subin- terval [k/L, (k + 1)/L] to the symbol Gk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The resulting temporal network S inherits, by construction, the properties of the scalar time series, and in particular can be used to generate chaotic TNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 3 Panel (a): Semi-log plot of the distance dk as a function of iteration index k, for two initially close network trajectories sampled from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We can appreciate an initial expo- nentially expanding phase, followed by a saturation phase, although the local expansion rate strongly fluctuates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Panel (b) Volume-averaged distance ⟨dk⟩volume as a function of time k, for N = 17 initial graph conditions inside a volume centered at an initial graph of n = 500 nodes and m = 2000 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Network dynamics evolve according to a logistic map as described in the text, whose true Lyapunov exponent is ln 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We can see how the volume enclosing the graphs on average expands exponentially fast –with an exponent close to ln 2, as expected– until it reaches the attractor size, what happens at the saturation time τ ≈ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In2·k n = 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 m = 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='01 (a) 0 10 20 30 kn = 500 m = 2000 N = 17 t (saturation time) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='01 In2·k (b) 0 5 10 15 20 25 30 kSpringer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 Results for the logistic map As a first validation, we consider the fully chaotic logistic map xt+1 = 4xt(1 − xt), xt ∈ [0, 1], (15) that generates chaotic trajectories with λMLE = ln 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Using the dic- tionary trick, from a signal extracted from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (15) we generate a temporal network trajectory S of |S| = 3000 network snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In this validation, networks have n = 500 nodes and m = 2000 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 4 Panel (a): Approximation to λW nMLE following Wolf’s approach (see text), computed by averaging ℓ over w randomly sampled initial conditions [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='6], as a function of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We can see that the exponent converges to the ground true exponent ln 2 as w increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Inset in (a): Probability distribution P(ℓ), sampled by estimating ℓ for w = 500 different initial graph conditions sampled randomly from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The mean of this empirical distribution is λW nMLE = ⟨Λ⟩At ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='685, very close to the true exponent ln 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Panel (b): Same as panel (a), but using Kantz’s approach (see the text), where we compute the volume and trajectory averaged expansion rate Λ for w initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Convergence properties are similar in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For illustration, in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 3 we plot in semi-log scales the (properly normalized) distance dk as a function of the iteration index k, for two initially close network trajectories sampled from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We can see an initial exponentially expanding phase (whose exponent is an estimation of ℓ) followed by a saturation, although the distance function shows strong fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' To cope with these, in panel (b) of the same figure we plot the volume-averaged expansion ⟨dk⟩volume vs k for a ball of radius ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='005 centered at a specific initial graph from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We can now clearly see the initial exponential phase followed by a cross-over to a saturation phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The cross-over marks the saturation time τ where the distance reaches the attractor size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Note that the slope of the exponential expansion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' the estimate of Λ) is close to ln 2, the true MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 (a) 2 W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='685 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='0 P1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 In 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='6 0 100 200 300 400 500 w (# initial conditions)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 3 (b) XNMLE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='685 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='0 2 P(Λ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='0 V In 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='7 0 100 200 300 400 500 w (# initial conditions)Based on Springer Nature LATEX template 14 Lyapunov Exponents for Temporal Networks Figure 4 shows the estimated of the nMLE obtained both using Wolf’s approach [panel (a)] and Kantz’s approach [panel (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' These are from aver- aging ℓ (Wolf) and Λ (Kantz) over w = 500 initial graph conditions sampled from S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In both cases, the average quickly stabilises for w ≈ 100, and we obtain estimates λW nMLE ≈ λK nMLE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='685, very close to the ground true λMLE = ln 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='693.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 5 Volume-averaged distance ⟨dk⟩volume as a function of time k, for a network dynamics evolving according to a chaotic logistic map xt+1 = 4xt(1 − xt), polluted with extrinsic Gaussian noise N(0, σ2) as described in the text, for four different noise intensities σ = 0, 10−3, 10−2, 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The exponential expansion phase –which systematically suggests the same exponent ln 2, as expected– is gradually erased as the noise intensity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='3 Noisy chaotic networks To explore how noise contamination can complicate the estimation of the nMLE, we proceed to generate a temporal network S from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (15) by using the dictionary trick, where before the network mapping, the original chaotic signal is contaminated by a certain amount of white Gaussian noise N(0, σ2)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' As we did in Section 3, we remove potential algorithmic biases by discarding ⟨d0⟩ for the computation of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Results are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The main observation is that noise pollution tends to reduce the extent of the exponen- tial phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', the saturation time τ decreases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For small amounts of noise, this phase is still observable, and the estimated nMLE continues to be con- sistent with that of the noise-free case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' When the noise intensity is above a certain threshold, noise effectively hides the chaotic signal, and the exponential phase can no longer be identified, resulting in an apparent vanishing nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' These results are consistent with intuition and with the typical phenomenology observed in noisy chaotic time series [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 5Note that we discard realizations of the noise that take the scalar variable outside the unit interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 10 10° =0 0=10-3 0=10-2 0=10-1 exp(In2 k) 10-4 2 4 6 8 10 12 14 kSpringer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='4 Results for the parametric logistic map Here we consider the logistic map xt+1 = rxt(1 − xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For each value of the parameter r > r∞, using the dictionary trick we generate a long sequence of networks Sr with the desired chaoticity properties, and proceed to estimate the network Lyapunov exponent using the method detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='6 we plot λW nMLE vs λMLE of the map, for a range of values of the parameter r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The agreement is excellent in the region of parameters where the temporal network is chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 6 Panel (a): Scatterplot of λW nMLE, estimated from a temporal network Sr generated via the dictionary trick (see the text) from a logistic map xt+1 = rxt(1 − xt) for a range of values of r, vs the ground true λMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The solid line is the diagonal of perfect agreement y = x, highlighting the good agreement found in the chaotic region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The legend states the goodness of fit metric R2 of the fit of dk to an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The method is unable to capture negative Lyapunov exponents (observe that in those cases the R2 of the exponential fit is very bad), but these cases can easily be identified as periodic orbits using the autocorrelation function [8], see text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Panel (b): Estimate of the negative nMLE using two initially close temporal networks generated via the dictionary trick from the logistic map at r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='4 (period-2 orbit), where one initial condition belongs to the period-2 attractor and the other is outside the attractor (see the text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 A note on negative Lyapunov exponents The classical approach to estimate the MLE from a single trajectory displayed by Wolf and Kantz algorithms –based on recurrences of the trajectory– is, by construction, unable to capture negative MLEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The reason is straightforward: once in the periodic attractor, the trajectory sequentially visits each element of the periodic orbit, and thus we won’t find recurrences that are close but away from the initial condition of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Accordingly, our method to esti- mate nMLE cannot work in that case for the same reasons, as confirmed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This drawback can be solved using two alternative approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' First, it is well known that a periodic time series has an autocorrelation function that peaks at the period of the time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Interestingly, a recent work [8] has operationalised a way to estimate the autocorrelation function 10-2 r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='4 入MLE = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='137 exp(-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='143k) 10-3 (b) 0 5 10 15 20 25 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='8 R?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='6 E (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='7) E [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='7,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='8] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='4 E [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='8,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='9] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 (a) + 0 ++ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 ΛMLEBased on Springer Nature LATEX template 16 Lyapunov Exponents for Temporal Networks of temporal networks, whereby temporal networks that display periodicity are well characterised by a network version of the autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Accordingly, from a practical point of view, before attempting to estimate the nMLE of a given temporal network, it is sensible to apply the procedure of [8] and exclude that the temporal network is periodic –which would typically6 mean a negative nMLE–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Once this test is done, it is sensible to conduct the nMLE analysis presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Second, it is indeed possible to estimate negative nMLEs if one has access to the latent graph dynamical system (GDS), as in this case one does not need to undergo a Wolf/Kantz approach and one can generate through the GDS temporal networks from close initial graph conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' To illustrate this, in panel (b) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 6 we plot the graph distance of two initially close networks evolving according to the logistic map for a value of the map’s parameter for which the orbit is periodic (the TNs are again generated via the dictionary trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One initial condition is set at one of the orbit elements, whereas the other initial condition is a network close in graph space (but outside the peri- odic attractor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' As we can see, there is an exponential shrinking of the initial distance, and the slope gives an estimate of the nMLE, which in this case is negative and in good agreement with the theoretical result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 5 High-dimensional chaotic networks We now consider the case of high-dimensional chaotic dynamics for temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We first introduce a generative model, based on coupled Map Lat- tices (CML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' These are high-dimensional dynamical systems with discrete time and continuous state variables, widely used to model complex spatio-temporal dynamics [20] in disparate contexts such as turbulence [21, 25], financial mar- kets [26], biological systems [27] or quantum field theories [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Globally Coupled Maps (GCMs) [29] are a mean-field version of CMLS, where the diffusive coupling between the entities in a CML is replaced with an all- to-all coupling, mimicking the effect of a mean-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We consider a globally coupled map of m entities, of the form xi(t + 1) = (1 − α)F[xi(t)] + α m m � j=1 F[xj(t)], i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' , m, (16) where F(x) = 4x(1 − x), x ∈ [0, 1], where α ∈ [0, 1] is the strength of the mean-field coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In the uncoupled case α = 0, the system is composed of m independent fully chaotic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Its attractor is thus high-dimensional and, since there are m Lyapunov exponents all equal to ln 2, we have λMLE = ln 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 6Some pathological cases exist for which we can have seemingly periodic behavior but not a negative MLE, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' when we have a disconnected attractor composed by a number of bands and a trajectory that periodically visits the different chaotic bands Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 17 At the other extreme, for complete coupling α = 1, the system is fully synchro- nized (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', for any time t we have xi(t) = xj(t) for all i, j), and the dynamics is reduced to the one-dimensional dynamics, again with λMLE = ln 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We add that complete synchronization is in fact known to occur for α > 1/2 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For intermediate coupling the system shows a number of different macroscopic phases [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Among these one finds high-dimensional chaos for weak coupling α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This is the so-called ‘turbulent state’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Interestingly, for CMLs with diffusive coupling, a scaling law has been established [22] λMLE = log 2 − βα1/p, (17) where p indicates the type of nonlinearity of F(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' p = 2 for the logistic map, p = 1 for tent maps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Results for GCM are less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' However, when the mean-field coupling can be considered ‘thermalized’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', independent of x) [23, 24] then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (17) holds for β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' However such thermalization is known to be true only for tent maps (p = 1) and not logistic maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Here we consider the range α ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', the turbulent state of the GCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We interpret the collection {xi}m i=1 as the (weighted) edge set of a fully connected undirected network backbone of n nodes and m = n(n−1)/2 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Once the time series of each edge {xi(t)}T t=1 has been computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (16), we proceed to binarise each edge activity by using a two-symbol generating partition as follows: values xi(t) < 1/2 are mapped into the symbol 0, and xi(t) ≥ 1/2 onto the symbol 1[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Note that the use of a generating partition ensures that the symbolised (binary) series preserves the chaotic properties of the original signal [31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Finally, we convert the (binary) evolution of the edges into a time-dependent adjacency matrix, thereby constructing a temporal network S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For values of α in the weak-coupling regime, we expect the temporal network to display sensitive dependence on initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In practice, the Wolf/Kantz methods of inferring the larges Lyapunov exponent proposed in the paper would require a very long sequence S for close enough recurrences to be observable in a system with large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' However, here we have access to the actual underlying GDS via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Given that the goal of this section is to show evidence that high-dimensional chaotic networks can be generated and their nMLE be estimated, we can use the GDS to generate the temporal network for any required initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Accordingly, for a given initial condition {xi(0)}m i=1, we construct a perturbed copy {x′ i(0)}m i=1 (where |x′ i(0) − xi(0)| < ϵ for some small choice of ϵ), generate temporal networks for both of these initial conditions, and track the network distance between the copies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We do this for 100 replicas to extract a volume- averaged distance, and then for 50 different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Observe that, at odds with the model developed in the previous section, here the number of edges in each network snapshot is not fixed, and thus the network phase space is substantially larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Similarly, the normalization factor of the distance function is now simply the total number of possible edges, n(n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Based on Springer Nature LATEX template 18 Lyapunov Exponents for Temporal Networks Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 7 Panel (a): Semi-log plot of the volume-averaged distance ⟨dk⟩volume as a function of the time step k, for a temporal network extracted from the GCM model with coupling constant α = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We observe an exponential phase, with different exponents for each value of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The solid lines are the best exponential fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Panel (b): Estimate of the network maximum Lyapunov exponent λnMLE vs the coupling constant α for temporal networks generated from a GCM of logistic maps (blue circles) and tent maps (black squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For each α, a total of 50 initial conditions were considered, and a ball of 100 points for each initial condition was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Error bars are standard deviation from the average over 50 different temporal network realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Blue lines report the theoretical predictions for logistic and tent CMLs [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (17)], whereas the black line reports the theoretical prediction for GCM with thermalised mean-field, applicable for tent GCMs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Results for a network of n = 100 nodes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In panel (a) we plot ⟨dk⟩volume vs time k, for three different coupling constants α = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 in the weak coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In every case we find a clear exponential phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The exponent in the uncoupled phase α = 0 is indeed equal to ln 2, as expected, further validating the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For increasing values of the coupling, interest- ingly, the nMLE seems to decrease and, as a byproduct, the saturation time τ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 7(b) we plot, as blue dots, the estimated λnMLE as a function of the coupling α ∈ [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2], indeed showing a clear decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Such decrease might be induced by the fact that the m degrees of freedom are now coupled in some nontrivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Blue lines correspond to the theoretical predictions for logistic and tent CMLs obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For completeness, we repeated the same analysis for network GCMs constructed from tent maps where λMLE is explicitly known (black line): F(x) = 1 − 2|x|, with x ∈ [−1, 1] and a sym- bolisation partition with x < 0 mapped to the symbol 0, and x ≥ 0 mapped to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Results for this case are plotted as black squares in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 7(b) We conclude that (i) the TN thereby generated exhibits high-dimensional chaos and its nMLE, reconstructed with the methods we have developed, shows the expected behaviour, and (ii) this validation shows that the method works with TNs where not only the position but also the total number of edges itself fluctuates over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (a) 10 10- 2 α=0 exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='68 k) (R² = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='99) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='05 exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='56 k) (R² = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='99) α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 10-3 exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='44 k) (R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='99) 0 5 10 15 k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='7 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='5 入nMLE E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='4 logistic map network log2 -βVa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='. log2 - βa tent map network 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 log2 + log(1-a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='2 aSpringer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 19 6 Discussion In this work, we propose to look at temporal networks as trajectories of a latent Graph Dynamical System (GDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This interpretation naturally leads us to explore whether these trajectories can show sensitive dependence on initial conditions, a fingerprint of chaotic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We have proposed a method to quantify this, and defined and computed the network Maximum Lyapunov Exponent (nMLE) for temporal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Since the latent GDS is rarely available in practice, our algorithm exploits the recurrences of the temporal network in graph space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It generalizes the classical approaches of Wolf and Kantz to networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We have validated the method by generating different synthetic GDS with known ground-truth nMLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Conceptually speaking, quantifying chaos in the trajectory of structured objects (in our case, mathematical graphs) is somewhat close in spirit to quan- tifying the dynamical stability of (lattice) spin systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Thus our approach shares some similarities with the damage-spreading [34] and self-overlap meth- ods [35] in statistical physics, and their applications to cellular automata [36] and random Boolean systems [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Observe that we have focused on exponential expansion on nearby con- ditions –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', sensitive dependence–, since one of the goals of the paper is to conceptually postulate the existence of chaotic networks and to potentially operationalise a way to measure this deterministic fingerprint in observed TNs, without needs to having access to the underlying GDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' However, our approach can be straightforwardly extended to non-exponential divergence, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' alge- braic or otherwise, simply by suitably modifying the definition of expansion rates, thus yielding a way to quantify other types of dynamical instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The rationale of this work is to consider graphs evolving over time as whole –yet not punctual– objects [8], and thus consider its evolution in graph space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is however true that this approach might have a limitation for (large) real- world temporal networks, as it is often difficult to observe recurrences in high- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' A possible solution is to extract suitable scalar variables from the network, analyse sensitive dependence on initial conditions in each of them, and extract a consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We leave this approach for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Observe that throughout this work we have considered labelled networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This choice was used for, convenience, illustration, and because TNs are usu- ally labelled, but we expect that a similar approach is possible for unlabelled TNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' graphs that evolve over time according to a certain graph dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' In this latter case, each network snapshot is no longer uniquely represented by a single adjacency matrix, in the sense that permutations of the rows and columns of the matrix lead to an equally valid description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is then clear that one needs to use graph distances showing invariance under permutation of rows and columns in the adjacency matrices [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This could be, for exam- ple, distances based on the network spectrum, or graph kernels [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We leave this interesting extension as a question for further research, as well as the Based on Springer Nature LATEX template 20 Lyapunov Exponents for Temporal Networks quantification of the full Lyapunov spectrum beyond the maximum one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Finally we would like to add that the fact that the method does not rely on knowing the GDS and instead directly estimates the nMLE from tempo- ral network trajectories enables the investigation of these matters in empirical temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We foresee a range of potentially interesting applications in physical, biological, economic and social sciences –as indeed temporal net- works pervade these disciplines–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This approach is specially appealing in those systems where we don’t have access to the ‘equations of motion’ but it is sen- sible to expect some underlying deterministic dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' physical systems, but the approach is also extensible to systems with socially or biologically- mediated interactions, for instance: do flocks of birds [40–42] or crowd behavior [43], adequately modelled as temporal proximity networks, show chaos?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Appendix: Graph distances Consider two adjacency matrices A, and B, each with binary entries (0 or 1), describing two simple unweighted graphs with n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The so-called edit distance [18] is a matrix distance defined as d(A, B) = n � i,j=1 |aij − bij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (18) The object d(A, B) counts the number of entries that are different in A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' For simple undirected graphs (symmetric adjacency matrices), we need to account for the fact that the number of edges is only half the number of positive entries of the adjacency matrix, and therefore d(A, B)/2 measures the number of edges that exist in one graph but not on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We have d(A, B)/2 = 0 if and only if A = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' It is also easy to see that d(A, B)/2 only takes integer values for symmetric adjacency matrices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' If A ̸= B, then 1 ≤ d(A, B)/2 ≤ n(n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' We have d(A, B)/2 = 1 when the two graphs are identical except for one edge, which is present in one graph and absent in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One can directly use this unnormalized distance (as we do in Section 3) or subsequently normalize d(A, B) using different strategies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' one can divide it over n(n − 1)/2 (as we do in Section 5), or just divide over the maximum possible distance, if further restrictions are imposed between A and B (as we do in Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' If we further impose that both graphs have the same number of edges, then the lower bound cannot be attained and 2 ≤ d(A, B)/2 when A ̸= B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This lower bound is reached when we only need a single edge rewiring to get from Springer Nature 2021 LATEX template Lyapunov Exponents for Temporal Networks 21 the first graph to the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One can thus define the rewiring distance d(A, B) = 1 4 n � i,j=1 |aij − bij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (19) applicable for simple graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' no self-links).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This quanity measures the total number of rewirings needed to transform A into B when the associated graphs are simple, unweighted, undirected (symmetric adjacency matrices) and have the same number of nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' The rewiring distance above is based on the concept of non-overlapping edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=', edges that are present in one graph, but not in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Thus, the edit and rewiring distances are based on |aij − bij| for the different edges, and hence assign the same importance to the presence or absence of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One can instead construct measures of distance based on the number of links that are present in both networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' If the edge ij is present in both graphs then aijbij = 1, while this product is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' One can prove that the following function is a distance [19]: d(A, B) = 1 − 1 2|E| n � i,j=1 aijbij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' (20) We replicated the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' 4 for the distance defined above, and results (resulting nMLE) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Acknowledgments We thank Federico Battiston for helpful comments on initial phases of this research, and Emilio Hern´andez-Fern´andez, Sandro Mel- oni, Lluis Arola-Fern´andez, Ernesto Estrada, Massimiliano Zanin, Diego Paz´o and Juan Manuel L´opez for helpful discussions around several aspects of the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' AC acknowledges funding by the Maria de Maeztu Programme (MDM- 2017-0711) and the AEI under the FPI programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' LL acknowledges funding from project DYNDEEP (EUR2021-122007), and LL and VME acknowledge funding from project MISLAND (PID2020-114324GB-C22), both projects funded by the Spanish Ministry of Science and Innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' This work has been partially supported by the Mar´ıa de Maeztu project CEX2021-001164-M funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Holme, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Saram¨aki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFOT4oBgHgl3EQf8TTm/content/2301.12966v1.pdf'} +page_content=' Temporal 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b/YdE3T4oBgHgl3EQfcQoq/content/tmp_files/2301.04523v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2835189290f6d5466cc5790f573abcc8457b5ecc --- /dev/null +++ b/YdE3T4oBgHgl3EQfcQoq/content/tmp_files/2301.04523v1.pdf.txt @@ -0,0 +1,962 @@ +1 + +Machine-learning-assisted environment-adaptive thermal +metamaterials +Peng Jin1, Liujun Xu2,3, Guoqiang Xu2, Jiaxin Li2, Cheng-Wei Qiu2,* and Jiping Huang1,* + +1Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro +and Nano Photonic Structures (MOE), Fudan University, Shanghai 200438, China +2Department of Electrical and Computer Engineering, National University of Singapore, +Singapore 117583, Singapore +3Graduate School of China Academy of Engineering Physics, Beijing 100193, China + + + + + + + + + + + + +*Corresponding authors. Emails: chengwei.qiu@nus.edu.sg; jphuang@fudan.edu.cn + +2 + +Abstract +Adaptive metamaterials have prevailed recently owing to their extraordinary features like +dynamic response to external interference. However, highly complicated parameters, narrow +working ranges, and supervised manual intervention are still long-term and tricky obstacles to +the most advanced self-adaptive metamaterials. To surmount these barriers, we present +environment-adaptive thermal metamaterials driven by machine learning, which can +automatically sense ambient temperatures and regulate thermal functions promptly and +continuously. Thermal functions are robust when external thermal fields change their directions, +and simulations and experiments exhibit excellent performance. Based on this, we further +design two metadevices with on-demand adaptability, performing distinctive features with +isotropic materials, wide working temperatures, and spontaneous response. This work provides +a paradigm for intelligent diffusion metamaterial design and can be extended to other diffusion +fields, responding to more complex and variable environments. + + + +3 + +Introduction +Metamaterials [1-5] have drawn intensive attraction due to their unprecedented ability to +manipulate physical fields. Profit from computer numerical control and three-dimensional +printing technology, metamaterials with novel functions are fabricated according to given +parameters and applied to laboratories or industries. Traditional metamaterials mainly focused +on static cases [6-15], lacking tunability for variable scenes. To tackle this issue, tunable +metamaterials with dynamic features have emerged, covering optics [16,17], acoustics [18], +and thermotics [19-21]. For example, many advanced thermal functions have been realized, +including macroscopic thermal diodes [19], tunable analog thermal materials [22], path- +dependent thermal metadevices [23], and tunable hybrid thermal metamaterials [24]. On the +other hand, adaptive thermal metadevices are presented to maintain the robustness of functions +against environmental changes or switch functions depending on application scenes [20,25-30]. +However, the achievement of state-of-the-art self-adaptive thermal metamaterials is confronted +with three longstanding and strong barriers. Firstly, adaptive thermal metadevices with robust +functions usually require extremely complicated parameters [25-27], which are difficult to +prepare from natural bulk materials. Secondly, existing adaptive metadevices, especially +macroscopic thermal diodes [19] and energy-free temperature trapping [20], are limited to a +specific temperature range related to the phase change temperature of shape memory alloys. +Finally, most tunable or adaptive metamaterials [16-30] need to be adjusted through manual +control rather than automatically, lacking self-cognitive ability. +Recently, intelligent materials, involving interdisciplinary research and combining +intelligent algorithms with material design, have motivated applications in optics [31-33], + +4 + +nanotechnology [34], theoretical physics [35], materials science [36], and thermal science [37]. +These advances inspire ideal self-adaptive thermal metamaterials with full embracement of +intelligence. In principle, ideal self-adaptive thermal metadevices should automatically +(without human aid) and timely adjust their dynamic components to keep function stable or +switch functions continuously in response to the broad range of ambient temperature change. +Developing such self-adaptive thermal metamaterials is highly desirable to be valuable in situ +scenes. However, the demanding technical performance requires an appropriate actuation +mechanism that integrates an algorithm-driven intelligent system with thermal metamaterial +design. Although these metamaterials have been used to design advanced self-adaptive optical +cloaks in wave systems [38], they fail in diffusion systems like heat transfer due to the lack of +controllable degrees of freedom. Existing machine-learning-based thermal metamaterials are +dictated by the inverse design method [39-42], which only calculates the parameters of +materials and sizes for desired functions. In addition, once such metamaterials are prepared, +their functions are not switchable, lacking response to various scenes. +Here, we introduce a machine-learning-assisted intelligent system and propose +environment-adaptive thermal metamaterials driven by big data. As a conceptual +implementation, we design an intelligent temperature gradient controller. We load the pre- +trained artificial neural network into a hardware system and combine it with the bilayer +structure [43]. Depending on the sensing-feedback ambient temperatures, the thermal +conductivity of a spinning component could be adjusted to achieve a tunable temperature +gradient in a target region, verified by finite-element simulations and experiments. We then +propose two applications with on-demand adaptability. One is a thermal signal modulator with + +5 + +functional robustness, making original thermal signals clearer. The other is an intelligent +thermoelectric generator with intelligent functional choice, which can automatically adjust the +electromotive force generated by thermoelectric materials [44] based on the ambient +temperature. The intelligent environment-adaptive thermal metamaterial features isotropic +materials, unlimited working temperatures, and cognitive responsiveness. A handy actuation +mechanism also integrates machine-learning-driven intelligent systems with diffusion +metamaterial design. Our work brings the design of self-adaptive diffusion metamaterials to a +new stage without human intervention. +Architecture +of +the +intelligent +temperature-gradient +controller +The architecture of the intelligent temperature-gradient controller is presented in Fig. 1. It +contains four main modules: a temperature acquisition module (micro infrared camera), a +computing system with a pre-trained artificial neural network (ANN), a stepper motor, and a +bilayer structure. We aim to manipulate the temperature gradient of the target region based on +the feedback of temperature information of its surroundings. Here, as a proof-of-concept +implementation, we consider a two-dimensional system with a bilayer metal structure. The +target region is the core region Ω! consisting of poly-dimethylsiloxane (PDMS). The +component of the inner layer (Silicone pad) is approximately adiabatic for precise control of +thermal fields of Ω!, and the outer layer as a compensation layer (Magnesium alloy) is intended +not to disturb the thermal fields of the background (Inconel alloy). The thermal conductivity +from the inside out is 0.15, 1, 72.7, and 9.8 W m-1 K-1, respectively. After setting R1 = 30 mm +and R2 = 53 mm, R3 can be calculated as 60 mm [43]. + +6 + +To characterize the temperature information of the bilayer structure’s surroundings, we +choose a series of discrete positions around the outer layer and extract their temperature using +a micro infrared camera. As shown in Fig. 1, the blue dashed circle is the chosen bilayer +structure’s surrounding, and the position marked 0◦ is the first position. Then, we take the +temperature of the amount of N equally spaced positions on the circumference in a +counterclockwise direction, serving as the input layer of ANN. +Thanks to the tunable analog thermal material [22], by spinning (angular velocity: 𝜔!) +the PDMS in the core region Ω!, the effective thermal conductivity of the spinning medium +can be tuned from near-zero (𝜔! = 0) to near-infinity (larger 𝜔!). On the other hand, the +effective thermal conductivity of Ω! is the crucial factor affecting the temperature-gradient +distributions of Ω!; see lower part of Fig. 1 for an intuitive description. Color denotes +temperature profiles of the core region, and the white lines represent the isotherms from finite- +element simulations. Here, sparser isotherms correspond to a smaller temperature gradient. +For the sake of “intelligence”, we utilize ANN to establish a mapping between extracted +temperature information (input data: 𝑻(#)) and the angular velocity of Ω! (output data: 𝜔!). +In Fig. 1, we show the structural component of the ANN, which is fully connected with four +hidden layers (50 neurons per layer). Activations of all neurons in the next layer are determined +by activations of those in the current layer (𝑯(%)), represented by + +⎩ +⎪ +⎨ +⎪⎧𝑯(%&!) = ReLU0𝑾(%)𝑻(%) + 𝒃(%&!)4, 𝑖 = 0 +𝑯(%&!) = ReLU0𝑾(%)𝑯(%) + 𝒃(%&!)4, 0 < 𝑖 < 4 +𝜔! = ReLU0𝑾(%)𝑯(%) + 𝒃(%&!)4, 𝑖 = 4 + +(1) +where ReLU (a) = max(0, a) is the rectified linear unit function. W and b are weights and +biases for the neurons. i is the ordinal number of layers, and 𝑖 = 0 represents the input layer. + +7 + +As the ANN is a data-driven-based model, we then prepare the dataset (see Supplementary Note +1). Finally, via the back propagation algorithm, the proposed ANN with selected +hyperparameters is well trained by the dataset (see Supplementary Note 1). +When the thermal field of the bilayer structure reaches equilibrium, the temperature +information (𝑇!, 𝑇', 𝑇(, ..., 𝑇)) at the circle with a radius of R3 = 60 mm is collected by micro +infrared camera and imported into the computing system as an array of input signals for the +pre-trained ANN. The computing system’s output signal is the stepper motor’s spinning angular +velocity 𝜔!. Considering a case where the output spinning angular velocity is 0 rad s-1, static +PDMS possesses thermal conductivity with 0.15 W m-1 K-1. At this time, the temperature +gradient in Ω! reaches its maximum value. +Intelligent response in omnidirectional simulations +Finite-element simulations are first used to demonstrate the performance of an intelligent +temperature-gradient controller. For proof-of-concept verification, we consider a two- +dimensional bilayer structure whose components and sizes are the same as mentioned above. +The system’s left (right) end connects to a hot (cold) source. In our simulations, the cold source +(𝑇* = 283 K) is fixed, while the hot source (𝑇+) is changeable. We first extract the temperature +data (𝑇𝒂 +(#), 𝑇𝒃 +(#), 𝑇𝒄 +(#)) of N = 36 equally spaced positions in the white dashed circle in three +cases (𝑇+ = 293, 303, 313 K) of a static bilayer structure, as shown in Fig. 2a. For each case, +the first data is the temperature of the position marked 0◦ in the dashed circle. The order of +taking the temperature of these positions is counterclockwise, serving as the input layer of the +pre-trained ANN. Hence, via the pre-trained ANN, the spinning angular velocity 𝜔! of PDMS +is calculated individually as 0.10, 0.00067, and 0 rad s-1. After setting the above parameters + +8 + +(𝑇+ and 𝜔!) in finite-element simulations, we show these three temperature profiles (color +distributions) of the bilayer structure; see Fig. 2b. No matter how the angular velocity 𝜔! of +PDMS changes, the temperature distributions of the background will not be disturbed (see +Supplementary Note 2). Finally, their corresponding temperature-gradient distributions in Ω! +are given in the right part of Fig. 2c. For comparison, we show the temperature-gradient +distributions in Ω! of pure background (size: 200 × 200 mm2) with three hot sources; see left +part of Fig. 2c. As anticipated, there is a mapping relationship between lowest/highest 𝑇+ − 𝑇* +and highest/lowest angular velocity 𝜔! (or say, lowest/highest |∇𝑇| in Ω!) in above scheme. +Further, the range of originally external temperature-gradient field [(𝑇+ − 𝑇*)/𝐿 : 50 to 150 K +m-1] can be adjusted to a wider range of temperature gradient (|∇𝑇|: 0 to 238.5 K m-1) in the +target region Ω!. +In the actual scene, we are unsure in which direction the external thermal field is exerted +on the bilayer structure. Therefore, we should ensure that the performance of the intelligent +temperature-gradient controller will not be affected by the change in the direction of the +external thermal field. In particular, we guarantee the hot and cold sources consistent with the +above and rotate them 30, 60, and 90◦ around the center of the bilayer structure. Apparently, +temperature distributions in the white dashed circle is different from each other when rotating +the same 𝑇+ and 𝑇*; see 𝑇(#), 𝑇/(#), 𝑇//(#), and 𝑇///(#) in Figs. 2a,d,g,j. Subsequently, we +calculate the angular velocity 𝜔! in cases with 𝑇+ = 293, 303, 313 K in the above four +directions using the pre-trained ANN and perform finite-element simulations. Finally, Figs. +2c,f,i,l verify that the performance of the intelligent temperature-gradient controller has good +robustness to the influence of external thermal flow’s direction. Such intelligent metadevice + +9 + +also maintains functional stability under other cold sources and non-uniform external thermal +fields (see Supplementary Note 3). In addition, the calculated 𝜔! from pre-trained ANN is +almost consistent with the set 𝜔!,123456 when external hot and cold sources are given, as +shown in Figs. 2m-o. +Experimental realization and measurements +The intelligent temperature-gradient controller contains four parts: a micro infrared +camera, a computing system with pre-trained ANN, a stepper motor, and a bilayer structure; +see the real experimental setup in Supplementary Figure S7. A bilayer structure with a thickness +of 2 mm is connected to a hot and cold container on the two sides, serving as heat baths. Its +components and sizes are the same as those in simulations. The micro infrared camera is +controlled by the computing system. Every time the infrared camera is started, it measures the +temperature distribution of the bilayer structure and transmits the temperature data to the +computing system. The computing system consists of power, a microcomputer Raspberry Pi, a +power of motor driver, and a motor driver. A pre-trained ANN program runs in the Raspberry +Pi. The input data 𝑇(#) is from the temperature data measured by the micro infrared camera. +After the program processing, the computing system extracts the temperature data of the bilayer +structure’s surroundings, provided to the input layer 𝑇(#) of ANN. When reading the input +data 𝑇(#), the pre-trained ANN program in the computing system calculates the angular +velocity 𝜔! of PDMS in the core region Ω! of the bilayer structure. The corresponding signal +of controlling 𝜔! is transmitted to the stepper motor via the motor driver. Finally, the PDMS +spins around the center, driven by the stepper motor, and the temperature-gradient distribution +in Ω! is regulated. To verify the performance of the intelligent device, we first set the + +10 + +temperatures of the hot and cold baths to 293 K and 283 K. After starting this intelligent system, +the temperature data 𝑇𝒂 +(#) are measured and the angular velocity 𝜔! of PDMS is calculated +as 0.118 rad s-1 via pre-trained ANN, see Fig. 3a. Fig. 3b shows the temperature profile of the +bilayer structure recorded by infrared camera Fotric 430. Note that in the core region, the +temperature distribution is uniform, and in the background region, the temperature field is +nearly undistorted. Subsequently, we fixed the temperature of the cold bath to 283 K and +changed the temperature of the hot bath to 303 K. When the temperature field is stable, we start +the intelligent device again. The measured temperature data 𝑇𝒃 +(#) and the calculated 𝜔! = + 0.0007 rad s-1 are shown in Fig. 3c. The temperature profile of the bilayer structure is shown +in Fig. 3d. At this time, the background temperature field is still undisturbed, and the uniformity +of the temperature distribution in the core region is slightly broken. Finally, with the same cold +bath, we further set the hot bath to 313 K. Fig. 3e displays the measured temperature data 𝑇𝒄 +(#). +As anticipated, the calculated 𝜔! is 0 rad s-1. Therefore, we get the temperature profile of the +bilayer structure, see Fig. 3f. We observe from Fig. 3f that the temperature distribution in the +core region has a maximal non-uniformity, also almost without disturbing the background +thermal field. Above temperature data 𝑇𝒂 +(#), 𝑇𝒃 +(#), and 𝑇𝒄 +(#), the relevant angular velocity +𝜔! , and their temperature profiles of the bilayer structure are consistent with the above +simulation results. For quantitative analysis, we use the central difference method to process +the discrete temperature data in Figs. 3b,d,f and obtain these temperature-gradient distributions +in the core region (see Fig. 3g), which is in good agreement with the simulation results in the +right part of Fig. 2c. As a result, we realize the manipulation of the core region’s temperature +gradient in the bilayer structure based on the feedback of temperature information 𝑻(#) of its + +11 + +surroundings. +Potential applications + +We realize a thermal signal modulator based on the intelligent temperature-gradient +controller for heat communications. The existing work adopted binary thermal spatial coding +to store information in heat communications (thermal signals) [45]. Binary 0 and 1 are +represented by encoding the temperature gradient in the working zone of the cloaking and +concentration with core-shell structures, where the core region is the working zone. For thermal +cloaking (concentration), there is a minimum (maximum) temperature-gradient distribution in +the working zone. Via the continuous arrangement of cloaking or concentration devices, +thermal signals could be stored and characterized by the temperature-gradient distributions in +the working zones of the arranged metadevices. Actually, binary encoding makes information +storage inefficient. Thermal signals should oscillate with space in a continuous mode to transmit +more encoding information simultaneously in heat communications. However, once the +continuous encoding is adopted, original thermal signals are easily disturbed and can only +oscillate with space in smaller amplitude due to thermal dissipation and thermal noise. Thanks +to the proposed thermal signal modulator, original disturbed thermal signals can be re- +modulated to oscillate with space in a larger amplitude; see the schematic in Fig. 4a. We select +a square area (see the dashed box marked in Fig. 4a) in the working zone of devices as the +encoding zone. The area size is consistent with the size of the intelligent temperature-gradient +controller. To ensure the temperature field in the working zone of the device is not disturbed, +we make the effective thermal conductivity inside and outside the encoding zone consistent. +The original temperature gradient in the encoding zone is considered the external thermal field + +12 + +of the intelligent temperature-gradient controller. In above simulations, external thermal fields +|∇𝑇| vary from (𝑇+,789 − 𝑇*)/𝐿 = 50 to (𝑇+,72: − 𝑇*)/𝐿 = 150 K m-1. Through the +intelligent temperature-gradient controller, the average temperature gradient of the modulated +zone (Ω!; see round dashed line marked in Fig. 4a) could be ranged from 0 to 238.5 K m-1. We +obtain a linear transformation relationship between the original temperature-gradient range and +modulated temperature-gradient range. Considering the oscillation of original thermal signals +driven by variable hot sources across the x direction in the sinusoidal law sin 2𝜋𝑥/𝜆, we apply +the linear transformation to original thermal signals and finally get the modulated thermal +signals, see Fig. 4b. We take 𝜆 as 10 m, and arrange 50 intelligent temperature-gradient +controllers (each size: 0.2 × 0.2 m2) in each range of 𝜆. Fig. 4c shows the temperature profiles +of several controllers and their original external thermal fields placed at x = 2.5, 17.5, 35, and +47.5 m (from finite-element simulations), respectively. Here, each x coordinate represents the +central position of each controller. We mark the original (modulated) temperature gradient +values in the encoding (modulated) zone of several controllers; see Fig. 4c. Such a thermal +signal modulator does not change the relative strength relationship between the original signals +but makes the difference between the original signals more obvious. +Conclusion and discussion +To sum up, we propose an environment-adaptive thermal metamaterial driven by a +machine-learning algorithm without manual intervention. As a conceptual verification, via pre- +trained ANN (artificial neural network), we achieve the property of intelligent adjustment of +the temperature gradient of the spinning component PDMS (poly-dimethylsiloxane) of a bilayer +structure based on its ambient temperature. For algorithm implementation, we consider the + +13 + +temperatures at a series of discrete positions on the outer contour of the bilayer structure +(ambient temperature) as the input data. Then, the pre-trained ANN outputs the PDMS’s +spinning angular velocity 𝜔!. 𝜔! can adjust the effective thermal conductivity of the spinning +PDMS and then adjust the temperature-gradient distribution in the core region. As for the +hardware implementation, we embed the pre-trained ANN into a microcomputer called +Raspberry Pi. One end is connected to the micro infrared camera to detect the ambient +temperature for the input data, and the other is connected to a stepper-motor driver to control +the 𝜔! of the motor, rotating the PDMS. Finite-element simulations and experiments have +confirmed this intelligent temperature-gradient controller. Meanwhile, the above thermal +manipulation is robust for the direction of external thermal fields. In addition, we design a +thermal signal modulator with functional robustness for the intelligent temperature-gradient +controller, which modulates original thermal signals clearer. +Incidentally, self-adaptive or intelligent devices involve two typical application scenarios. +The first is that the device displays stable function in a changeable environment like the thermal +signal modulator. The second is the device with intelligent functional choice responding to +environmental changes. We still design an application based on the intelligent temperature- +gradient controller for the second typical scenario, an intelligent thermoelectric generator with +intelligent functional choice in a changeable environment. The adjustable temperature gradient +in the core region enables thermoelectric materials like Bi2Te3 to generate tunable electromotive +force, verified by finite-element simulations (see Supplementary Note 4). Finally, we make +metamaterials have the ability to perceive the environment. The proposed concept combines +diverse domains, such as artificial intelligence, metamaterials, energy utilization, heat + +14 + +communications, and thermal management. We promise the interdisciplinary work to provide +new inspiration for progress in various areas, for example, intelligent thermal management in +chips. + +Additional information +All study data are included in the article and/or Supplementary Information. +Acknowledgements +This work was supported by the National Natural Science Foundation of China to J.H. +(11725521 and 12035004), the Science and Technology Commission of Shanghai Municipality +to J.H. (20JC1414700), and the Singapore Ministry of Education to C.-W.Q. (R-263-000-E19- +114). +Author contributions +P.J., C.-W.Q., and J.H. conceived of the idea. P.J. proposed the methodology, designed the +algorithm and hardware programs, and conducted the experiments. 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Appl. 10, 054032 (2018). + + + +20 + + +Fig. 1 The architecture of the intelligent temperature-gradient controller. The intelligent +controller comprises a micro infrared camera, a pre-trained artificial neural network, a stepper +motor, and a bilayer structure. The infrared camera measures the temperature data of the bilayer +structure’s surroundings. The measured temperature data is input into the pre-trained artificial +neural network, calculating the angular velocity 𝜔! of the spinning component in the core +region Ω! of the bilayer structure. The stepper motor reads the angular velocity and drives the +spinning component to spin (through the spin disk). Finally, thermal functions in the core region +are regulated. + +Input layer +Hidden layers +Output layer +Ti +T2 +T3 +01 +2 +Spin disk +Stepper motor +n +Artificialneuralnetwork(ANN) +emperature +Surroundings +Angle +Surroundings +Variable +Variable +R. +R +Microinfraredcamera +Bilayerstructure21 + + +Fig. 2 Response of intelligent temperature-gradient controller in simulations. a N = 36 equally +spaced temperature data 𝑇𝒂 +(#) , 𝑇𝒃 +(#) , 𝑇𝒄 +(#) in the white dashed circle in static bilayer +structures. The hot source is set as 293, 303, 313 K, respectively. The cold source is fixed at +283 K. The first data is the temperature of the position marked 0◦. Each temperature data is +taken every 10◦ in the counterclockwise direction. b Temperature profiles of the bilayer + +a +b +d +e +90° +.06 +120° +60° +120° +60° +T +150° +30° +150° +30° +0.136 +30° +Angle +180° +0° +Angle +180° +0° +@1= 0.0006 +@1= 0.00089 +313 K +210° +330° +210° +330° +T(O) +T (0) +240° +300° +240° +300° +270° +270° +c +f +250 +250 +CE +250 +250 +0=0 +@=0 +200 +200 +200 +Original +150 +3 T. - 313 K +150 +150 +Original +2n T--313 K +150 +，=0.00067 +Modt +@1= 0.00089 +100 +c T. - 303 K +100 +100 +T.-303K +100 +Modu +50 +2 T. - 293 K +50 +50 +A2 T--293 K +@, = 0.10 +@, = 0.136 +0.02 +0.02 +0.059 +0.02 +0.02 +x (m) +-0.02~0.02 +y (m) +x (m) +-0.02-0.02 +y (m) +x (m) +-0.02~0.02 +y (m) +x (m) +y(m) +g +h +j +k +90° +90° +120° +60° +120° +60° +T "(0) +150° +30° +=0.105 +150° +30° +@-0.115 +(0).. +T"(0) +.09 +.06 +283K +4(0) +Angle +180° +0 +Angle +180° +00 +01= 0.0008 ++ +@=0.00085 +210° +330° +210° +330° +240° +300° +240° +300° +270° +270° +0 +00 +- +250 +250 +0=0 +250 +250 +0m +IVTI(K mr) +200 +ated +200 +200 +Original +ted +150 +3·T-313K +150 +150 +C8 T, - 313 K +150 +Modu +0=0.0008 +w1=0.00085 +100 +6T-303K +100 +100 - +2 T, - 303 K +100 +50 +2·T=293K +2 T, -293 K +@, = 0.105 +50 +@, = 0.115 +0.02 +0.02 +0.02 +0.02 +0.001 +0.09 +0.02 +-0.020.02 +y(m) +-0.020.02 +0 +y (m) +0 +x (m) +y (m) +-0.02-0.02 +x (m) +x (m) +x (m) +y(m) +T= 293K T.=283K +T. = 303 K T= 283 K +T=313KT=283K +0.14 +1 ×10-3 +m +Wi.Target +n +0.12 +山 +W1,Target +0.8 +.0.81 +山 +0.1 +0.08 +0.6 +Wi,Target +(rad +0F +0.06 +0.4 +0.04 +0.2 +0.02 +0 +0 +0° +30° +60° +90° +0° +30° +60° +90° +0° +30° +60° +90° +Direction of external thermal field +Direction of external thermal field +Direction of external thermal field22 + +structure with spinning angular velocity 𝜔! = 0.10, 0.00067, 0 rad s-1, respectively. Left part +of c Temperature-gradient distributions |∇𝑇| in core region Ω! of static pure background +with three hot sources. Right part of c Temperature-gradient distributions |∇𝑇| in core region +Ω! of the bilayer structure with 𝜔! = 0.10, 0.00067, 0 rad s-1, respectively. d-f Same +characterization with a-c when the external thermal field rotates 30◦ around the center of the +bilayer structure in the counterclockwise direction. g-i Same characterization with a-c when +the external thermal field rotates 60◦ around the center of the bilayer structure in the +counterclockwise direction. j-l Same characterization with a-c when the external thermal field +rotates 90◦ around the center of the bilayer structure in the counterclockwise direction. m-o +Comparison of calculated 𝜔! and targeted 𝜔!,123456 in above four directions of the external +thermal field in cases with 𝑇+ = 293, 303, 313 K, respectively. + + + + + + + + + +23 + + +Fig. 3 Realization of the intelligent temperature-gradient controller. a,c,e Experimental +temperature data 𝑇𝒂 +(#), 𝑇𝒃 +(#), 𝑇𝒄 +(#) in the dashed circle with radius R3 = 60 mm, marked in +b,d,f, when hot bath is set to 293, 303, 313 K, respectively. The cold bath is fixed to 283 K. +b,d,f Measured temperature profile of the bilayer structure with spinning angular velocity +𝜔! = 0.118, 0.0007, 0 rad s-1, respectively. g Calculated temperature-gradient distributions +|∇𝑇| in core region Ω! of the bilayer structure with 𝜔! = 0.118, 0.0007, 0 rad s-1, +respectively. + + + +Ta +(0)(K) +θ (∘) +Tb +(0)(K) +θ (∘) +θ (∘) +Tc +(0)(K) +Exp. +Simu. +Exp. +Simu. +Exp. +Simu. +293 K +283 K +303 K +283 K +313 K +283 K +a +b +c +d +e +f +g +ω1,Target +ω1(rad s-1) +0.118 +0.12 +ω1,Target +ω1(rad s-1) +0.0007 +0.001 +ω1,Target +ω1(rad s-1) +0 +0 +0∘ +θ∘ +0∘ +0∘ +Ta +(0) +Tb +(0) +Tc +(0) +R3= 60 mm +0 +50 +-0.015 +100 +150 +200 +0.015 +250 +0 +0 +0.015 +-0.015 +y (m) +x (m) +|∇T| (K m-1) +ω1 = 0 +ω1 = 0.0007 +ω1 = 0.118 + +24 + + +Fig. 4 Intelligent temperature-gradient controller for robust modulation of thermal signals. a +Schematic of the thermal signal modulator. b Comparison of modulated and original thermal +signals. Modulated (original) thermal signals are denoted by distributions of averaged +temperature gradients in the modulated zones of the controllers (only external thermal fields) +across the x direction. Each x coordinate represents the central position of each controller. c +Simulated temperature profiles of several controllers and their external thermal fields placed at +𝑥 = 2.5, 17.5, 35, and 47.5 m, respectively. + + + + + + + + + +Modulated zone +Cold source (TC) +Encoding zone +x +y +z +|∇T| +|∇T′| +Original +|∇T| = TH − TC +L +T(0) +Designed +intelligent system +ω1 +Modulated +|∇T′| +|∇T| +|∇T′| +Original thermal signal +Modulated thermal signal +a +238.5 +b + (K m-1) +0 +10 +20 +30 +40 +50 +0 +50 +100 +150 +200 +250 +0 +50 +100 +150 +200 +x (m) +Original +Modulated +c +x = 2.5 m +x = 17.5 m +x = 35 m +x = 47.5 m +|∇T| +Encoding zone +TH +TC +T(0) +ω1 +L +x +y +Modulated +zone +Ω1 +L = 0.2 m +Original +150 K m-1 +Modulated |∇T′| Ω1 +Unit: K +283 +293 +303 +313 +|∇T| +238.5 K m-1 +50 K m-1 +0 K m-1 +100 K m-1 +120 K m-1 +50 K m-1 +0 K m-1 +TC +(fixed) +TH +(variable) +Hot source (TH) + diff --git a/YdE3T4oBgHgl3EQfcQoq/content/tmp_files/load_file.txt b/YdE3T4oBgHgl3EQfcQoq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e82a6ee96c63e2921fd5b43a547ec0de765940df --- /dev/null +++ b/YdE3T4oBgHgl3EQfcQoq/content/tmp_files/load_file.txt @@ -0,0 +1,791 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf,len=790 +page_content='1 Machine-learning-assisted environment-adaptive thermal metamaterials Peng Jin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Liujun Xu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Guoqiang Xu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Jiaxin Li2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Cheng-Wei Qiu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='* and Jiping Huang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='* 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' State Key Laboratory of Surface Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' and Key Laboratory of Micro and Nano Photonic Structures (MOE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Fudan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Shanghai 200438,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' China 2Department of Electrical and Computer Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' National University of Singapore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Singapore 117583,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Singapore 3Graduate School of China Academy of Engineering Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Beijing 100193,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' China Corresponding authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Emails: chengwei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='qiu@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' jphuang@fudan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='cn 2 Abstract Adaptive metamaterials have prevailed recently owing to their extraordinary features like dynamic response to external interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' However, highly complicated parameters, narrow working ranges, and supervised manual intervention are still long-term and tricky obstacles to the most advanced self-adaptive metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' To surmount these barriers, we present environment-adaptive thermal metamaterials driven by machine learning, which can automatically sense ambient temperatures and regulate thermal functions promptly and continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Thermal functions are robust when external thermal fields change their directions, and simulations and experiments exhibit excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Based on this, we further design two metadevices with on-demand adaptability, performing distinctive features with isotropic materials, wide working temperatures, and spontaneous response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' This work provides a paradigm for intelligent diffusion metamaterial design and can be extended to other diffusion fields, responding to more complex and variable environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3 Introduction Metamaterials [1-5] have drawn intensive attraction due to their unprecedented ability to manipulate physical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Profit from computer numerical control and three-dimensional printing technology, metamaterials with novel functions are fabricated according to given parameters and applied to laboratories or industries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Traditional metamaterials mainly focused on static cases [6-15], lacking tunability for variable scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' To tackle this issue, tunable metamaterials with dynamic features have emerged, covering optics [16,17], acoustics [18], and thermotics [19-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For example, many advanced thermal functions have been realized, including macroscopic thermal diodes [19], tunable analog thermal materials [22], path- dependent thermal metadevices [23], and tunable hybrid thermal metamaterials [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' On the other hand, adaptive thermal metadevices are presented to maintain the robustness of functions against environmental changes or switch functions depending on application scenes [20,25-30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' However, the achievement of state-of-the-art self-adaptive thermal metamaterials is confronted with three longstanding and strong barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Firstly, adaptive thermal metadevices with robust functions usually require extremely complicated parameters [25-27], which are difficult to prepare from natural bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Secondly, existing adaptive metadevices, especially macroscopic thermal diodes [19] and energy-free temperature trapping [20], are limited to a specific temperature range related to the phase change temperature of shape memory alloys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, most tunable or adaptive metamaterials [16-30] need to be adjusted through manual control rather than automatically, lacking self-cognitive ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Recently, intelligent materials, involving interdisciplinary research and combining intelligent algorithms with material design, have motivated applications in optics [31-33], 4 nanotechnology [34], theoretical physics [35], materials science [36], and thermal science [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' These advances inspire ideal self-adaptive thermal metamaterials with full embracement of intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In principle, ideal self-adaptive thermal metadevices should automatically (without human aid) and timely adjust their dynamic components to keep function stable or switch functions continuously in response to the broad range of ambient temperature change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Developing such self-adaptive thermal metamaterials is highly desirable to be valuable in situ scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' However, the demanding technical performance requires an appropriate actuation mechanism that integrates an algorithm-driven intelligent system with thermal metamaterial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Although these metamaterials have been used to design advanced self-adaptive optical cloaks in wave systems [38], they fail in diffusion systems like heat transfer due to the lack of controllable degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Existing machine-learning-based thermal metamaterials are dictated by the inverse design method [39-42], which only calculates the parameters of materials and sizes for desired functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In addition, once such metamaterials are prepared, their functions are not switchable, lacking response to various scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Here, we introduce a machine-learning-assisted intelligent system and propose environment-adaptive thermal metamaterials driven by big data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As a conceptual implementation, we design an intelligent temperature gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We load the pre- trained artificial neural network into a hardware system and combine it with the bilayer structure [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Depending on the sensing-feedback ambient temperatures, the thermal conductivity of a spinning component could be adjusted to achieve a tunable temperature gradient in a target region, verified by finite-element simulations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We then propose two applications with on-demand adaptability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' One is a thermal signal modulator with 5 functional robustness, making original thermal signals clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The other is an intelligent thermoelectric generator with intelligent functional choice, which can automatically adjust the electromotive force generated by thermoelectric materials [44] based on the ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The intelligent environment-adaptive thermal metamaterial features isotropic materials, unlimited working temperatures, and cognitive responsiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' A handy actuation mechanism also integrates machine-learning-driven intelligent systems with diffusion metamaterial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Our work brings the design of self-adaptive diffusion metamaterials to a new stage without human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Architecture of the intelligent temperature-gradient controller The architecture of the intelligent temperature-gradient controller is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' It contains four main modules: a temperature acquisition module (micro infrared camera), a computing system with a pre-trained artificial neural network (ANN), a stepper motor, and a bilayer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We aim to manipulate the temperature gradient of the target region based on the feedback of temperature information of its surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Here, as a proof-of-concept implementation, we consider a two-dimensional system with a bilayer metal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The target region is the core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' consisting of poly-dimethylsiloxane (PDMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The component of the inner layer (Silicone pad) is approximately adiabatic for precise control of thermal fields of Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', and the outer layer as a compensation layer (Magnesium alloy) is intended not to disturb the thermal fields of the background (Inconel alloy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The thermal conductivity from the inside out is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='15, 1, 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='7, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='8 W m-1 K-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' After setting R1 = 30 mm and R2 = 53 mm, R3 can be calculated as 60 mm [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 6 To characterize the temperature information of the bilayer structure’s surroundings, we choose a series of discrete positions around the outer layer and extract their temperature using a micro infrared camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 1, the blue dashed circle is the chosen bilayer structure’s surrounding, and the position marked 0◦ is the first position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Then, we take the temperature of the amount of N equally spaced positions on the circumference in a counterclockwise direction, serving as the input layer of ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Thanks to the tunable analog thermal material [22], by spinning (angular velocity: 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=') the PDMS in the core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', the effective thermal conductivity of the spinning medium can be tuned from near-zero (𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 0) to near-infinity (larger 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' On the other hand, the effective thermal conductivity of Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' is the crucial factor affecting the temperature-gradient distributions of Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see lower part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 1 for an intuitive description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Color denotes temperature profiles of the core region, and the white lines represent the isotherms from finite- element simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Here, sparser isotherms correspond to a smaller temperature gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For the sake of “intelligence”, we utilize ANN to establish a mapping between extracted temperature information (input data: 𝑻(#)) and the angular velocity of Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' (output data: 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 1, we show the structural component of the ANN, which is fully connected with four hidden layers (50 neurons per layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Activations of all neurons in the next layer are determined by activations of those in the current layer (𝑯(%)), represented by ⎩ ⎪ ⎨ ⎪⎧𝑯(%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=') = ReLU0𝑾(%)𝑻(%) + 𝒃(%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' )4, 𝑖 = 0 𝑯(%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=') = ReLU0𝑾(%)𝑯(%) + 𝒃(%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' )4, 0 < 𝑖 < 4 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = ReLU0𝑾(%)𝑯(%) + 𝒃(%&!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' )4, 𝑖 = 4 (1) where ReLU (a) = max(0, a) is the rectified linear unit function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' W and b are weights and biases for the neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' i is the ordinal number of layers, and 𝑖 = 0 represents the input layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 7 As the ANN is a data-driven-based model, we then prepare the dataset (see Supplementary Note 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, via the back propagation algorithm, the proposed ANN with selected hyperparameters is well trained by the dataset (see Supplementary Note 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' When the thermal field of the bilayer structure reaches equilibrium, the temperature information (𝑇!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=", 𝑇', 𝑇(, ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', 𝑇)) at the circle with a radius of R3 = 60 mm is collected by micro infrared camera and imported into the computing system as an array of input signals for the pre-trained ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The computing system’s output signal is the stepper motor’s spinning angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='. Considering a case where the output spinning angular velocity is 0 rad s-1, static PDMS possesses thermal conductivity with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='15 W m-1 K-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' At this time, the temperature gradient in Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' reaches its maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Intelligent response in omnidirectional simulations Finite-element simulations are first used to demonstrate the performance of an intelligent temperature-gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For proof-of-concept verification, we consider a two- dimensional bilayer structure whose components and sizes are the same as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The system’s left (right) end connects to a hot (cold) source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In our simulations, the cold source (𝑇* = 283 K) is fixed, while the hot source (𝑇+) is changeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We first extract the temperature data (𝑇𝒂 (#), 𝑇𝒃 (#), 𝑇𝒄 (#)) of N = 36 equally spaced positions in the white dashed circle in three cases (𝑇+ = 293, 303, 313 K) of a static bilayer structure, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For each case, the first data is the temperature of the position marked 0◦ in the dashed circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The order of taking the temperature of these positions is counterclockwise, serving as the input layer of the pre-trained ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Hence, via the pre-trained ANN, the spinning angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of PDMS is calculated individually as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00067, and 0 rad s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' After setting the above parameters 8 (𝑇+ and 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=') in finite-element simulations, we show these three temperature profiles (color distributions) of the bilayer structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' No matter how the angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of PDMS changes, the temperature distributions of the background will not be disturbed (see Supplementary Note 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, their corresponding temperature-gradient distributions in Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' are given in the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For comparison, we show the temperature-gradient distributions in Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of pure background (size: 200 × 200 mm2) with three hot sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see left part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As anticipated, there is a mapping relationship between lowest/highest 𝑇+ − 𝑇* and highest/lowest angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' (or say, lowest/highest |∇𝑇| in Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=') in above scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Further, the range of originally external temperature-gradient field [(𝑇+ − 𝑇*)/𝐿 : 50 to 150 K m-1] can be adjusted to a wider range of temperature gradient (|∇𝑇|: 0 to 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 K m-1) in the target region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='. In the actual scene, we are unsure in which direction the external thermal field is exerted on the bilayer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Therefore, we should ensure that the performance of the intelligent temperature-gradient controller will not be affected by the change in the direction of the external thermal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In particular, we guarantee the hot and cold sources consistent with the above and rotate them 30, 60, and 90◦ around the center of the bilayer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Apparently, temperature distributions in the white dashed circle is different from each other when rotating the same 𝑇+ and 𝑇*;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see 𝑇(#), 𝑇/(#), 𝑇//(#), and 𝑇///(#) in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2a,d,g,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Subsequently, we calculate the angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' in cases with 𝑇+ = 293, 303, 313 K in the above four directions using the pre-trained ANN and perform finite-element simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2c,f,i,l verify that the performance of the intelligent temperature-gradient controller has good robustness to the influence of external thermal flow’s direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Such intelligent metadevice 9 also maintains functional stability under other cold sources and non-uniform external thermal fields (see Supplementary Note 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In addition, the calculated 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' from pre-trained ANN is almost consistent with the set 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=',123456 when external hot and cold sources are given, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2m-o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Experimental realization and measurements The intelligent temperature-gradient controller contains four parts: a micro infrared camera, a computing system with pre-trained ANN, a stepper motor, and a bilayer structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see the real experimental setup in Supplementary Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' A bilayer structure with a thickness of 2 mm is connected to a hot and cold container on the two sides, serving as heat baths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Its components and sizes are the same as those in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The micro infrared camera is controlled by the computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Every time the infrared camera is started, it measures the temperature distribution of the bilayer structure and transmits the temperature data to the computing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The computing system consists of power, a microcomputer Raspberry Pi, a power of motor driver, and a motor driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' A pre-trained ANN program runs in the Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The input data 𝑇(#) is from the temperature data measured by the micro infrared camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' After the program processing, the computing system extracts the temperature data of the bilayer structure’s surroundings, provided to the input layer 𝑇(#) of ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' When reading the input data 𝑇(#), the pre-trained ANN program in the computing system calculates the angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of PDMS in the core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of the bilayer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The corresponding signal of controlling 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' is transmitted to the stepper motor via the motor driver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, the PDMS spins around the center, driven by the stepper motor, and the temperature-gradient distribution in Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' is regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' To verify the performance of the intelligent device, we first set the 10 temperatures of the hot and cold baths to 293 K and 283 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' After starting this intelligent system, the temperature data 𝑇𝒂 (#) are measured and the angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of PDMS is calculated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='118 rad s-1 via pre-trained ANN, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3b shows the temperature profile of the bilayer structure recorded by infrared camera Fotric 430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Note that in the core region, the temperature distribution is uniform, and in the background region, the temperature field is nearly undistorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Subsequently, we fixed the temperature of the cold bath to 283 K and changed the temperature of the hot bath to 303 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' When the temperature field is stable, we start the intelligent device again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The measured temperature data 𝑇𝒃 (#) and the calculated 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0007 rad s-1 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The temperature profile of the bilayer structure is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' At this time, the background temperature field is still undisturbed, and the uniformity of the temperature distribution in the core region is slightly broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, with the same cold bath, we further set the hot bath to 313 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3e displays the measured temperature data 𝑇𝒄 (#).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As anticipated, the calculated 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' is 0 rad s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Therefore, we get the temperature profile of the bilayer structure, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We observe from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3f that the temperature distribution in the core region has a maximal non-uniformity, also almost without disturbing the background thermal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Above temperature data 𝑇𝒂 (#), 𝑇𝒃 (#), and 𝑇𝒄 (#), the relevant angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' , and their temperature profiles of the bilayer structure are consistent with the above simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For quantitative analysis, we use the central difference method to process the discrete temperature data in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3b,d,f and obtain these temperature-gradient distributions in the core region (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3g), which is in good agreement with the simulation results in the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As a result, we realize the manipulation of the core region’s temperature gradient in the bilayer structure based on the feedback of temperature information 𝑻(#) of its 11 surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Potential applications We realize a thermal signal modulator based on the intelligent temperature-gradient controller for heat communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The existing work adopted binary thermal spatial coding to store information in heat communications (thermal signals) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Binary 0 and 1 are represented by encoding the temperature gradient in the working zone of the cloaking and concentration with core-shell structures, where the core region is the working zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For thermal cloaking (concentration), there is a minimum (maximum) temperature-gradient distribution in the working zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Via the continuous arrangement of cloaking or concentration devices, thermal signals could be stored and characterized by the temperature-gradient distributions in the working zones of the arranged metadevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Actually, binary encoding makes information storage inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Thermal signals should oscillate with space in a continuous mode to transmit more encoding information simultaneously in heat communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' However, once the continuous encoding is adopted, original thermal signals are easily disturbed and can only oscillate with space in smaller amplitude due to thermal dissipation and thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Thanks to the proposed thermal signal modulator, original disturbed thermal signals can be re- modulated to oscillate with space in a larger amplitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see the schematic in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We select a square area (see the dashed box marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4a) in the working zone of devices as the encoding zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The area size is consistent with the size of the intelligent temperature-gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' To ensure the temperature field in the working zone of the device is not disturbed, we make the effective thermal conductivity inside and outside the encoding zone consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The original temperature gradient in the encoding zone is considered the external thermal field 12 of the intelligent temperature-gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In above simulations, external thermal fields |∇𝑇| vary from (𝑇+,789 − 𝑇*)/𝐿 = 50 to (𝑇+,72: − 𝑇*)/𝐿 = 150 K m-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Through the intelligent temperature-gradient controller, the average temperature gradient of the modulated zone (Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see round dashed line marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4a) could be ranged from 0 to 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 K m-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We obtain a linear transformation relationship between the original temperature-gradient range and modulated temperature-gradient range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Considering the oscillation of original thermal signals driven by variable hot sources across the x direction in the sinusoidal law sin 2𝜋𝑥/𝜆, we apply the linear transformation to original thermal signals and finally get the modulated thermal signals, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We take 𝜆 as 10 m, and arrange 50 intelligent temperature-gradient controllers (each size: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='2 m2) in each range of 𝜆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4c shows the temperature profiles of several controllers and their original external thermal fields placed at x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5, 35, and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 m (from finite-element simulations), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Here, each x coordinate represents the central position of each controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We mark the original (modulated) temperature gradient values in the encoding (modulated) zone of several controllers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Such a thermal signal modulator does not change the relative strength relationship between the original signals but makes the difference between the original signals more obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Conclusion and discussion To sum up, we propose an environment-adaptive thermal metamaterial driven by a machine-learning algorithm without manual intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As a conceptual verification, via pre- trained ANN (artificial neural network), we achieve the property of intelligent adjustment of the temperature gradient of the spinning component PDMS (poly-dimethylsiloxane) of a bilayer structure based on its ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' For algorithm implementation, we consider the 13 temperatures at a series of discrete positions on the outer contour of the bilayer structure (ambient temperature) as the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Then, the pre-trained ANN outputs the PDMS’s spinning angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='. 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' can adjust the effective thermal conductivity of the spinning PDMS and then adjust the temperature-gradient distribution in the core region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' As for the hardware implementation, we embed the pre-trained ANN into a microcomputer called Raspberry Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' One end is connected to the micro infrared camera to detect the ambient temperature for the input data, and the other is connected to a stepper-motor driver to control the 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of the motor, rotating the PDMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finite-element simulations and experiments have confirmed this intelligent temperature-gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Meanwhile, the above thermal manipulation is robust for the direction of external thermal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' In addition, we design a thermal signal modulator with functional robustness for the intelligent temperature-gradient controller, which modulates original thermal signals clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Incidentally, self-adaptive or intelligent devices involve two typical application scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The first is that the device displays stable function in a changeable environment like the thermal signal modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The second is the device with intelligent functional choice responding to environmental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We still design an application based on the intelligent temperature- gradient controller for the second typical scenario, an intelligent thermoelectric generator with intelligent functional choice in a changeable environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The adjustable temperature gradient in the core region enables thermoelectric materials like Bi2Te3 to generate tunable electromotive force, verified by finite-element simulations (see Supplementary Note 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, we make metamaterials have the ability to perceive the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The proposed concept combines diverse domains, such as artificial intelligence, metamaterials, energy utilization, heat 14 communications, and thermal management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' We promise the interdisciplinary work to provide new inspiration for progress in various areas, for example, intelligent thermal management in chips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Additional information All study data are included in the article and/or Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Acknowledgements This work was supported by the National Natural Science Foundation of China to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' (11725521 and 12035004), the Science and Technology Commission of Shanghai Municipality to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' (20JC1414700), and the Singapore Ministry of Education to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' (R-263-000-E19- 114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Author contributions P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' conceived of the idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' proposed the methodology, designed the algorithm and hardware programs, and conducted the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='- W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' made the visualization and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' supervised the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' All authors contributed to the discussion and finalization of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Competing interests The authors declared no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 15 References [1] Chen, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 7, 021407 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [38] Qian, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Deep-learning-enabled self-adaptive microwave cloak without human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 14, 383–390 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [39] Jin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Particle swarm optimization for realizing bilayer thermal sensors with bulk isotropic materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 172, 121177 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [40] Lu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', Jain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=', Ang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' & Ong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Deep learning techniques elucidate and modify the shape factor to extend the effective medium theory beyond its original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 184, 122305 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [41] Ji, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Design of thermal cloaks with isotropic materials based on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 189, 122716 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [42] Ji, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Deep learning based design of thermal metadevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Heat Mass Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 196, 123149 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [43] Han, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Experimental demonstration of a bilayer thermal cloak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 112, 19 054302 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [44] Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Seebeck-driven transverse thermoelectric generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 20, 463– 467 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' [45] Hu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Binary Thermal Encoding by Energy Shielding and Harvesting Units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 10, 054032 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 1 The architecture of the intelligent temperature-gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The intelligent controller comprises a micro infrared camera, a pre-trained artificial neural network, a stepper motor, and a bilayer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The infrared camera measures the temperature data of the bilayer structure’s surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The measured temperature data is input into the pre-trained artificial neural network, calculating the angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of the spinning component in the core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of the bilayer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The stepper motor reads the angular velocity and drives the spinning component to spin (through the spin disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Finally, thermal functions in the core region are regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Input layer Hidden layers Output layer Ti T2 T3 01 2 Spin disk Stepper motor n Artificialneuralnetwork(ANN) emperature Surroundings Angle Surroundings Variable Variable R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' R Microinfraredcamera Bilayerstructure21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 2 Response of intelligent temperature-gradient controller in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' a N = 36 equally spaced temperature data 𝑇𝒂 (#) , 𝑇𝒃 (#) , 𝑇𝒄 (#) in the white dashed circle in static bilayer structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The hot source is set as 293, 303, 313 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The cold source is fixed at 283 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The first data is the temperature of the position marked 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Each temperature data is taken every 10◦ in the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' b Temperature profiles of the bilayer a b d e 90° .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='06 120° 60° 120° 60° T 150° 30° 150° 30° 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='136 30° Angle 180° 0° Angle 180° 0° @1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0006 @1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00089 313 K 210° 330° 210° 330° T(O) T (0) 240° 300° 240° 300° 270° 270° c f 250 250 CE 250 250 0=0 @=0 200 200 200 Original 150 3 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' - 313 K 150 150 Original 2n T--313 K 150 ，=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00067 Modt @1= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00089 100 c T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' - 303 K 100 100 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='-303K 100 Modu 50 2 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' - 293 K 50 50 A2 T--293 K @, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='10 @, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 x (m) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 y (m) x (m) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 y (m) x (m) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 y (m) x (m) y(m) g h j k 90° 90° 120° 60° 120° 60° T "(0) 150° 30° =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='105 150° 30° @-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='115 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='. T"(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='09 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='06 283K 4(0) Angle 180° 0 Angle 180° 00 01= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0008 + @=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00085 210° 330° 210° 330° 240° 300° 240° 300° 270° 270° 0 00 - 250 250 0=0 250 250 0m IVTI(K mr) 200 ated 200 200 Original ted 150 3·T-313K 150 150 C8 T, - 313 K 150 Modu 0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0008 w1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00085 100 6T-303K 100 100 - 2 T, - 303 K 100 50 2·T=293K 2 T, -293 K @, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='105 50 @, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 y(m) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0 y (m) 0 x (m) y (m) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 x (m) x (m) x (m) y(m) T= 293K T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='=283K T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 303 K T= 283 K T=313KT=283K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='14 1 ×10-3 m Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='Target n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='12 山 W1,Target 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='81 山 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='6 Wi,Target (rad 0F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='02 0 0 0° 30° 60° 90° 0° 30° 60° 90° 0° 30° 60° 90° Direction of external thermal field Direction of external thermal field Direction of external thermal field22 structure with spinning angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00067, 0 rad s-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Left part of c Temperature-gradient distributions |∇𝑇| in core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of static pure background with three hot sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Right part of c Temperature-gradient distributions |∇𝑇| in core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of the bilayer structure with 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='10, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='00067, 0 rad s-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' d-f Same characterization with a-c when the external thermal field rotates 30◦ around the center of the bilayer structure in the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' g-i Same characterization with a-c when the external thermal field rotates 60◦ around the center of the bilayer structure in the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' j-l Same characterization with a-c when the external thermal field rotates 90◦ around the center of the bilayer structure in the counterclockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' m-o Comparison of calculated 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' and targeted 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=',123456 in above four directions of the external thermal field in cases with 𝑇+ = 293, 303, 313 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 23 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 3 Realization of the intelligent temperature-gradient controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' a,c,e Experimental temperature data 𝑇𝒂 (#), 𝑇𝒃 (#), 𝑇𝒄 (#) in the dashed circle with radius R3 = 60 mm, marked in b,d,f, when hot bath is set to 293, 303, 313 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' The cold bath is fixed to 283 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' b,d,f Measured temperature profile of the bilayer structure with spinning angular velocity 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='118, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0007, 0 rad s-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' g Calculated temperature-gradient distributions |∇𝑇| in core region Ω!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' of the bilayer structure with 𝜔!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='118, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0007, 0 rad s-1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Ta (0)(K) θ (∘) Tb (0)(K) θ (∘) θ (∘) Tc (0)(K) Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Simu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Simu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Simu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 293 K 283 K 303 K 283 K 313 K 283 K a b c d e f g ω1,Target ω1(rad s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='12 ω1,Target ω1(rad s 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='001 ω1,Target ω1(rad s 1) 0 0 0∘ θ∘ 0∘ 0∘ Ta (0) Tb (0) Tc (0) R3= 60 mm 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='015 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='015 250 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='015 y (m) x (m) |∇T| (K m 1) ω1 = 0 ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='0007 ω1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='118 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' 4 Intelligent temperature-gradient controller for robust modulation of thermal signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' a Schematic of the thermal signal modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' b Comparison of modulated and original thermal signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Modulated (original) thermal signals are denoted by distributions of averaged temperature gradients in the modulated zones of the controllers (only external thermal fields) across the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Each x coordinate represents the central position of each controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' c Simulated temperature profiles of several controllers and their external thermal fields placed at 𝑥 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5, 35, and 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content=' Modulated zone Cold source (TC) Encoding zone x y z |∇T| |∇T′| Original |∇T| = TH − TC L T(0) Designed intelligent system ω1 Modulated |∇T′| |∇T| |∇T′| Original thermal signal Modulated thermal signal a 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 b (K m-1) 0 10 20 30 40 50 0 50 100 150 200 250 0 50 100 150 200 x (m) Original Modulated c x = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 m x = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 m x = 35 m x = 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 m |∇T| Encoding zone TH TC T(0) ω1 L x y Modulated zone Ω1 L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='2 m Original 150 K m-1 Modulated |∇T′| Ω1 Unit: K 283 293 303 313 |∇T| 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} +page_content='5 K m-1 50 K m-1 0 K m-1 100 K m-1 120 K m-1 50 K m-1 0 K m-1 TC (fixed) TH (variable) Hot source (TH)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE3T4oBgHgl3EQfcQoq/content/2301.04523v1.pdf'} diff --git a/_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/2301.02643v1.pdf.txt b/_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/2301.02643v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1620a44e697e1b9c5006a7c152b993bd539f1904 --- /dev/null +++ b/_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/2301.02643v1.pdf.txt @@ -0,0 +1,717 @@ +Auto-Assembly: a framework for automated robotic +assembly directly from CAD. +Fedor Chervinskii§, Sergei Zobov§, Aleksandr Rybnikov§, Danil Petrov§, Komal Vendidandi§ +Λ Γ Γ I V Λ L +Abstract—In this work, we propose a framework called Auto- +Assembly for automated robotic assembly from design files and +demonstrate a practical implementation on modular parts joined +by fastening using a robotic cell consisting of two robots. We show +the flexibility of the approach by testing it on different input +designs. Auto-Assembly consists of several parts: design analysis, +assembly sequence generation, bill-of-process (BOP) generation, +conversion of the BOP to control code, path planning, simulation, +and execution of the control code to assemble parts in the physical +environment. +Index +Terms—industry +4.0, +smart +manufacturing, +cyber- +physical systems, smart factory, manufacturing automation, ma- +nipulators, cellular manufacturing, digital twins, robotic assem- +bly +I. INTRODUCTION +Assembly planning is one of the most laborious tasks +when releasing a new product for manufacturing. Thus, many +algorithms and methods around computer-aided design (CAD) +and digital twins of the factories have emerged in recent +years that help process engineers to prepare a new design +for assembly (Computer-aided Assembly Process Planning +techniques [1]). An emerging trend of Industry 4.0 [2] suggests +that a digital, highly automated factory should be able to infer +the process from the design. In practice, even for an automated +factory, assembly planning has to be followed by an offline- +programming of all the robots and devices to perform the +assembly plan. +Additive manufacturing technology (3D printing [3]) at the +same time has achieved a much higher rate of process design +automation. One can simply load a CAD file into a machine +that will yield a part of the desired design. The main question +we are trying to address in this paper is ”Could a robotic +cell or even the whole factory work just as a 3D printer?”. +When loaded with target assembly CAD design and given +input parts in specified conditions (e.g. placed in special input +jigs) - would it perform the required assembly? +In this work, we show how this can be achieved under +specific constraints, paving the road for future experiments +towards a more general approach and wider applications. +However, the framework we propose is general enough to +accommodate more complex designs and conditions, like +many types of tooling and different joining technologies. +§Equal contribution. +[chervinskii, zobov, rybnikov, danil.petrov, vendidandi]@arrival.com +Fig. 1: Experimental setup: robotic cell with two UR5e manip- +ulators. Left: UR5e with a screwdriver Likratec EH2 R1030- +A and Right: UR5e with gripper Robotiq 2F-85 with custom +designed gripper clamps. On the table: custom-designed 3D- +printed jigs. +II. RELATED WORK AND BACKGROUND +An Assembly Planning for a given design typically starts +from identifying the mating features or joints and suggesting +a feasible Assembly Sequence, which could be automated as +seen in [4], [5], [6]. +To proceed to the process planning, a virtual environment, +also known as Digital Twin [7] is necessary. There are at- +tempts to develop a common ontology, e.g. [8], [9] and unify +interfaces between systems [10] to support process design +automation. +Sierla, Seppo, et al. [11] discuss the conceptual framework +of automated assembly planning using a digital twin. It uses +the XML-based AutomationML [12] data modeling frame- +work. This framework aggregates different data exchange +formats like CAEX for plant description, COLLADA for +geometry and kinematics of 3D models, etc. +There is still not sufficient work in joining together process +planning, motion planning and execution using a common +framework. In [13] authors used artificial intelligence to solve +a Tooling Matching problem and developed an add-on for +Octopuz [14] to do a Motion Planning and Robot Program +Generation for disassembly, but not testing in physical cells. +In another work, [15] a similar pipeline is described for an +arXiv:2301.02643v1 [cs.RO] 6 Jan 2023 + +Screwdriver +Gripper +Assembly +Assembly Jig +Parts Unload Jigs +Screws Unload Jigarchitectural domain, mainly focusing on parametric design +and modular assembly. +We claim that Auto-Assembly is the first proposed frame- +work that can generate and execute robotic assembly process +for an arbitrary input CAD design. +III. PROBLEM STATEMENT AND METHOD OVERVIEW +The main objective of our work is to create a framework +that enables a closed loop between design and robotic manu- +facturing. A target framework should analyse the design and +provide a simulation of assembly, executable programs (when +possible) and other feedback. The primary aim of the feedback +is to help in adapting the design and manufacturing to better +correspond to each other. +The feedback we should provide can be split into two +categories: +• Successful simulation and its’ artefacts can be directly +used to decide on physical manufacturing. Users can +choose between different processes to choose the one, +based on the key performance indicators (KPI) they want +to optimize: time, tooling price, energy consumption, etc. +• In case of a failure, the system should provide all neces- +sary feedback that helps to change the design, robot’s +position, choose the robots with better parameters or +different cell configuration. Such feedback can be: failed +operations, missing appropriate tooling, parts or tools in +collision, unreachable states. +To achieve this, we implement a framework described in +detail in Section IV. Section IV-A gives an overview of +our system and its components. Section IV-B discusses 3D +modelling of the assembly design files that form the base of +our data extraction pipeline. Section IV-C reviews the usage +of this extracted data to produce a set of possible assembly +sequences. +Each operation in the assembly sequence is enriched with +tooling information as discussed in section IV-D. Finding a +specific cell that contains all the resources like jigs, robots, +and their tooling, etc to execute all the operations needed for +an assembly is explained in section IV-E. Section IV-F tells +about the generation of the control code that moves the robots +to grasp, place and fasten parts in a cell. +In section V, we test our framework on different assemblies +and discuss the results. In section VI, we review our findings +from the experiments and future work. +IV. FRAMEWORK +A. System architecture +Auto-Assembly framework can be divided into two parts as +shown in Fig. 2. The first part, called ”Artefacts generation”, +works with CAD files provided by a design engineer. It is +intended to run once on the input data and provide artefacts, +which can then be stored and re-used to run the assembly +process in the simulated and physical environments. This +part includes Assembly Sequence Generation, Tool and Cell +Matching, Bill-of-Process (BOP) Generation and Control Code +generation. +The second part can be seen as a deployed environment. +It is represented as a system where we have many services, +providing “abilities” which can be called from the domain- +specific Process language (PL). An example of the PL script +can be seen in a Listing 3. Here we describe the most important +services and their respective abilities: +• Robot Controller +– Abilities to control the robots on a low level. As +input, it takes a trajectory as a list of a robot’s joint +states, and as output, interpolates the trajectory and +moves the robot. +– Abilities to control tooling connected to the robot, +like grippers, screwdrivers, etc. +• Motion Planner +– Ability to plan a trajectory in the cell to move a +robot to a target pose with cell objects taken as the +collisions. +• Jig Controller +– Ability to return a pose of a part in a jig with respect +to the jig origin. +• Assembly Service +– Ability to retrieve the information about fasteners +and resulting parts’ pose with respect to the cell +origin. +• Transform Service +– Ability to get the position of any object inside a cell +with respect to any object in the cell. +• 3D Simulator +– Abilities to load objects from cell description and +visualize cell state. +• Database and Message Bus +– Abilities to publish and retrieve JSON objects. This +component is used as a message bus and data storage. +All system parts exchange the data in a special format +called Factory Control Model (FCM). It can be considered +as a schema and also is a vital part of our system since it lets +all the components speak the same language. +B. CAD Data Preparation and Extraction +For any given assembly, our framework needs two design +files. +• Design file containing part assembly with joints. Fasten- +ers are labelled as separate joints in order to distinguish +them from other parts. +• Design file containing the jigs and gripper at different +stages of assembly like grasping, placing, etc. +The examples of these files for an assembly are depicted in +Figs 6 and 7 and are created by us in Fusion 360. Our method +is CAD-software agnostic as long as we can extract the CAD +data using an API. +From the design file in Fig. 6, we extract the joints and part +occurrences information using Fusion 360 API [16]. Using +this data, a joint register is created that maps every joint to its +parts. The joint register follows the FCM schema. + +Fig. 2: System architecture +Fig. 3: Listing of PL-code implementing high-level of robot +control. Abilities get cell state and plan trajectory are imple- +mented by Motion Planner and execute trajectory by Robot +Controller +From the design file in Fig. 7, we extract the pose of gripper +occurrence relative to the part during grasping it from the jig +and placing it at the assembly state using the Fusion 360 API. +We call this data as recipes. +C. Assembly Sequence Generation +A CAD design contains a lot of important information about +the part’s geometries, relations, and absolute poses. But what +it lacks - the right assembling order - is the key information +to move towards the assembled product. Assembly sequence +encodes the order of operations needed to be performed on +parts by the robotic cell. Although the operations can be +executed sequentially, assembly sequences are represented by +polytree (directed acyclic graph whose underlying undirected +graph is a tree). Not any such tree represents a valid and +feasible assembly sequence: +• only directly joined parts should be neighbours; +• the order of operations should take into account the +geometrical limitations; +• the number of generated assembly sequences should be +reasonably limited. Naturally, it grows exponentially with +the number of parts involved. This makes it hard to check +all the generated sequences to pick the best one according +to some criteria. +The assembly sequence generation step aims to solve all three +aforementioned issues, providing a limited number of valid +sequences. The whole process can be divided into three steps: +• a liaison graph generation; +• assembly sequences generation based on the obtained +liaison graph; + +Artefacts generation +CAD file: +parts geometries +Assembly +- joints +Sequence +- labelling +Assembly +CAD Data +BOP +Control Code +Sequences +Extraction +generation +Generation +CAD file: +Generation +- tooling placing + 3D-model files +absolute +BOP +- jigs placing +positions and +occurrencesof +parts +Tool and +- joints +- fastening +Cell +features +Matching +recipes +Artefacts set +Objects +FCM +PL script +objects +geometries +Deploy +graph +Physical +PL script +Parts +FCM objects +Best Run +environment +geometries +graph +Selection +Unsuccessful simulation runloop +deploy +Succesful run +Databaseand +artefacts set +messagebus +Services +Simulation +Robot +Motion +Assembly +Transform +3D +controller +planner +Controller +Service +Service +Simulator1-move_robot_to_position(cell, motion_group, manipulator_service, +2 +planning_object,position,move_type, +3 - +ignored_collisions,ignored_collision_pairs)( +4 - +rules ( +5 +~ get_cell_state(cell = cell, out cell_state = cell_state) +6 - +~plan_trajectory( +7 +position= position, +8 +motion_group=motion_group +9 +planning_object= planning_object +10 +planning_socket_name="eef", +11 +move_type=move_type, +12 +ignoredcollision pairs = ignored collisionpairs +13 +ignored_collisions = ignored_collisions, +14 +cell_state = cell_state, +15 +out result_motion_plan= trajectory +16 +) +17 +execute_trajectory(trajectory = trajectory) +18 +} seq +19 - +constraints ( +20 +execute_trajectory.@provider_id.resource_id== @manipulator_service.id +21 +22• geometry feasibility checking based on parts geometries +This approach we used is described in [5]. Further, the high- +level steps, important implementation details, and differences +with the original paper are described. +1) Liaison graph generation: The CAD file consists of the +individual parts combined together with joints and fasteners. +The information about the joints is crucial to accurately +determine parts connectivity. Considering the parts as liaison +graph nodes, connectivity information transfers into edges in +this graph. We extracted the information about joints and +fasteners from the design in CAD software to build up a liaison +graph to further analyze it and generate assembly sequences. +2) Sequences generation: An assembly sequence deter- +mines the order of operations on parts. The liaison graph +itself, being undirected, doesn’t set the order of operations +in general. But the order should be based on the liaison graph +since the latter contains the information about the connectivity +in the resulting assembly. Usually, there are many sequences of +operations. [5] describes the approach of extracting all possible +assembly sequences from the liaison graph. We followed the +suggested approach. +3) Geometry feasibility checking: The geometrical feasibil- +ity of an assembly process is the fundamental property, which +should be checked first to eliminate irrelevant sequences. +These irrelevant sequences could contain, for example, one +part to be joined with another part, which is trapped already +inside the sub-assembly. To prevent this, geometrical analysis +of sub-assemblies is used. One sub-assembly is translated step- +by-step w.r.t another sub-assembly in one of chosen directions +until the bounding boxes of the sub-assemblies still intersect +and the solid bodies’ intersection is checked. If the intersection +represents a volume, it’s impossible to join the sub-assemblies +in the chosen direction, and the remaining directions should +be checked. +Choosing the directions of translations alongside step size +is important for the result. Due to the nature of assembly +parts and their orientation alignment, directions along the +main coordinate axes work well in the tested assemblies. In +other cases, information from joints from the CAD file could +be used to determine the potential directions. Step size is +computed based on the minimal size of the part across both +sub-assemblies. Precisely, the step size is computed as a 0.75 +ratio of the diagonal of the smallest bounding box part. The +idea behind this value is to exclude the possibility of going +completely through the smallest part with a single translation +step. +D. Tooling Matching +The assembly sequence in itself doesn’t require specific +tooling models, but this is information is required for the +next steps in the assembling process. Given a graph of the +assembly sequence from the previous section, we traverse +this graph, considering the type of operation and parts used, +assigning all the tooling models and adding recipes to process +this operation. +To archive this, we extract the following information from +the CAD files: +• For grippers: +– Model of the part gripper can be applied. +– List of positions for grasping the part, calculated with +respect to the part origin. We use the information +from “joints”, such as JointAxis, to extract the vec- +tor of connection. Based on this vector poses are +calculated. +– States of digital inputs register to control the gripper. +• For jigs: +– Model of the part jig can hold. +– Position of a part in a jig. +• For screwdrivers: +– Screw-picking requirements, such as type of screw- +holder. +We store this data in Tooling Database. In our approach +Tooling Database is a storage with an API which allows +adding, matching and visualizing of the tooling. +By analyzing the dataset of the tooling used in physical +world production in the automotive field, we concluded that +the same information is stored in the tooling design files and +propagated to the tooling integration in the physical cells. We +decided to formalize the requirements and then store this data. +For the cases where it can’t be calculated from the design files, +we can manually put that information into the tooling database. +E. Cell matching +Cell description includes all the information representing +an assembly cell. Cell description is used to deploy both +environments (virtual and physical) and to choose a cell to +execute Assembly Sequence. We topologically sort a graph of +the Assembly Sequence and assign a level for every operation. +The level is required to assign resources for the parallel +operations when we should use different resources of the +same model. Then by traversing each operation, we check the +resources’ models required for this operation and find their +representation in the cell. If there are no cells satisfying all +the resource requirements for the Assembly Sequence, we fail, +providing feedback with the exact operation and the resources +model we were not able to assign. As a result of the execution +of the described algorithm, we have an assembly sequence to +be converted into a BOP. +F. Control code generation +For each operation in the BOP, we match a specific PL- +script, which is self-containing to perform this type of oper- +ation, and pass the operation, its resources and parts as the +parameters, creating one PL-script, to assemble a product. +An example of PL-script implementing unload operation is +presented on the Listing 4 +V. EXPERIMENTS AND RESULTS +The objectives of our experiments are: +• To evaluate the framework. + +Fig. 4: Listing of PL-code for unload operation +• To evaluate the assembly BOPs in the physical environ- +ment to provide metrics and feedback on the assembly. +The assembly we chose to test is shown in Fig. 6 and its +tooling, and jig design are shown in Fig.7. +• Data Preparation: +– we extract the joint register and recipes as mentioned +in section IV-B. +– taking the joint register and part models files, as- +sembly sequence generator produced 8 assembly +sequences for this assembly. These sequences are +mentioned in Fig. 5. +– we enrich the assembly sequences using tool match- +ing mentioned in section IV-D. +– we convert the enriched assembly sequences to BOP +using cell matching as mentioned in section IV-E. +Fig. 5: A diagram of assembly sequences generation process +for the design used in the experiments. Square blocks represent +parts while circles represent (sub-)assemblies(D and E). The +possible sequences are Left: ABDC, BADC, CABD, CBAD +and Right: BCEA, CBEA, ABCE, ACBE. +• Scene Preparation: Before starting the assembly, if it’s +a simulated environment, the jigs are unloaded at the +same poses as in the physical world in Fig. 1. If it’s +the physical environment, the jigs and parts are placed in +their respective poses. +• Simulation +Fig. 6: A simple assembly containing 3 parts. Profiles: A, C +and connector: B +Fig. 7: Assembly Design file showing the assembly jig, +custom-designed gripper adapters, grasping, and insertion +states of the gripper. Assembly state: Center of the table. Jig +state: Top and bottom right of the table. +1) Start simulation deployment with services and a +database and message bus instance as mentioned in +Fig. 2. +2) We trigger execution of the PL code, which starts +from running operations of type ”unload” on input +parts. This operation effectively initializes part in- +stances on respective positions in the input jigs, so +that the parts are now represented in the digital twin +of the cell, as active objects with poses, visible for +the simulator as well as for the motion planner. +3) The rest of the PL code is executed, sequentially +reading necessary gripper positions, planning and +executing trajectories, and triggering gripper control +programs for grasping/releasing/fastening, all by +calling respective PL functions that use abilities of +underlying systems described in IV-A. +4) The user can observe the execution of assembly +in the 3D simulator. During the execution of the +assembly process, the motion planner gives us direct +feedback, on whether it can reach a certain pose in +the assembly or not. +For our example assembly, the initial results were the +following: +– Of all the possible 8 assembly sequences, only one +assembly sequence (ABDC) passed through the cell +matching, as the cell resource descriptions (in this +case jigs) support this. Many sequences get filtered + +1- unload_operation(operation, cell, jig, out part_instance_id)( +2- rules { +3 ~ +~ read_fcm#find_part( +4 +object_id = @operation, +5 +level = 2, +6 +format ="array" +7 +object_filters=[{"path":"type","operation":"EQ","value":"part"}] +8 +link_filters=[{"path":"type","operation":"EQ","value":"uses"}], +9 +out id = part) +10 - + get_part_pose_wrt_jig( +11 +jig_id=@jig.id, +12 +absoccurrenceid=@part.abs_occurrence_object, +13 +out pose_from_jig) +14 - +~ unload_part( +15 +part_object = @part, +16 +cell_object = @cell, +17 +pose = @pose_from_jig, +18 +out part_instance_id) +19 + seq +20 +;assembly +assembly +C +D +A +E +1 +I +- +A +B +B +cC +AQ0060000600000000000 +0000000060000000000 +Q0000000000000000 +e00000000000000 +Q00000000 +00000008 +9000000000000 +000000 +00000000000 +000000000 +0000000 +00000 +00000000 +QC0000000000000 +0000000000 +0000000 +0000000 +QDO0000000008based on the cell resources (jigs, robot tooling, etc.). +All the sequences for any given assembly are fea- +sible, if the cell has the resources to hold sub- +assemblies, For example, the assembly sequence +CABD is possible when the cell has the jig that +supports moving part C first to the assembly pose, +then creating a sub-assembly D by moving parts A +and B in the same order. Now the question here is, +how to make the decision on which resources in this +case jigs are needed to be designed to hold the sub- +assemblies. If there are cycles in the graph, multiple +BOPs pass through the cell matching which needs +the same cell resources, which enables us to simulate +and select the best one based on the metrics. +– We also noticed that our initial design failed due to +fastening robot reachability, we took this feedback +from the framework and changed the fastening posi- +tion in the assembly to assemble a product. +To adjust the design, we changed the positions of the +screws in the assembly to other holes without losing the +structure stability. After this design adjustment, we were +able to successfully simulate the one feasible assembly +sequence. +This is one of the main features of the proposed frame- +work - to get this kind of feedback about the prod- +uct/tooling/cell design compatibility as soon as possible +with minimal manual input. +• Running assembly on physical robotic cell: Once we +find an assembly sequence that passes in the simulation, +we can proceed to the physical assembly process. +This is achieved by running the same generated PL code +as before but now in a physical robotic environment. The +only thing that differs compared to the previous pipeline +in simulation is the first step - deployment of the systems. +For a physical assembly we deploy the robot and the +gripper drivers to be connected to robots and devices, +such that in parallel with updating the state of the digital +twin, these controllers will be changing the states of the +tools in the physical world, such as robots moving along +precomputed trajectories, gripper opening/closing and the +screwdriver fastening the screws. +We evaluated this assumption on the physical robotic cell +with two collaborative robots the layout of which can be +seen in Fig. 1. The results of these experiments are two +folds: +1) On one side, we can see that as soon as the +digital twin is accurate enough, all computed gripper +positions allow performing most of the operations, +such as picking a screw, grasping and releasing a +part, and in some cases to fasten a screw. +2) On another side, some operations show that an ac- +cumulated tolerance stack of robot calibration, tool +accuracy, and parts accuracy leads to the inability +to perform the joint operation such as fastening +successfully, and the screwing position requires cor- +rection. +The example of running assembly in the virtual and phys- +ical environments can be seen in Fig. 8. The process of +assembly of the provided CAD by running the generated +PL code can be seen in the accompanying video. +Fig. 8: Assembly process. Left: Physical environment and +Right: Virtual environment. +VI. CONCLUSION AND FUTURE SCOPE +In this paper, we implemented and tested a framework to +run a robotic assembly of a product by using only CAD +files as input. We were able to use the feedback provided +by the framework to change the original design and achieve +successful assembly. We re-iterated the whole pipeline and +transferred the assembly from the virtual to the physical +world. We conclude that this transfer can be done only if the +digital twin matches the physical cell precisely, which requires +additional work, such as robots and cell calibration, but it’s out +of the scope of this paper. The choice of design of our system +proved its flexibility since we were able to analyze and change +artefacts produced during the different steps of the execution. +The system is general enough to support new products and +cell configurations. +The next step is to validate our framework on more complex +assemblies, including new types of operations and operations +which involve more than two parts. +In our experiment, we relied only on the parts’ dimensional +precision and the accuracy of the robots. While it could work +for some parts, and partially worked in our case, it would +likely fail on many other parts and materials. To address this +problem, computer vision and other perception methods should +be introduced into the framework to deal with variations in the +real assembly process. +In the section IV-C3 the constraint we chose could lead to +some possible assembly sequences being rejected. To solve +this issue, we plan to implement a geometrical feasibility +check based on joints from the CAD files or other optimization +algorithms. +The method used in the section IV-E can lead to a sub- +optimal configuration or even to setups where some robots +can’t reach parts. This approach was chosen as the easiest to +track and implement. In future works, we plan to implement a +more sophisticated scheduling algorithm based on geometrical +and utilization constraints. + +REFERENCES +[1] Henrioud, J. M., and A. 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IEEE, +2006. +[10] Rust, Romana, et al. ”COMPAS FAB: Robotic fabrication package for +the COMPAS Framework” Gramazio Kohler Research, ETH Zurich. +https://github.com/compas-dev/compas fab (2018) +[11] Sierla, Seppo, et al. ”Automatic assembly planning based on digital +product descriptions.” Computers in Industry 97 (2018): 34-46 +[12] Drath, Rainer, ed. AutomationML: the industrial cookbook. Walter de +Gruyter GmbH & Co KG, 2021. +[13] Beck, Joshua, Alexander Neb, and Katharina Barbu. ”Towards a CAD- +based Automated Robot Offline-Programming Approach for Disassem- +bly.” Procedia CIRP 104 (2021): 1280-1285. +[14] Octopuz® https://octopuz.com +[15] Huang, Yijiang. Automated motion planning for robotic assembly of +discrete architectural structures. Diss. Massachusetts Institute of Tech- +nology, 2018. +[16] Autodesk® Fusion 360® https://www.autodesk.com/ + diff --git a/_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/load_file.txt b/_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7edd2bb7c5588d86b817a26d2df69594330ecb47 --- /dev/null +++ b/_NE0T4oBgHgl3EQfxQFJ/content/tmp_files/load_file.txt @@ -0,0 +1,376 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf,len=375 +page_content='Auto-Assembly: a framework for automated robotic assembly directly from CAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Fedor Chervinskii§, Sergei Zobov§, Aleksandr Rybnikov§, Danil Petrov§, Komal Vendidandi§ Λ Γ Γ I V Λ L Abstract—In this work, we propose a framework called Auto- Assembly for automated robotic assembly from design files and demonstrate a practical implementation on modular parts joined by fastening using a robotic cell consisting of two robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We show the flexibility of the approach by testing it on different input designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Auto-Assembly consists of several parts: design analysis, assembly sequence generation, bill-of-process (BOP) generation, conversion of the BOP to control code, path planning, simulation, and execution of the control code to assemble parts in the physical environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Index Terms—industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='0, smart manufacturing, cyber- physical systems, smart factory, manufacturing automation, ma- nipulators, cellular manufacturing, digital twins, robotic assem- bly I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' INTRODUCTION Assembly planning is one of the most laborious tasks when releasing a new product for manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Thus, many algorithms and methods around computer-aided design (CAD) and digital twins of the factories have emerged in recent years that help process engineers to prepare a new design for assembly (Computer-aided Assembly Process Planning techniques [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' An emerging trend of Industry 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='0 [2] suggests that a digital, highly automated factory should be able to infer the process from the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In practice, even for an automated factory, assembly planning has to be followed by an offline- programming of all the robots and devices to perform the assembly plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Additive manufacturing technology (3D printing [3]) at the same time has achieved a much higher rate of process design automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' One can simply load a CAD file into a machine that will yield a part of the desired design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The main question we are trying to address in this paper is ”Could a robotic cell or even the whole factory work just as a 3D printer?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' When loaded with target assembly CAD design and given input parts in specified conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' placed in special input jigs) - would it perform the required assembly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In this work, we show how this can be achieved under specific constraints, paving the road for future experiments towards a more general approach and wider applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' However, the framework we propose is general enough to accommodate more complex designs and conditions, like many types of tooling and different joining technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' §Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' [chervinskii, zobov, rybnikov, danil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='petrov, vendidandi]@arrival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='com Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 1: Experimental setup: robotic cell with two UR5e manip- ulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Left: UR5e with a screwdriver Likratec EH2 R1030- A and Right: UR5e with gripper Robotiq 2F-85 with custom designed gripper clamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' On the table: custom-designed 3D- printed jigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' RELATED WORK AND BACKGROUND An Assembly Planning for a given design typically starts from identifying the mating features or joints and suggesting a feasible Assembly Sequence, which could be automated as seen in [4], [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To proceed to the process planning, a virtual environment, also known as Digital Twin [7] is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' There are at- tempts to develop a common ontology, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' [8], [9] and unify interfaces between systems [10] to support process design automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Sierla, Seppo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' [11] discuss the conceptual framework of automated assembly planning using a digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' It uses the XML-based AutomationML [12] data modeling frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This framework aggregates different data exchange formats like CAEX for plant description, COLLADA for geometry and kinematics of 3D models, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' There is still not sufficient work in joining together process planning, motion planning and execution using a common framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In [13] authors used artificial intelligence to solve a Tooling Matching problem and developed an add-on for Octopuz [14] to do a Motion Planning and Robot Program Generation for disassembly, but not testing in physical cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In another work, [15] a similar pipeline is described for an arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='02643v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='RO] 6 Jan 2023 Screwdriver Gripper Assembly Assembly Jig Parts Unload Jigs Screws Unload Jigarchitectural domain, mainly focusing on parametric design and modular assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We claim that Auto-Assembly is the first proposed frame- work that can generate and execute robotic assembly process for an arbitrary input CAD design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' PROBLEM STATEMENT AND METHOD OVERVIEW The main objective of our work is to create a framework that enables a closed loop between design and robotic manu- facturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' A target framework should analyse the design and provide a simulation of assembly, executable programs (when possible) and other feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The primary aim of the feedback is to help in adapting the design and manufacturing to better correspond to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The feedback we should provide can be split into two categories: Successful simulation and its’ artefacts can be directly used to decide on physical manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Users can choose between different processes to choose the one, based on the key performance indicators (KPI) they want to optimize: time, tooling price, energy consumption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In case of a failure, the system should provide all neces- sary feedback that helps to change the design, robot’s position, choose the robots with better parameters or different cell configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Such feedback can be: failed operations, missing appropriate tooling, parts or tools in collision, unreachable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To achieve this, we implement a framework described in detail in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Section IV-A gives an overview of our system and its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Section IV-B discusses 3D modelling of the assembly design files that form the base of our data extraction pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Section IV-C reviews the usage of this extracted data to produce a set of possible assembly sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Each operation in the assembly sequence is enriched with tooling information as discussed in section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Finding a specific cell that contains all the resources like jigs, robots, and their tooling, etc to execute all the operations needed for an assembly is explained in section IV-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Section IV-F tells about the generation of the control code that moves the robots to grasp, place and fasten parts in a cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In section V, we test our framework on different assemblies and discuss the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In section VI, we review our findings from the experiments and future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' FRAMEWORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' System architecture Auto-Assembly framework can be divided into two parts as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The first part, called ”Artefacts generation”, works with CAD files provided by a design engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' It is intended to run once on the input data and provide artefacts, which can then be stored and re-used to run the assembly process in the simulated and physical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This part includes Assembly Sequence Generation, Tool and Cell Matching, Bill-of-Process (BOP) Generation and Control Code generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The second part can be seen as a deployed environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' It is represented as a system where we have many services, providing “abilities” which can be called from the domain- specific Process language (PL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' An example of the PL script can be seen in a Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Here we describe the most important services and their respective abilities: Robot Controller – Abilities to control the robots on a low level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' As input, it takes a trajectory as a list of a robot’s joint states, and as output, interpolates the trajectory and moves the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – Abilities to control tooling connected to the robot, like grippers, screwdrivers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Motion Planner – Ability to plan a trajectory in the cell to move a robot to a target pose with cell objects taken as the collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Jig Controller – Ability to return a pose of a part in a jig with respect to the jig origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Assembly Service – Ability to retrieve the information about fasteners and resulting parts’ pose with respect to the cell origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Transform Service – Ability to get the position of any object inside a cell with respect to any object in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 3D Simulator – Abilities to load objects from cell description and visualize cell state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Database and Message Bus – Abilities to publish and retrieve JSON objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This component is used as a message bus and data storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' All system parts exchange the data in a special format called Factory Control Model (FCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' It can be considered as a schema and also is a vital part of our system since it lets all the components speak the same language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' CAD Data Preparation and Extraction For any given assembly, our framework needs two design files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Design file containing part assembly with joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Fasten- ers are labelled as separate joints in order to distinguish them from other parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Design file containing the jigs and gripper at different stages of assembly like grasping, placing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The examples of these files for an assembly are depicted in Figs 6 and 7 and are created by us in Fusion 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Our method is CAD-software agnostic as long as we can extract the CAD data using an API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' From the design file in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 6, we extract the joints and part occurrences information using Fusion 360 API [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Using this data, a joint register is created that maps every joint to its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The joint register follows the FCM schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2: System architecture Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 3: Listing of PL-code implementing high-level of robot control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Abilities get cell state and plan trajectory are imple- mented by Motion Planner and execute trajectory by Robot Controller From the design file in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 7, we extract the pose of gripper occurrence relative to the part during grasping it from the jig and placing it at the assembly state using the Fusion 360 API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We call this data as recipes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Assembly Sequence Generation A CAD design contains a lot of important information about the part’s geometries, relations, and absolute poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' But what it lacks - the right assembling order - is the key information to move towards the assembled product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Assembly sequence encodes the order of operations needed to be performed on parts by the robotic cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Although the operations can be executed sequentially, assembly sequences are represented by polytree (directed acyclic graph whose underlying undirected graph is a tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Not any such tree represents a valid and feasible assembly sequence: only directly joined parts should be neighbours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' the order of operations should take into account the geometrical limitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' the number of generated assembly sequences should be reasonably limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Naturally, it grows exponentially with the number of parts involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This makes it hard to check all the generated sequences to pick the best one according to some criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The assembly sequence generation step aims to solve all three aforementioned issues, providing a limited number of valid sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The whole process can be divided into three steps: a liaison graph generation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' assembly sequences generation based on the obtained liaison graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Artefacts generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='CAD file: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='parts geometries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Assembly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='joints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Sequence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='labelling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Assembly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='CAD Data ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='BOP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Control Code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Sequences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Extraction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='CAD file: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='tooling placing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='3D-model files ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='absolute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='BOP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='jigs placing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='positions and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='occurrencesof ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='parts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Tool and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='joints ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='fastening ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Cell ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='recipes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Artefacts set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='FCM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='PL script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='geometries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Deploy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Physical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='PL script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Parts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='FCM objects ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Best Run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='environment ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='geometries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='graph ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Unsuccessful simulation runloop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='deploy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Succesful run ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Databaseand ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='artefacts set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='messagebus ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Services ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Simulation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Robot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Motion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Assembly ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Transform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='3D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='planner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Controller ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Service ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='Simulator1-move_robot_to_position(cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' motion_group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' manipulator_service,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2 planning_object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='move_type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 3 - ignored_collisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='ignored_collision_pairs)( 4 - rules ( 5 ~ get_cell_state(cell = cell,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' out cell_state = cell_state) 6 - ~plan_trajectory( 7 position= position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 8 motion_group=motion_group 9 planning_object= planning_object 10 planning_socket_name="eef",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 11 move_type=move_type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 12 ignoredcollision pairs = ignored collisionpairs 13 ignored_collisions = ignored_collisions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 14 cell_state = cell_state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 15 out result_motion_plan= trajectory 16 ) 17 execute_trajectory(trajectory = trajectory) 18 } seq 19 - constraints ( 20 execute_trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' @provider_id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='resource_id== @manipulator_service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='id 21 22• geometry feasibility checking based on parts geometries This approach we used is described in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Further, the high- level steps, important implementation details, and differences with the original paper are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 1) Liaison graph generation: The CAD file consists of the individual parts combined together with joints and fasteners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The information about the joints is crucial to accurately determine parts connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Considering the parts as liaison graph nodes, connectivity information transfers into edges in this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We extracted the information about joints and fasteners from the design in CAD software to build up a liaison graph to further analyze it and generate assembly sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2) Sequences generation: An assembly sequence deter- mines the order of operations on parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The liaison graph itself, being undirected, doesn’t set the order of operations in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' But the order should be based on the liaison graph since the latter contains the information about the connectivity in the resulting assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Usually, there are many sequences of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' [5] describes the approach of extracting all possible assembly sequences from the liaison graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We followed the suggested approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 3) Geometry feasibility checking: The geometrical feasibil- ity of an assembly process is the fundamental property, which should be checked first to eliminate irrelevant sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' These irrelevant sequences could contain, for example, one part to be joined with another part, which is trapped already inside the sub-assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To prevent this, geometrical analysis of sub-assemblies is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' One sub-assembly is translated step- by-step w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='t another sub-assembly in one of chosen directions until the bounding boxes of the sub-assemblies still intersect and the solid bodies’ intersection is checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' If the intersection represents a volume, it’s impossible to join the sub-assemblies in the chosen direction, and the remaining directions should be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Choosing the directions of translations alongside step size is important for the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Due to the nature of assembly parts and their orientation alignment, directions along the main coordinate axes work well in the tested assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In other cases, information from joints from the CAD file could be used to determine the potential directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Step size is computed based on the minimal size of the part across both sub-assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Precisely, the step size is computed as a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='75 ratio of the diagonal of the smallest bounding box part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The idea behind this value is to exclude the possibility of going completely through the smallest part with a single translation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Tooling Matching The assembly sequence in itself doesn’t require specific tooling models, but this is information is required for the next steps in the assembling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Given a graph of the assembly sequence from the previous section, we traverse this graph, considering the type of operation and parts used, assigning all the tooling models and adding recipes to process this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To archive this, we extract the following information from the CAD files: For grippers: – Model of the part gripper can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – List of positions for grasping the part, calculated with respect to the part origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We use the information from “joints”, such as JointAxis, to extract the vec- tor of connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Based on this vector poses are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – States of digital inputs register to control the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' For jigs: – Model of the part jig can hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – Position of a part in a jig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' For screwdrivers: – Screw-picking requirements, such as type of screw- holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We store this data in Tooling Database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In our approach Tooling Database is a storage with an API which allows adding, matching and visualizing of the tooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' By analyzing the dataset of the tooling used in physical world production in the automotive field, we concluded that the same information is stored in the tooling design files and propagated to the tooling integration in the physical cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We decided to formalize the requirements and then store this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' For the cases where it can’t be calculated from the design files, we can manually put that information into the tooling database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Cell matching Cell description includes all the information representing an assembly cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Cell description is used to deploy both environments (virtual and physical) and to choose a cell to execute Assembly Sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We topologically sort a graph of the Assembly Sequence and assign a level for every operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The level is required to assign resources for the parallel operations when we should use different resources of the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Then by traversing each operation, we check the resources’ models required for this operation and find their representation in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' If there are no cells satisfying all the resource requirements for the Assembly Sequence, we fail, providing feedback with the exact operation and the resources model we were not able to assign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' As a result of the execution of the described algorithm, we have an assembly sequence to be converted into a BOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Control code generation For each operation in the BOP, we match a specific PL- script, which is self-containing to perform this type of oper- ation, and pass the operation, its resources and parts as the parameters, creating one PL-script, to assemble a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' An example of PL-script implementing unload operation is presented on the Listing 4 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' EXPERIMENTS AND RESULTS The objectives of our experiments are: To evaluate the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 4: Listing of PL-code for unload operation To evaluate the assembly BOPs in the physical environ- ment to provide metrics and feedback on the assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The assembly we chose to test is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 6 and its tooling, and jig design are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Data Preparation: – we extract the joint register and recipes as mentioned in section IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – taking the joint register and part models files, as- sembly sequence generator produced 8 assembly sequences for this assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' These sequences are mentioned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – we enrich the assembly sequences using tool match- ing mentioned in section IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – we convert the enriched assembly sequences to BOP using cell matching as mentioned in section IV-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 5: A diagram of assembly sequences generation process for the design used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Square blocks represent parts while circles represent (sub-)assemblies(D and E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The possible sequences are Left: ABDC, BADC, CABD, CBAD and Right: BCEA, CBEA, ABCE, ACBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Scene Preparation: Before starting the assembly, if it’s a simulated environment, the jigs are unloaded at the same poses as in the physical world in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' If it’s the physical environment, the jigs and parts are placed in their respective poses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Simulation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 6: A simple assembly containing 3 parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Profiles: A, C and connector: B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 7: Assembly Design file showing the assembly jig, custom-designed gripper adapters, grasping, and insertion states of the gripper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Assembly state: Center of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Jig state: Top and bottom right of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 1) Start simulation deployment with services and a database and message bus instance as mentioned in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2) We trigger execution of the PL code, which starts from running operations of type ”unload” on input parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This operation effectively initializes part in- stances on respective positions in the input jigs, so that the parts are now represented in the digital twin of the cell, as active objects with poses, visible for the simulator as well as for the motion planner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 3) The rest of the PL code is executed, sequentially reading necessary gripper positions, planning and executing trajectories, and triggering gripper control programs for grasping/releasing/fastening, all by calling respective PL functions that use abilities of underlying systems described in IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 4) The user can observe the execution of assembly in the 3D simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' During the execution of the assembly process, the motion planner gives us direct feedback, on whether it can reach a certain pose in the assembly or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' For our example assembly, the initial results were the following: – Of all the possible 8 assembly sequences, only one assembly sequence (ABDC) passed through the cell matching, as the cell resource descriptions (in this case jigs) support this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Many sequences get filtered 1- unload_operation(operation, cell, jig, out part_instance_id)( 2- rules { 3 ~ ~ read_fcm#find_part( 4 object_id = @operation, 5 level = 2, 6 format ="array" 7 object_filters=[{"path":"type","operation":"EQ","value":"part"}] 8 link_filters=[{"path":"type","operation":"EQ","value":"uses"}], 9 out id = part) 10 - get_part_pose_wrt_jig( 11 jig_id=@jig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='id, 12 absoccurrenceid=@part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='abs_occurrence_object, 13 out pose_from_jig) 14 - ~ unload_part( 15 part_object = @part, 16 cell_object = @cell, 17 pose = @pose_from_jig, 18 out part_instance_id) 19 seq 20 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content='assembly assembly C D A E 1 I A B B cC AQ0060000600000000000 0000000060000000000 Q0000000000000000 e00000000000000 Q00000000 00000008 9000000000000 000000 00000000000 000000000 0000000 00000 00000000 QC0000000000000 0000000000 0000000 0000000 QDO0000000008based on the cell resources (jigs, robot tooling, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' All the sequences for any given assembly are fea- sible, if the cell has the resources to hold sub- assemblies, For example, the assembly sequence CABD is possible when the cell has the jig that supports moving part C first to the assembly pose, then creating a sub-assembly D by moving parts A and B in the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Now the question here is, how to make the decision on which resources in this case jigs are needed to be designed to hold the sub- assemblies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' If there are cycles in the graph, multiple BOPs pass through the cell matching which needs the same cell resources, which enables us to simulate and select the best one based on the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' – We also noticed that our initial design failed due to fastening robot reachability, we took this feedback from the framework and changed the fastening posi- tion in the assembly to assemble a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To adjust the design, we changed the positions of the screws in the assembly to other holes without losing the structure stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' After this design adjustment, we were able to successfully simulate the one feasible assembly sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This is one of the main features of the proposed frame- work - to get this kind of feedback about the prod- uct/tooling/cell design compatibility as soon as possible with minimal manual input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Running assembly on physical robotic cell: Once we find an assembly sequence that passes in the simulation, we can proceed to the physical assembly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This is achieved by running the same generated PL code as before but now in a physical robotic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The only thing that differs compared to the previous pipeline in simulation is the first step - deployment of the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' For a physical assembly we deploy the robot and the gripper drivers to be connected to robots and devices, such that in parallel with updating the state of the digital twin, these controllers will be changing the states of the tools in the physical world, such as robots moving along precomputed trajectories, gripper opening/closing and the screwdriver fastening the screws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We evaluated this assumption on the physical robotic cell with two collaborative robots the layout of which can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The results of these experiments are two folds: 1) On one side, we can see that as soon as the digital twin is accurate enough, all computed gripper positions allow performing most of the operations, such as picking a screw, grasping and releasing a part, and in some cases to fasten a screw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 2) On another side, some operations show that an ac- cumulated tolerance stack of robot calibration, tool accuracy, and parts accuracy leads to the inability to perform the joint operation such as fastening successfully, and the screwing position requires cor- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The example of running assembly in the virtual and phys- ical environments can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The process of assembly of the provided CAD by running the generated PL code can be seen in the accompanying video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' 8: Assembly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' Left: Physical environment and Right: Virtual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' CONCLUSION AND FUTURE SCOPE In this paper, we implemented and tested a framework to run a robotic assembly of a product by using only CAD files as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We were able to use the feedback provided by the framework to change the original design and achieve successful assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We re-iterated the whole pipeline and transferred the assembly from the virtual to the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' We conclude that this transfer can be done only if the digital twin matches the physical cell precisely, which requires additional work, such as robots and cell calibration, but it’s out of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The choice of design of our system proved its flexibility since we were able to analyze and change artefacts produced during the different steps of the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The system is general enough to support new products and cell configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The next step is to validate our framework on more complex assemblies, including new types of operations and operations which involve more than two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In our experiment, we relied only on the parts’ dimensional precision and the accuracy of the robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' While it could work for some parts, and partially worked in our case, it would likely fail on many other parts and materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To address this problem, computer vision and other perception methods should be introduced into the framework to deal with variations in the real assembly process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' In the section IV-C3 the constraint we chose could lead to some possible assembly sequences being rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' To solve this issue, we plan to implement a geometrical feasibility check based on joints from the CAD files or other optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' The method used in the section IV-E can lead to a sub- optimal configuration or even to setups where some robots can’t reach parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NE0T4oBgHgl3EQfxQFJ/content/2301.02643v1.pdf'} +page_content=' This approach was 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b/_NE1T4oBgHgl3EQf8gXW/content/tmp_files/2301.03547v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da7bf8ffd22040e6dc815fff88c1adabbf7314c7 --- /dev/null +++ b/_NE1T4oBgHgl3EQf8gXW/content/tmp_files/2301.03547v1.pdf.txt @@ -0,0 +1,466 @@ +ON PERTURBATIONS RETAINING CONSERVATION +LAWS OF DIFFTRENTIAL EQUATIONS +ALEXEY SAMOKHIN +Abstract. The paper deals with perturbations of the equation +that have a number of conservation laws. When a small term is +added to the equation its conserved quantities usually decay at in- +dividual rates, a phenomenon known as a selective decay. These +rates are described by the simple law using the conservation laws’ +generating functions and the added term. Yet some perturbation +may retain a specific quantity(s), such as energy, momentum and +other physically important characteristics of solutions. We intro- +duce a procedure for finding such perturbations and demonstrate +it by examples including the KdV-Burgers equation and a system +from magnetodynamics. Some interesting properties of solutions +of such perturbed equations are revealed and discussed. +Keywords: conservation laws, perturbed equations, selective de- +cay, traveling waves. +MSC[2010]: 35Q53, 35B36. +1. Introduction +Many physical systems are modeled using equations that have a sig- +nificant number of conservation laws. Yet when an additional (usually +dissipative) term is added to the equation its conserved quantities decay +at individual rates, which are connected to their generating functions +[1]. The famous example is the KdV equation (it has infinitely many +conservation laws) and the KdV-Burgers equation (with additional, +with respect to KdV, dissipative term and only one conservation law). +To be precise let E(u) = 0 be a system of equations describing an +ideal (unperturbed) media state. A scalar H depending on u and its +derivatives is a conservation law if for ⟨H⟩, the integral of H over some +fixed spatial domain, ∂⟨H⟩ +∂t +���� +E += 0. +For the perturbed equation the quantity H is constant no more and +∂⟨H⟩ +∂t +̸= 0 is called the decay rate of H, cf. [2]. +A perturbed state usually satisfies the equation E(u) + LF(u) = 0, +where L is a small parameter diagonal matrix diag(λi); for L = 0 we +get the ideal state equation. The decay rate depends on the additional +term L F(u). The connection between decay rate and LF(u) was called +a ’balance law’ in [3]. +1 +arXiv:2301.03547v1 [nlin.PS] 9 Jan 2023 + +2 +ALEXEY SAMOKHIN +This law expresses ∂t⟨H⟩ in terms of scalar product of LF(u) and +the generating function g of the conserved quantity H, [1]: +∂⟨H⟩ +∂t += ⟨g · LF⟩ +(1) +Remarks +• The right-hand side of (1) is not unique: e.g, one can get a +different but equivalent form integrating by parts. +• In the case of the integrand in the right-hand side of (1) is null or +an exact form we get the situation when the conserved quantity +⟨H⟩ is conserved as well for the correspondent perturbed state. +• Let us restrict considerations to R[u], the ring of differential +polynoms of u. Then all perturbations F retaining the conser- +vation law with the generating function g must satisfy +g · LFdx1 . . . dxn ∈ Im(d), +where d : Λn−1 → Λn and Λk are k-forms of spatial variables. +Of course, the intersection of the principal ideal g · R[u] with +Im(d) is huge. +A considerable difference in decay rates leads to a simple method, +first discovered by Taylor, [4], for finding quasi-stationary states of +plasma which are of great practical importance. +He studied the model where the decay of energy E is monotonic but +those of momentum M and helicity are not necessarily so. Such an +inequality in decay rates leads to a distinct physical phenomenon of +’self–organization’ or quasi–stable states. +There exist a very simple procedure for finding solutions of the above +described behavior. It was suggested in [4], and is known as ’Taylor +trick’. The procedure is as follows. +Taking into consideration their comparative decay rates, minimize +E with M as constrain. Put δ(E + λM = 0), M and presumed con- +stant, λ being Lagrange multiplier. This Euler–Lagrange equation is +not necessarily compatible with the initial equation but nevertheless it +gives a way for good approximations of self-organization phenomena. +There is a considerable number of publication in the field, see a recent +paper [5] for recent developments. +Another application of selective decay is given in [6]. The problem +is the behavior of the soliton which, while moving in non-dissipative +and dispersion-constant medium encounters a finite-width barrier with +varying dissipation and/or dispersion; beyond the layer dispersion is +constant (but not necessarily of the same value) and dissipation is null. +The transmitted wave either retains the form of a soliton (though of +different parameters) or scatters a into a number of them. Using the +relative decay of the KdV conserved quantities a simple algorithm to +predict the number and amplitudes of resulting solitons was obtained. + +ON PERTURBATIONS RETAINING CONSERVATION LAWS +3 +In [7] the selective decay approach was applied to some well-known +equations of mathematical physics (KdV and KdV-Burgers equation, +BBM and its dissipative generalization, two-dimensional generalized +shallow water wave equation). It have showed that the Taylor trick +extremals are associated with first-order PDEs and travelling wave so- +lutions. +In this paper we search, for some popular equations, their low-order +perturbations which retain a chosen conservation law (in a sense that +the perturbed equation has the same conserved quantity as initial one). +Examples include KdV and its conserved energy or momentum and the +Kadomtsev-Pogutse system of equation from magnetohydrodynamics +with its three known conserved quantities. Some interesting properties +of solutions of such perturbed equations are revealed and discussed. +2. KdV and KdV-Burgers +The generalized KdV equation (KdV-Burgers equation) considered +here is of the form +ut = 2uux + uxxx + λuxx; +(2) +The classical KdV equation corresponds to λ = 0. +The first three conserved quantities for KdV are +m += +� +∞ +−∞ +u(x, t) dx — mass, +M += +� +∞ +−∞ +u2(x, t) dx — momentum, +E += +� +∞ +−∞ +� +2u3(x, t) − 3(ux(x, t))2� +dx — energy, +and there are infinite number of them. +The generating functions for the above conservation laws of the KdV +are, up to multiplication constants, 1, u and u2 + uxx correspondingly. +As for the equation (2), it has a form of a conservation law, ut = Fx, +the ”mass” +� +∞ +−∞ +u dx is a conserved quantity. For a soliton this mass +is equal to 12aγ. +But the impulse ⟨u2⟩ = +� +∞ +−∞ +u2 dx declines monotonically: +Mt = 1 +2⟨u2⟩t = ⟨uut⟩ = ⟨u(u2 + uxx + λux)x⟩ = 2 +3u3��+∞ +−∞ − u2 +x|+∞ +−∞ − λ⟨u2 +x⟩ += +(3) +By analogy, for the energy + +4 +ALEXEY SAMOKHIN +Et = ⟨ +� +2u3(x, t) − 3(ux(x, t))2� +⟩t = 6λ⟨uxx(u2 + uxx)⟩ +(4) +Thus the energy does not necessary declines. +2.1. Transformations of KdV that retain momentum. Now let +us find perturbations of the form F(u, ux, uxx) that retain momen- +tum. Accordingly to the remark 2 above, the differential form λu · +F(u, ux, uxx)dx must be exact. Thus +u · F(u, ux, uxx) = Dx(A(u, ux)) +(5) +for some A(u, ux). Here +Dx = ∂ +∂x + +∞ +� +n=0 +uxn+1 +∂ +∂uxn +is the operator of the full differentiation with respect to x. +Below we restrict the search to polynomials of u and its derivatives. +Then in (5) the polynomial Dx(A(u, ux)) is divisible by u, so A(u, ux) = +u2B(u, ux). +On the other hand +Dx(u2B(u, ux)) = 2uuxB(u, ux) + u2(ux +∂B +∂u + uxx +∂B +∂ux +). +Hence the second order retaining momentum perturbation is defined +by +F(u, ux, uxx) = 2uxB(u, ux) + u(ux +∂B +∂u + uxx +∂B +∂ux +) +for an arbitrary B. Note that F is linear in uxx. +For instance, if B = ux the λ transformation of the KdV equation +ut = 2uux + uxxx + λ(2u2 +x + uuxx) +(6) +retains ⟨u2⟩ as its conserved quantity. +Remark 1. This construction can be generalized. If g is the gen- +erating function for some conserved quantity Cl of an one-spational +equation E, then F = g−1Dx(g2Φ) is the addendum to E which re- +tains Cl, Φ being a arbitrary function of u and its derivatives. +Remark 2. The equation (6) has travelling wave solutions, in par- +ticular shock waves of the form +3 +2λ +� +a tanh +�a3λ2 + 3a +λ2 +t + ax +� ++ 1 +λ +� +. +(7) +This shock moves to the left. If require u|−∞ = 0 then (7) becomes +the shock wave +3 +2λ2 +� +1 + tanh +� 4 +λ2t + 1 +λx +�� + +ON PERTURBATIONS RETAINING CONSERVATION LAWS +5 +with the velocity 4/λ, see figure 1, left. +Figure 1. The travelling wave solution +Left: for the equation (6). Right: For the equation (9), +a = 1/2; λ = 1. +Remark 4. The perturbed equation has only translations in x and +t as its point symmetries, but a lot of conservation laws. +2.2. Transformations of KdV that retain energy. Now for energy +saving transformations of KdV. Since the generating function of energy +is, up to a constant multiplier, u2 + uxx, one must solve +(u2 + uxx) · F(u, ux, uxx, uxxx) = Dx(A(u, ux, uxx)) +(8) +for some A(u, ux, uxx), to find an low-order F(u, ux, uxx), the suitable +transformation term. By analogy to the momentum case, the one pos- +sibility is A = (u2 + uxx)2B +F(u, ux, uxx) = 2Dx(u2+uxx)B+(u2+uxx)(ux +∂B +∂u +uxx +∂B +∂ux ++uxxx +∂B +∂uxx +), +for an arbitrary B = B(u, ux, uxx). If B = u then F = 5u2ux+2uuxxx+ +uxuxx +The corresponding transformed equation is +ut = 2uux + uxxx + λ(5u2ux + 2uuxxx + uxuxx). +(9) +Its point symmetries are only translations in x and t. +Remark 5. The equation (9) has travelling wave solutions, in par- +ticular — solutons of the form of a vertically shifted soliton +u(x, t) = −6a2 tanh2(a(4a4·λt+x))+4a2 = 6a2 sech2(a(4a4·λt+x))−2a2 +(10) +found by Maple, with the velocity V = 4a4λ, see figure 1, right. + +6 +ALEXEY SAMOKHIN +Yet it is not the whole answer. Computer experiments demonstrate +that an arbitrary initial datum for this equation scatters into a number +of solitary peaks of different but constant height and velocity and a ’tail’ +(see figures 2 and 3) — in a manner of the KdV itself, cf. [6]. +Figure 2. Left: Initial profile 1.5 sech2(0.5x) for the +equation (9), λ = 1. +Right: Resulting profile at t = 6: single soliton-like +peak of a constant form and velocity and an oscillating +tail moving in opposite direction +Figure 3. Left: Initial profile sech2(0.1x) for the equa- +tion (9), λ = 1. +Right: Resulting profile at t = 40: multiple soliton-like +peaks of a constant form and velocity and (seemingly) +no tail. +The analytical description of these peaks is so far unknown. The +reason is that the equation on travelling waves, u = u(x + V t), here +V u′ = 2uu′ + u′′ + λ(5u2u′ + 2uu′′′ + u′u′′ +can be readily integrated introducing the new dependent variable u′ = +p(u) which leads to a linear first order ordinary differential equation on +z(u) = p(u)p′(u), +(2uλ + 1)z′ + λz = V − 5λu2 − 2u. + +ON PERTURBATIONS RETAINING CONSERVATION LAWS +7 +But the resulting general solution looks hopelessly implicit. The likes +of (10) arise in the case of a very special combination of the arbitrary +constants entering this general solution, and such combinations are +hard to discover. +3. Two-dimensional MHD System +Consider the Kadomtsev-Pogutse sysnem of equations +� +∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy += +0 +vt + uxvy − uyvx += +0 +(11) +which describes quasi-stationary states of plasma. It has three conser- +vation laws, that is there are three non–trivial conserved densities (two +of them depending on arbitrary functions): the total energy E (mag- +netic plus kinetic energy), generalized ’cross helicity’ Hc and mean +magnetic potential A, +E += +1 +2⟨u2 +x + u2 +y + v2 +x + v2 +y⟩ +H += +⟨f ′(v) · (uxvx + uyvy)⟩ +A += +⟨Φ(v)⟩ +(12) +Their generating functions are, respective order, +� +u +∆v +� +, +� +f(v) +f ′(v)∆u +� +, +� +0 +Φ′(v) +� +(13) +where f and Φ are arbitrary functions. +Let us seek transformations of (11) of the form +� +∆ut + ux∆uy − uy∆ux + vy∆vx − vx∆vy += +νF(u, v) +vt + uxvy − uyvx += +ηG(u, v) +(14) +Here F, G are functions of u(x, y, t), v(x, y, t) and their derivatives. +3.1. Energy-retaining transformations. In this instance +∂⟨E⟩/∂t = 0 implies +(−νu · F − η∆v · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) = +(DxA(u, v) + DyB(u, v))dx ∧ dy. +(15) +There are a lot of solutions to (15). We restrict ourselves to some +low-order examples. +3.1.1. Ortogonal transformations. One can always get zero right hand +side in equation (15): just put F = η∆ and G = −νu. The vector +(F, G) is orthogonal to the generating function so ∂⟨E⟩/∂t = 0. It +works if the number of any system of equations is greater than one. + +8 +ALEXEY SAMOKHIN +3.1.2. Splitted sum transformations. Another solution may be obtained +assuming +− νu · F(u, v) = DxA(u, v), +η∆v · G(u, v) = DyB(u, v). +(16) +Here again A, B are functions of u(x, y, t), v(x, y, t) and their deriva- +tives. This equations may be solved by analogy to the KdV case. +One of numerous solutions here is A = νun, +B = η(∆v)2, so F = +−νnun−2ux, G = 2η∆vy +3.1.3. {ν = η}—case transformations. Take A = Gvx, B = Gvy. +Then uF = vxDxG + vyDyG. For instance, choose G = u2; it fol- +lows that F = 2(uxvx + uyvy). +3.2. Mean magnetic potential retaining transformations. Here +∂⟨A⟩/∂t = 0 implies +(−ν0 · F − ηΦ′(v) · G)dx ∧ dy = d(A(u, v)dy − B(u, v)dx) = +(DxA(u, v) + DyB(u, v))dx ∧ dy. +(17) +Thus F is an arbitrary function. Then one possible solution is +−ηΦ′(v) · Φ(v)(αvx + βvy) = DxαΦ2 + DyβΦ2, α, β ∈ R. +That is, to retain the mean magnetic potential of (11), its first equation +may be transformed in arbitrary way and the second one by ηG = +−ηΦ(v)(αvx + βvy) for all α, β ∈ R. +3.3. Cross helicity retaining transformations. Here ∂⟨Hc⟩/∂t = 0 +implies +−νf(v)·F(u, v)−ηf ′(v)∆(u)·G(u, v) = DxA(u, v)+DyB(u, v). (18) +In the case ηη = ν it is not hard to find some suitable transformations +(F, G). Namely, take +A = −ηf 2(v)f ′(v)ux, B = −ηf 2(v)f ′(v)uy; +It follows +F = [2f ′2(v) + f(v)f ′′(v)](vxux + uyvy), G = f ′(v)f(v)∆u. +For f(v) = v it comes to +F = −2η(vxux + uyvy) G = −ηv∆u. + +ON PERTURBATIONS RETAINING CONSERVATION LAWS +9 +Conclusion +The paper deals with perturbations of the equation that have a num- +ber of conservation laws. When a small term is added to the equation +its conserved quantities usually decay at individual rates, a phenome- +non known as a selective decay. These rates are described by the simple +law using the conservation laws’ generating functions and the added +term. Yet some perturbation may retain a specific quantity(s), such +as energy, momentum and other physically important characteristics +of solutions. We introduced a procedure for finding such perturbations +and demonstrated it by examples including the KdV-Burgers equation +and a system from magnetodynamics. +Our worked out examples show that the perturbed equations retain- +ing a specific conservation law frequently also retain additional alge- +braic properties such as travelling wave solutions or a presence of other +conservation laws. +Thus the present paper as well as [5] and our previous research of +the KdV solitons in nonhomogeneous media, [6], persuades that the +selective decay approach is a valid and effective instrument to obtain +qualitative approximations and estimates for behavior of solutions. +The figures in this paper were generated numerically using Maple +PDETools package. +The mode of operation uses the default Euler +method, which is a centered implicit scheme, and can be used to find +solutions to PDEs that are first order in time, and arbitrary order in +space, with no mixed partial derivatives. +References +[1] A. V. Samokhin, Decay velocity of conservation laws for nonevolution equa- +tions,Acta Applicanda Math., v. 41 n. 1, 1–11 (1995) +[2] A. C. Ting, M. H. Matthaeus, D. Montgomery, Turbulent relaxation processes +in magnetohydrodynamics Phys. Fluids, v.29, 3261–3274 (1986) +[3] E. van Groesen, F. Mainardi, Balance laws and centro velocity in dissipative +systems, J. Math. Phys.v. 31 (11), 2136–2140 (1990) +[4] J. B. Taylor. Relaxation of toroidal plasma and generation of reverse magnetic +fields, Phys. Rev.Lett., v. 33, 1139–1141 (1974) +[5] R. Brecht1, W. Bauer, A. Bihlo, F. Gay-Balmaz, S. MacLachlan. Selective decay +for the rotating shallow-water equations with a structure-preserving discretiza- +tion Phys. Fluids, v.33, 116604 (2021); https://doi.org/10.1063/5.0062573 +[6] A. V. Samokhin, The KdV soliton crosses a dissipative and dispersive border, +Journal of Differential Geometry and its Applications.75, Part A, 11 pages(April +2021) https://doi.org/10.1016/j.difgeo.2021.101723 +[7] A. V. Samokhin, Taylor Trick and Travelling Wave Solutions, Lobachevskii +Journal +of +Mathematics, +2022, +43, +n. +10, +2808—2815, +(2022). +DOI: +10.1134/S1995080222130406 +Institute of Control Sciences of Russian Academy of Sciences 65 +Profsoyuznaya street, Moscow 117997, Russia +Email address: +samohinalexey@gmail.com + diff --git a/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/2301.11408v1.pdf.txt b/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/2301.11408v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8e0d98edd6ad81837db05fe4cacf8c0b6b93246 --- /dev/null +++ b/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/2301.11408v1.pdf.txt @@ -0,0 +1,1425 @@ +Proceedings of Machine Learning Research – Under Review:1–20, 2023 +Full Paper – MIDL 2023 submission +DBGDGM: Dynamic Brain Graph Deep Generative Model +Alexander Campbell∗1,2 +ajrc4@cl.cam.ac.uk +Simeon Spasov∗1 +ses88@cl.cam.ac.uk +Nicola Toschi1 +Pietro Li`o3,4 +1 Department of Computer Science and Technology, University of Cambridge, United Kingdom +2 The Alan Turing Institute, United Kingdom +3 University of Rome Tor Vergata, Italy +4 A.A. Martinos Center for Biomedical Imaging, Harvard Medical School, United States +Editors: Under Review for MIDL 2023 +Abstract +Graphs are a natural representation of brain activity derived from functional magnetic +imaging (fMRI) data. It is well known that clusters of anatomical brain regions, known as +functional connectivity networks (FCNs), encode temporal relationships which can serve +as useful biomarkers for understanding brain function and dysfunction. Previous works, +however, ignore the temporal dynamics of the brain and focus on static graphs. In this +paper, we propose a dynamic brain graph deep generative model (DBGDGM) which simul- +taneously clusters brain regions into temporally evolving communities and learns dynamic +unsupervised node embeddings. Specifically, DBGDGM represents brain graph nodes as +embeddings sampled from a distribution over communities that evolve over time. +We +parameterise this community distribution using neural networks that learn from subject +and node embeddings as well as past community assignments. Experiments demonstrate +DBGDGM outperforms baselines in graph generation, dynamic link prediction, and is com- +parable for graph classification. Finally, an analysis of the learnt community distributions +reveals overlap with known FCNs reported in neuroscience literature. +Keywords: Dynamic graph, generative model, functional magnetic resonance imaging +1. Introduction +Functional magnetic resonance imaging (fMRI) is a non-invasive imaging technique pri- +marily used to measure blood-oxygen level dependent (BOLD) signal in the brain (Huettel +et al., 2004). A natural representation of fMRI data is as a discrete-time graph, henceforth +referred to as a dynamic brain graph (DBG), consisting of a set of fixed nodes correspond- +ing to anatomically separated brain regions and a set of time-varying edges determined by +a measure of dynamic functional connectivity (dFC) (Calhoun et al., 2014). DBGs have +been widely used in graph-based network analysis for understanding brain function (Hirsch +and Wohlschlaeger, 2022; Raz et al., 2016) and dysfunction (Alonso Mart´ınez et al., 2020; +Dautricourt et al., 2022; Yu et al., 2015). +∗ Contributed equally +© 2023 A. Campbell, S. Spasov, N. Toschi & P. Li`o. +arXiv:2301.11408v1 [cs.LG] 26 Jan 2023 + +Campbell Spasov Toschi Li`o +Recently, there is growing interest in using deep learning-based methods for learning +representations of graph-structured data (Goyal and Ferrara, 2018; Hamilton, 2020). A +graph representation typically consists of a low-dimensional vector embedding of either the +entire graph (Narayanan et al., 2017) or a part of it’s structure such as nodes (Grover and +Leskovec, 2016), edges (Gao et al., 2019), or sub-graphs (Adhikari et al., 2017). Although +originally formulated for static graphs (i.e. not time-varying), several existing methods have +been extended (Mahdavi et al., 2018; Goyal et al., 2020), and new ones proposed (Zhou +et al., 2018; Sankar et al., 2020), for dynamic graphs. The embeddings are usually learnt +in either a supervised or unsupervised fashion and typically used in tasks such as node +classification (Pareja et al., 2020) and dynamic link prediction (Goyal et al., 2018). +To date, very few deep learning-based methods have been designed for, or existing +methods applied to, representation learning of DBGs. Those that do, tend to use graph +neural networks (GNNs) that are designed for learning node- and graph-level embeddings +for use in graph classification (Kim et al., 2021; Dahan et al., 2021). Although node/graph- +level embeddings are effective at representing local/global graph structure, they are less +adept at representing topological structures in-between these two extremes such a clusters +of nodes or communities (Wang et al., 2017). Recent methods that explicitly incorporate +community embeddings alongside node embeddings have shown improved performance for +static graph representation learning tasks (Sun et al., 2019; Cavallari et al., 2017). How to +leverage the relatedness of graph, node, and community embeddings in a unified framework +for DBG representation learning remains under-explored. We refer to Appendix A for a +summary of related work. +Contributions +To address these shortcomings, we propose DBGDGM, a hierarchical +deep generative model (DGM) designed for unsupervised representing learning on DBGs +derived from multi-subject fMRI data. Specifically, DBGDGM represents nodes as embed- +dings sampled from a distribution over communities that evolve over time. The community +distribution is parameterized using neural networks (NNs) that learn from graph and node +embeddings as well as past community assignments. +We evaluate DBGDGM on multi- +ple real-world fMRI datasets and show that it outperforms state-of-the-art baselines for +graph reconstruction, dynamic link prediction, and achieves comparable results for graph +classification. +2. Problem formulation +We consider a dataset of multi-subject DBGs derived from fMRI data D ≡ G(1:S, 1:T) = +{G(s, t)}S, T +s, t=1 that share a common set of nodes V = {v1, . . . , vN} over T ∈ N timepoints +for S ∈ N subjects. Each G(s, t) ∈ G(1:S, 1:T) denotes a non-attributed, unweighted, and +undirected brain graph snapshot for the s-th subject at the t-th timepoint. We define a +brain graph snapshot as a tuple G(s, t) = (V, E(s, t)) where E(s, t) ⊆ V × V denotes an edge +set. The i-th edge for the s-th subject at the t-th timepoint e(s, t) +i +∈ E(s, t) is defined e(s, t) +i += +(w(s,t) +i +, c(s,t) +i +) where w(s,t) +i +is a source node and c(s,t) +i +is a target node. We assume each node +corresponds to a brain region making the number of nodes |V| = V ∈ N fixed over subjects +and time. We also assume edges correspond to a measure of dFC allowing the number of +edges |E(s, t)| = E(s, t) ∈ N vary over subjects and time. We further assume there exists +2 + +DBGDGM: Dynamic Brain Graph Deep Generative Model +K ∈ N clusters of nodes, or communities, the membership of which dynamically changes +over time for each subject. Let z(s, t) +i +∈ [1 : K] denote the latent community assignment of +the i-th edge for the s-th subject at the t-th timepoint. For each subject’s DBG our aim +is to learn, in an unsupervised fashion, graph α(s) ∈ RHα, node φ(s, t) +1:N = [φ(s, t) +n +] ∈ RN×Hφ, +and community ψ(s, t) +1:K = [ψ(s, t) +k +] ∈ RK×Hψ representations of dimensions Hα, Hφ, Hψ ∈ N, +respectively, for use in a variety of downstream tasks. +3. Method +Figure 1: Plate diagram for DBGDGM. La- +tent and observed variables are denoted by +white-and gray-shaded circles, respectively. +Solid black squares denote non-linear map- +pings parameterized by NNs. +DBGDGM defines a hierarchical deep gen- +erative model and inference network for +the end-to-end learning of graph, node, +and community embeddings from multi- +subject DBG data. Specifically, DBGDGM +treats the embeddings and edge commu- +nity assignments as latent random vari- +ables collectively denoted Ω(s, t) = {α(s), +φ(s, t) +1:N , +ψ(s, t) +1:K , {z(s, t) +i +}E(s, t) +i=1 +}, +which along +with the observed DBGs, defines a proba- +bilistic latent variable model with joint den- +sity pθ(G1:S, 1:T , Ω1:S, 1:T ). +3.1. Generative model +Graph embeddings +We begin the gen- +erative process by sampling graph embed- +dings from a prior α(s) ∼ pθα(α(s)) imple- +mented as a normal distribution following +pθα(α(s)) = Normal(0Hα, IHα) +(1) +where 0Hα is a matrix of zeros and IHα is a identity matrix. Each embedding is a vector +α(s) ∈ RHα representing subject-specific information that remains fixed over time. +Node and community embeddings +Next, let φ(s, t) +n +∈ RHφ and ψ(s, t) +k +∈ RHψ denote +the n-th node and the k-th community embedding, respectively. To incorporate tempo- +ral dynamics, we assume node and community embeddings are related through Markov +chains with prior transition distributions φ(s, t) +n +∼ pθφ(φ(s, t) +n +|φ(s, t−1) +n +, α(s)) and ψ(s, t) +k +∼ +pθψ(ψ(s, t) +k +|ψ(s, t−1) +k +, α(s)). We specify each prior to be a normal distribution following +pθφ(φ(s, t) +n +|φ(s, t−1) +n +, α(s)) = Normal(φ(s, t−1) +n +, σφIHφ) +(2) +pθψ(ψ(s, t) +k +|ψ(s, t−1) +k +, α(s)) = Normal(ψ(s, t−1) +k +, σψIHψ) +(3) +where the graph embeddings are used for initializing the means, i.e., φ(s, 0) +n += α(s), ψ(s, 0) +k += +α(s) and the standard deviations σφ, σψ ∈ R>0 are hyperparameters controlling how smoothly +each embedding changes between consecutive timepoints. +3 + +Campbell Spasov Toschi Li`o +Edge generation +We next describe the edge generative process of a graph snapshot +G(s, t) ∈ G(1:S, 1:T). Similar to Sun et al. (2019), for each edge e(s, t) +i += (w(s, t) +i +, c(s, t) +i +) ∈ E(s, t) +we first sample a latent community assignment z(s, t) +i +∈ [1 : K] from a conditional prior +z(s, t) +i +∼ pθz(z(s, t) +i +|w(s, t) +i +) implemented as a categorical distribution +pθz(z(s, t) +i +|w(s, t) +i +) = Categorical(π(s, t) +θz +), +π(s, t) +θz += MLPθz(φ(s, t) +wi +) +(4) +where MLPθz : RHφ → RK is a Lz-layered multilayered perception (MLP) that parame- +terizes community probabilities using node embeddings indexed by w(s, t) +i +. In other words, +each source node w(s, t) +i +is represented as a mixture of communities. A linked target node +c(s, t) +i +∈ [1 : N] is then sampled from the conditional likelihood c(s, t) +i +∼ pθc(c(s, t) +i +|z(s, t) +i +) which +is also implemented as a categorical distribution +pθc(c(s, t) +i +|z(s, t) +i +) = Categorical(π(s, t) +θc +), +π(s, t) +θc += MLPθc(ψ(s, t) +zi +) +(5) +where MLPθc : RHψ → RN is a Lc-layered MLP that parameterizes node probabilities using +community embeddings indexed by z(s, t) +i +. That is, each community assignment z(s, t) +i +is +represented as a mixture of nodes. By integrating out the latent community assignment +variable +p(c(s, t) +i +|w(s, t) +i +) = +� +z(s, t) +i +∈[1:K] +pθc(c(s, t) +i +|z(s, t) +i +)pθz(z(s, t) +i +|w(s, t) +i +) +(6) +we define the likelihood of node c(s, t) +i +being a linked neighbor of node w(s, t) +i +, in a given +graph snapshot. +Factorized generative model +Given this model specification, the joint probability of +the observed data and the latent variables can be factorized following +pθ(G1:S 1:T , Ω1:S,1:T ) = +S +� +s=1 +� +pθα(α(s)) +T +� +t=1 +� +V� +n=1 +pθφ(φ(s, t) +n +|φ(s, t−1) +n +) +K +� +k=1 +pθψ(ψ(s,t) +k +|ψ(s,t−1) +k +) +E(s, t) +� +i=1 +pθz(z(s, t) +i +|φ(s, t) +wi +)pθc(c(s, t) +i +|ψ(s, t) +zi +) +�� +(7) +where θ = {θc , θz} is the set of generative model parameters, i.e., NN weights. The gener- +ative model of DBGDGM summarized in Appendix B +3.2. Inference network +To learn the embeddings, we must infer the posterior distribution over all latent variables +conditioned on the observed data pθ(Ω(1:S, 1:T)|G(1:S, 1:T)). However, exact inference is in- +tractable due the log marginal likelihood requiring integrals that are hard to evaluate, i.e., +4 + +DBGDGM: Dynamic Brain Graph Deep Generative Model +log pθ(G(1:S, 1:T)) = +� +Ω log pθ(G(1:S, 1:T), Ω(1:S, 1:T))dΩ. As a result, we use variational infer- +ence (Jordan et al., 1999) to approximate the true posterior with a variational distribution +qλ(Ω(1:S,1:T)) with parameters λ. To do this, we maximize a lower bound on the log marginal +likelihood of the DBGs, referred to as the ELBO (evidence lower bound), defined as +LELBO(θ, λ) = Eqλ +� +log pθ(G1:S, 1:T , Ω1:S, 1:T ) +qλ(Ω(1:S, 1:T)) +� +≤ log pθ(G(1:S, 1:T)) +(8) +where Eqλ[·] denotes the expectation taken with respect to the variational distribution +qλ(Ω(1:S, 1:T)). By maximizing the ELBO with respect to the generative and variational +parameters θ and λ we train our generative model and perform Bayesian inference, respec- +tively. +Structured variational distribution +To ensure a good approximation to true posterior, +we retain the Markov properties of the node and community embeddings. This results in a +structured variational distribution (Hoffman and Blei, 2015; Saul and Jordan, 1995) which +factorizes following +qλ(Ω(1:S, 1:T)) = +S +� +s=1 +� +qλα(α(s)) +T +� +t=1 +� +V� +n=1 +qλφ(φ(s, t) +n +| φ(s, t−1) +n +) +K +� +k=1 +qλψ(ψ(s, t) +k +| ψ(s, t−1) +k +) +E(s, t) +� +i=1 +qλz(z(s, t) +i +| φ(s, t) +wi +, φ(s, t) +ci +) +�� +(9) +where each distribution is specified to mimic the structure of the generative model so that +qλα(α(s)) = Normal(µ(s) +λα, σ(s) +λα ) +(10) +qλφ(φ(s, t) +n +|φ(s, t−1) +n +) = Normal(µ(s, t) +λφ , σ(s, t) +λφ +) +{µ(s, t) +λφ , σ(s, t) +λφ +} = GRUλφ(φ(s, t−1) +n +) (11) +qλψ(ψ(s, t) +k +|ψ(s, t−1) +n +) = Normal(µ(s, t) +λψ , σ(s, t) +λψ ) +{µ(s, t) +λψ , σ(s, t) +λψ } = GRUλψ(ψ(s, t−1) +k +) (12) +qλz(z(s, t) +i +|φ(s, t) +wi +, φ(s, t) +ci +) = Categorical(π(s, t) +λz +) +π(s, t) +λz += MLPλz(φ(s, t) +wi +⊙ φ(s, t) +ci +) +(13) +where GRUλj : RHj → RHj is a Lj-layered GRU for each j ∈ {φ, ψ} and MLPλz : +RHφ → RK is Lz-layered MLP. Furthermore, we use MLPs to initialize the GRUs with +the graph embeddings such that φ(s, 0) +n += MLPλφ(α(s)) and ψ(s, 0) +k += MLPλψ(α(s)) where +MLPλj : RNα → RNj. This allows for subject-specific variation to be incorporated in the +temporal dynamics of the node and community embeddings. Another difference with the +generative model is now the variational distribution of the community assignment qλz(·) in- +cludes information from neighboring nodes via c(s, t) +i +. Finally, we use the same NN from the +generative model to parameterize the variational distribution of the community assignment, +i.e., λz = θz. This not only spares additional trainable parameters for the variational dis- +tribution but also further links the variational parameters of qλ(·) to generative parameters +of pθ(·) resulting in more robust learning (Farnoosh and Ostadabbas, 2021). The set of pa- +rameters for the inference network is therefore λ = {λα = {µ(s) +λα, σ(s) +λα }S +s=1, λφ, λψ, λz = θz}. +5 + +Campbell Spasov Toschi Li`o +Model +HCP +UKB +NLL (↓) +MSE (↓) +NLL (↓) +MSE (↓) +CMN +5.999 ± 0.029 * +0.050 ± 0.005 * +5.861 ± 0.017 * +0.050 ± 0.003 * +VGAE +5.857 ± 0.017 * +0.051 ± 0.002 * +5.851 ± 0.027 * +0.061 ± 0.002 * +OSBM +5.808 ± 0.026 * +0.051 ± 0.003 * +5.726 ± 0.039 * +0.052 ± 0.003 * +VGRAPH +5.569 ± 0.046 * +0.022 ± 0.004 * +5.716 ± 0.037 * +0.020 ± 0.003 * +VGRNN +5.674 ± 0.034 * +0.011 ± 0.003 * +5.649 ± 0.035 * +0.014 ± 0.002 * +ELSM +5.924 ± 0.040 * +0.081 ± 0.002 * +5.809 ± 0.024 * +0.115 ± 0.003 * +DBGDGM +4.587 ± 0.045 +0.001 ± 0.002 +4.586 ± 0.084 +0.004 ± 0.003 +AUROC (↑) +AP (↑) +AUROC (↑) +AP (↑) +CMN +0.665 ± 0.007 * +0.654 ± 0.006 * +0.678 ± 0.004 * +0.668 ± 0.005 * +VGAE +0.661 ± 0.010 * +0.674 ± 0.008 * +0.688 ± 0.010 * +0.607 ± 0.009 * +OSBM +0.655 ± 0.027 * +0.675 ± 0.024 * +0.678 ± 0.032 * +0.682 ± 0.033 * +VGRAPH +0.689 ± 0.004 * +0.682 ± 0.002 * +0.664 ± 0.002 * +0.621 ± 0.001 * +VGRNN +0.689 ± 0.007 * +0.698 ± 0.006 * +0.698 ± 0.009 * +0.696 ± 0.007 * +ELSM +0.669 ± 0.004 * +0.662 ± 0.002 * +0.661 ± 0.001 * +0.662 ± 0.002 * +DBGDGM +0.768 ± 0.026 +0.732 ± 0.032 +0.786 ± 0.040 +0.762 ± 0.038 +Table 1: Graph reconstruction (top) and dynamic link prediction (bottom) results (mean +± standard deviation over 5 runs). +First and second-best results shown in bold and +underlined. Statistically significant difference from DBGDGM marked *. +Training objective +Substituting the variational distribution from (9) and the joint dis- +tribution from (7) into the ELBO (8) gives the full training objective which can be optimized +using stochastic gradient descent. We estimate all gradients using the reparameterization +trick (Kingma and Welling, 2013) and the Gumbel-softmax trick (Jang et al., 2016; Mad- +dison et al., 2016). We refer to Appendix B further details on the ELBO and learning the +parameters. +4. Experiments +We evaluate DBGDGM against baseline models on the tasks of graph reconstruction, dy- +namic link prediction, and graph classification. Each task is designed to evaluate the use- +fulness of the learnt embeddings. +Datasets +We construct two multi-subject DBG datasets using publicly available fMRI +scans from the Human Connectome Project (HCP) (Van Essen et al., 2013) and UK Biobank +(UKB) (Sudlow et al., 2015). We randomly sample S = 300 subjects ensuring an even +male/female split. +To create DBGs, we parcellate each scan into V = 360 region-wise +BOLD signals using the Glasser atlas (Glasser et al., 2016), apply sliding-window Pearson +correlation (Calhoun et al., 2014) with a non-overlapping window of size and stride of 30, +and threshold the top 5% values of the lower triangle of each correlation matrix as connected +following Kim et al. (2021). The described procedure gives T = 16 graph snapshots for each +subject. Biological sex is taken as graph-level labels. We refer to Appendix C for further +details on each dataset. +6 + +DBGDGM: Dynamic Brain Graph Deep Generative Model +Baselines +We compare DBGDGM against a range of different unsupervised probabilistic +baseline models. For static baselines, we include variational graph autoencoder (VGAE) (Kipf +and Welling, 2016b), a deep generative version of the overlapping stochastic block model +(OSBM) (Mehta et al., 2019), and vGraph (VGRAPH) (Sun et al., 2019). For dynamic +baselines we include variational graph recurrent neural network (VGRNN) (Hajiramezanali +et al., 2019) and evolving latent space model (ELSM) (Gupta et al., 2019). For the graph re- +construction and link prediction tasks, we also include a heuristic baseline based on common +neighbors between nodes at previous snapshots (CMN). Finally, for graph classification we +include a support vector machine which takes as import static FC matrices (FCM) (Abra- +ham et al., 2017). Further details about baseline model can be found in Appendix D. +Implementation +We split both datasets into 80/10/10% training/validation/test data +along the time dimension. We train all models using the Adam optimizer (Kingma and +Ba, 2014) with decoupled weight decay (Loshchilov and Hutter, 2017). All baseline hy- +perparameters are set following their original implementations. For DBGDGM, choose the +number of communities K based on validation NLL. Finally, we train all models 5 times +using different random seeds. Implementation details can be found in Appendix E. +Evaluation metrics +For graph reconstruction, we evaluate the probability of the edges +in the test dataset using negative log-likelihood (NLL). We also compare the mean-squared +error (MSE) between actual and reconstructed node degree over all test snapshots. For dy- +namic link prediction, we sample an equal number of positive and negative edges in the test +dataset and measure performance using area under the receiver operator curve (AUROC) +and average precision (AP). Finally, for graph classification we predict the biological sex +for each subjects’ DBG and evaluate on accuracy. To predict graph labels, we average node +embeddings per subject for the baselines and the community embeddings for DBGDGM +before training a SVM using 10-fold cross-validation. For comparing models, we use the al- +most stochastic order (ASO) test (Dror et al., 2019) with significance level 0.05 and correct +for multiple comparisons (Bonferroni, 1936). +5. Results +Dynamic graph reconstruction and link prediction. +We summarize the average test +results of all models over 5 runs using optimally tuned hyperparameters. From Table 1, +it is clear that DBGDGM outperforms baselines on both tasks. For graph reconstruction, +DBGDGM shows an 18% and 30% relative improvement in NLL on HCP and UKB, re- +spectively, compared to the second-best baselines. For dynamic link prediction, the relative +improvement is > 11% in AUCROC and > 5% in AP compared to second-best baselines de- +pending on dataset. We attribute these statistically significant gains to DBGDGM’s ability +to learn dynamic brain connectivity more effectively. +Graph classification +For graph classification, DBGDGM achieves ∼ 75% accuracy for +HCP and ∼ 73% for UKB (see Fig. 2). We outperform 4 baselines and show indiscernible +performance to VGAE and OSBM. To show the interpretative power of DBGDGM, we re- +run the graph classification experiment for HCP with the embeddings of each community +separately. We find a community which comprises brain regions in the Cingulo-opercular +7 + +Campbell Spasov Toschi Li`o +Figure 2: Graph classification results (5 runs). Statistical significance from DBGDGM marked *. +Figure 3: Overlap between communities learned by DBGDGM and FCNs from Ji et al. (2019). +(CON) and the Somatomotor (SMN) networks, which achieves 68% accuracy. This finding +is in agreement with studies that show SMN is predictive of gender (Zhang et al., 2018). +Interpretability analysis +We use the learnt distributions over the nodes to calculate +overlap between each community and known functional connectivity networks (FCNs) from Ji +et al. (2019) (see Appendix F). Figure 3 shows that DBGDGM finds communities that sig- +nificantly overlap with existing FCNs. In particular, nodes in community 1 almost fully +corresponds to the visual network (VIS1 + VIS2), which is in keeping with the nature of +the experiment (the resting state data was acquired with eyes open and cross-hair fixation). +Remarkably, the second and third most homogeneous communities correspond to a large +degree to the DMN, which is well known to dominate resting state activity as a whole +(Yeshurun et al., 2021). The inspection of additional communities and respective predictive +power, along with their evolution in time at the region-of-interest granularity, has the poten- +tial to unveil the yet largely unexplored relationships between dynamic brain connectivity +changes and, e.g. psychiatric or neurological disorders (Heitmann and Breakspear, 2017). +8 + +HCP +100% +AUD +CON +DAN +80% +DMN +FPN +LAN +ORA +60% +PMM +SMN +VIS1 +VIS2 +40% +VMM +20% +0% +5 +4 +6 +7 +8 +9 10 11 12 13 14 15 16UKB +100% +AUD +CON +DAN +80% +DMN +FPN +LAN +ORA +60% +PMM +SMN +VIS1 +VIS2 +40% +VMM +20% +0% +5 +3 +4 +6 +7 +8 +910 1112 13 14 15 16HCP +UKB +90% +90% +* +* +85% +80% +80% +75% +70% +2 +Accura +70% +65% +60% +60% +50% - +55% +50% +VGAE +OSBM +ELSM +VGRAPH +FCM +VGAE +OSBM +VGRNN +ELSM +DBGDGM +VGRAPH +DBGDGM +VGRNN +FCMDBGDGM: Dynamic Brain Graph Deep Generative Model +6. Conclusion +We propose DBGDGM, a hierarchical DGM designed for unsupervised representing learning +of DBGs. Specifically, DBGDGM jointly learns graph-, community-, and node-level embed- +dings that outperform baselines on classification, interpretability, and dynamic link predic- +tion with statistical significance. Moreover, an analysis of the learnt dynamic community- +node distributions shows significant overlap with existing FCNs from neuroscience literature +further validating our method. +Acknowledgments +This work is supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1. +Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium +(Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by +the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Re- +search; and by the McDonnell Center for Systems Neuroscience at Washington University. +References +Alexandre Abraham, Michael P Milham, Adriana Di Martino, R Cameron Craddock, Dim- +itris Samaras, Bertrand Thirion, and Gael Varoquaux. 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In Proceedings of the 2021 SIAM International Conference on Data +Mining (SDM), pages 738–746. SIAM, 2021. +Lekui Zhou, Yang Yang, Xiang Ren, Fei Wu, and Yueting Zhuang. +Dynamic network +embedding by modeling triadic closure process. In Proceedings of the AAAI conference +on artificial intelligence, volume 32, 2018. +Appendix A. Related work +Dynamic graph generative models +Classic generative models for graph-structured +data are designed for capturing a small set of specific properties (e.g., degree distribution, +eigenvalues, modularity) of static graphs (Erdos et al., 1960; Barab´asi and Albert, 1999; +Nowicki and Snijders, 2001). DGMs that exploit the learning capacity of NNs are able to +learn more expressive graph distributions (Mehta et al., 2019; Kipf and Welling, 2016b; +Sarkar et al., 2020). Recent DGMs for dynamic graphs are majority VAE-based (Kingma +and Welling, 2013) and cannot learn community representations (Hajiramezanali et al., +2019; Gracious et al., 2021; Zhang et al., 2021). The few that do, are designed for static +graphs (Sun et al., 2019; Khan et al., 2021; Cavallari et al., 2017). +Learning representations of dynamic brain graphs +Unsupervised representation +learning methods for DBGs tend to focus on clustering DBGs into a finite number of con- +nectivity patterns that recur over time (Allen et al., 2014; Spencer and Goodfellow, 2022). +Community detection is another commonly used method but mainly applied to static brain +graphs (Pavlovi´c et al., 2020; Esfahlani et al., 2021). Extensions to DBGs are typically not +end-to-end trainable and do not scale to multi-subject datasets (Ting et al., 2020; Martinet +et al., 2020). Recent deep learning-based methods are predominately GNN-based (Kim +et al., 2021; Dahan et al., 2021). Unlike DBGDGM, these methods are supervised and +focus on learning deterministic node- and graph-level representations. +Appendix B. Method +B.1. Generative model +Algorithm 1 summarizes the generative model for DBGDGM. +15 + +Campbell Spasov Toschi Li`o +Algorithm 1: DBGDGM generative model +Input: {E(s, t)}S, T +s, t=1 +Hyperparameters: K, Hα, Hψ, Hφ, Lψ, Lφ, Lz, σ2 +ψ, σ2 +φ, +Initialize: D ← ∅ +for s ← 1 to S do +α(s) ∼ p(α(s)) = Normal(0Hα, IHα) +for t ← 1 to T do +for k ← 1 to K do +ψ(s,t) +k +∼ p(ψ(s, t) +k +|ψ(s, t−1) +k +) = Normal(ψ(s, t−1) +k +, σψIHψ) +end +for n ← 1 to V do +φ(s,t) +n +∼ p(φ(s, t) +n +|φ(s, t−1) +n +) = Normal(φ(s, t−1) +k +, σφIHφ) +end +˜E(s, t) ← ∅ +for i ← 1 to |E(s, t)| do +z(s, t) +i +∼ p(z(s, t) +i +|w(s, t) +i +) = Categorical(fθπ(φ(s, t) +wi +)) +c(s, t) +i +∼ p(c(s, t) +i +|z(s, t) +i +) = Categorical(fθπ(ψ(s, t) +zi +)) +˜E(s, t) ← ˜E(s, t) ∪ {(w(s, t) +i +, c(s, t) +i +)} +end +G(s, t) ← (V, ˜E(s, t)) +D ← D ∪ {G(s, t)} +end +end +B.2. Training objective and learning the parameters +Substituting the variational distribution from (9) and the joint distribution from (7) into +the ELBO (8) gives the full training objective defined as +LELBO(θ, λ) = +S +� +s=1 +T +� +t=1 +E(s, t) +� +i=1 +� +Eqλz qλψ +� +log pθ(c(s, t) +i +|w(s, t) +i +, ψ(s, t) +zi +) +� +− Eqλφ +� +DKL[qλz(z(s, t) +i +| φ(s, t) +wi +, φ(s, t) +ci +)||pθz(z(s, t) +i +| φ(s, t) +wi +)] +�� +− +S +� +s=1 +� +DKL[qλα(α(s))||pθα(α(s))] +T +� +t=1 +� +(14) +− +V +� +n=1 +Eqλφ +� +DKL[qλφ(φ(s, t) +n +| φ(s, t−1) +n +)||pθφ(φ(s, t) +n +| φ(s, t−1) +n +)] +� +− +K +� +k=1 +Eqλψ +� +DKL[qλψ(ψ(s, t) +k +| ψ(s, t−1) +k +)||pθψ(ψ(s, t) +k +| ψ(s, t−1) +k +)] +��� +16 + +DBGDGM: Dynamic Brain Graph Deep Generative Model +where DKL[·||·] denotes the Kullback-Leibler (KL) divergence. By maximizing (14), the +parameters (θ, λ) of the generative model and inference network can be jointly learnt. +Learning the parameters +In order to use efficient stochastic gradient-based optimiza- +tion techniques (Robbins and Monro, 1951) for learning (θ, λ), the gradient of the ELBO +has to be estimated. The main challenge of this is obtaining gradients of the variables under +expectation, i.e., Eq∗[·], since they are sampled. To allow gradients to flow through these +sampling steps, we use the reparameterization trick (Kingma and Welling, 2013; Rezende +et al., 2014) for the normal distributions and the Gumbel-softmax trick (Jang et al., 2016; +Maddison et al., 2016) for the categorical distributions. All gradients are now easily com- +puted via back-propagation (Rumelhart et al., 1986) making DBGDGM end-to-end train- +able. In addition, we analytically calculate the KL terms for both normal and categorical +distributions, which leads to lower variance gradient estimates and faster training as com- +pared to noisy Monte Carlo estimates. +Appendix C. Datasets +To create multi-subject DBG datasets, we use real fMRI scans from the UK Biobank (Sud- +low et al., 2015) and Human Connectome Project (Van Essen et al., 2013). Both data +sources represent well-characterized population cohorts that have undergone standardized +neuroimaging and clinical assessments to ensure high quality. +UK Biobank1 (UKB) +The UKB dataset consists of S = 300 resting-rate fMRI scans +(i.e. 3D image of the brain taken over consecutive timepoints) randomly sampled from the +v1.3 January 2017 release ensuring an equal male/female split (i.e. sex balanced) with an +age range of 44 − 57 years. The total number of images for each scan is 490 timepoints (6 +minutes duration with a repetition time of 0.74s). The dataset is minimally preprocessed +following the pipeline described in Alfaro-Almagro et al. (2018). +Human Connectome Project2 (HCP) +The HCP dataset similarly consists of S = 300 +sex balanced resting-state fMRI scans randomly sampled from the S1200 release with an +age range of 22 − 35 years. Only images from the first scanning-session using left-right +phase encoding are used. The total number of images for each scan is 1, 200 timepoints (15 +minutes duration with a repetition time of 0.72s). The dataset is minimally preprocessed +following the pipeline described in Glasser et al. (2013) +Further preprocessing +The fMRI scans from each dataset are further preprocessed to +create DBGs. Firstly, each scan is transformed into a multivariate timeseries of BOLD +signals using the Glasser atlas (Glasser et al., 2016) to average voxels within V = 360 brain +regions. Next, to ensure comparability with UKB, we truncate the length of HCP timeseries +to 490 timepoints. Following the commonly used sliding-window method (Calhoun et al., +2014), we use Pearson correlation to calculate FC matrices within non-overlapping windows +of length 1 < W ≤ 490 along the temporal dimension. At every window, we create an +edge set of a unweighted and undirected graph with no self-edges by thresholding the top +1 ≤ ϵ < 100 percentile values of the lower triangle of the FC matrix (excluding the principal +1. https://www.ukbiobank.ac.uk +2. https://www.humanconnectome.org +17 + +Campbell Spasov Toschi Li`o +diagonal) as connected following Kim et al. (2021). For both datasets, we choose W = 30 +and ϵ = 5 resulting in T = ⌊490/30⌋ = 16 graph snapshots each with E(s, t) = ⌊(360(360 − +1)/2)(5/100)⌋ = 3, 231 edges. +Appendix D. Baselines +We compare DBGDGM against a range of static and dynamic unsupervised graph repre- +sentation learning baseline models, all with publicly available code. In particular, we focus +on baselines that are generative and can quantify uncertainty. We leave comparisons to +popular deterministic baselines such as DynamicTriad (Zhou et al., 2018), DySAT (Sankar +et al., 2020), and DynNode2Vec (Mahdavi et al., 2018) for future work. Furthermore, since +all of the baselines were originally designed to model large single-graph datasets, we had to +adapt each implementation to work with smaller multi-graph datasets. +Variational graph auto encoder3 (VGAE) (Kipf and Welling, 2016b) +An extension +of the variational autoencoder (Kingma and Welling, 2013) (VAE) for graph structured +data. Specifically, VGAE uses a graph convolutional network (GCN) (Kipf and Welling, +2016a) to learn a distribution over node embeddings. Originally designed for static graphs, +we train VGAE on each dynamic graph snapshot independently. +Overlapping stochastic block model4 (OSBM) (Mehta et al., 2019) +A deep gener- +ative version of the overlapping stochastic block model (Miller et al., 2009). In particular, +OSBM places a stick-breaking prior over the number of communities which allows the model +to automatically infer the optimal number of communities from the data during training. +Similar to VGAE, OSBM uses a GCN to parameterize the distribution over node embed- +dings and is designed for static graphs. +Variational graph RNN5 (VGRNN) (Hajiramezanali et al., 2019) +An extension of +VGAE for dynamic graphs. Using a modified graph RNN architecture, VGRNN is able +to learn dependencies between and within changing graph topology over time. +Similar +to DBGDGM, the prior distribution over node embeddings is parameterized using hidden +states from previous timepoints. +Evolving latent space model6 (ELSM) (Gupta et al., 2019) +A generative model for +dynamic graphs that learns node embeddings and performs community detection. In par- +ticular, node embeddings are initially sampled from a Gaussian mixture model over com- +munities and then evolved over time using an LSTM. Unlike the previous baselines, ELSM +does not use a GNNs to parameterize model distributions. +vGraph7 (VGRAPH) (Sun et al., 2019) +Similar to DBGDGM, VGRAPH simultane- +ously learns node embeddings and community assignments by modeling nodes as being +3. https://github.com/tkipf/gae +4. https://github.com/nikhil-dce/SBM-meet-GNN +5. https://github.com/VGraphRNN/VGRNN +6. https://github.com/sh-gupta/ELSM +7. https://github.com/fanyun-sun/vGraph +18 + +DBGDGM: Dynamic Brain Graph Deep Generative Model +generated from a mixture of communities. The generative process of VGRAPH also re- +lies on edge information. Since VGRAPH only models static graphs, we train it on each +dynamic graph snapshot independently. +Common neighbors (CMN) +In light of recent work demonstrating that heuristic meth- +ods are able to outperform deep-learning based models on dynamic link prediction tasks (Skard- +ing et al., 2022; Poursafaei et al., 2022), we include our own heuristic-based generative model +baseline. More formally, let π(t) +vi ∈ [0, 1]V denote a vector of Jaccard index scores for node +v(t) +i +∈ V with all other nodes v(t) +j +∈ V for i ̸= j. The Jaccard index between two nodes +v(t) +i , v(t) +j +∈ V is defined |Γ(v(t) +i ) ∩ Γ(v(t) +j )|/|Γ(v(t) +i ) ∪ Γ(v(t) +j )| where Γ(v(t) +i ) denotes the set of +neighbors of node v(t) +i . We define the probability of node v(t) +i +having a linked neighbor v(t) +j +at snapshot t as +p(v(t) +j |v(t) +i ) = Categorical(π(t−1) +vi +). +(15) +This simple generative model captures the intuition that nodes are more likely to form links +if they had common neighbors in a previous snapshot. +Appendix E. Implementation details +Software and hardware +All models are developed in Python 3.7 (Python Core Team, +2019) using scikit-learn 1.1.1 (Pedregosa et al., 2011), PyTorch(Paszke et al., 2019), and +numpy 1.1.1 (Harris et al., 2020). Statistical significance tests are carried out using deep- +significance 1.1.1 (Ulmer et al., 2022). Experiments are performed on a Linux server (Debian +5.10.113-1) with a NVIDIA RTX A6000 GPU with 48 GB memory and 16 CPUs. +Training and testing +All baselines are implemented as per the original paper and/or +code repository given in Appendix D. For the static graph baselines VGAE, OSBM, VGRAPH +we train on each snapshot independently and use the node and/or community embeddings +at the last training snapshot to make predictions. +Hyperparameter optimization +We use model and training hyperparameter values de- +scribed in the original implementation of each baseline as a starting point for tuning on +the validation dataset. Since searching for optional values for each hyperparameter con- +figuration was outside the scope of this paper, we focus mainly on tuning the dimensions +of hidden layers. For DBGDGM, we use a learning rate of 1e-4 with a weight decay of +0. We choose the number of communities K ∈ {3, 6, 8, 12, 16, 24} based on lowest average +validation NLL (see Figure 4). In the generative model, we fix the temporal smoothness +hyperparameters σφ = σψ = 0.01. In the inference network, we fix the number of layers +for all NNs to Lφ = Lψ = Lz = 1. For the Gumbel-softmax reparameterization trick we +anneal the softmax temperature parameter starting from a maximum of 1 to a minimum +of 0.05 at a rate of 3e-4. Finally, we train all models for 1, 000 epochs using early-stopping +with a patience of 15 based on the lowest validation NLL. +Appendix F. Interpretability analysis +Using DBGDGM, for each community we average the node distributions across subjects +and timepoints and take the top 10% most probable nodes. We use these high probability +19 + +Campbell Spasov Toschi Li`o +3 +6 +8 +12 +16 +24 +Number of communities +4.45 +4.50 +4.55 +4.60 +4.65 +4.70 +4.75 +4.80 +Validation nll +hcp +3 +6 +8 +12 +16 +24 +Number of communities +4.35 +4.40 +4.45 +4.50 +4.55 +4.60 +4.65 +4.70 +4.75 +Validation nll +ukb +Figure 4: Elbow plot for finding the optimal number of communities K. +nodes to calculate overlap between each community and the brain regions that comprise +each functional network from Ji et al. (2019). More specifically, the coloured proportions in +Figure 3 represent the proportion of top nodes in each community, which belong to a given +functional network. +Abbreviation +Functional network +AUD +Auditory network +CON +Cingulo-opercular network +DAN +Dorsal-attention network +DMN +Default mode network +FPN +Frontoparietal network +LAN +Language network +ORA +Orbito-affective network +PMM +Posterior-multimodal network +SMN +Somatomotor network +VIS1 +Visual network 1 +VIS2 +Visual network 2 +VMM +Ventral-multimodal network +Table 2: Functional connectivity networks (FCNs) from Ji et al. (2019) +20 + diff --git a/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/load_file.txt b/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b207a23a4f674eee8c0856b17a3ef8d93d199b9d --- /dev/null +++ b/_dFIT4oBgHgl3EQf9yvA/content/tmp_files/load_file.txt @@ -0,0 +1,996 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf,len=995 +page_content='Proceedings of Machine Learning Research – Under Review:1–20, 2023 Full Paper – MIDL 2023 submission DBGDGM: Dynamic Brain Graph Deep Generative Model Alexander Campbell∗1,2 ajrc4@cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='uk Simeon Spasov∗1 ses88@cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='uk Nicola Toschi1 Pietro Li`o3,4 1 Department of Computer Science and Technology, University of Cambridge, United Kingdom 2 The Alan Turing Institute, United Kingdom 3 University of Rome Tor Vergata, Italy 4 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Martinos Center for Biomedical Imaging, Harvard Medical School, United States Editors: Under Review for MIDL 2023 Abstract Graphs are a natural representation of brain activity derived from functional magnetic imaging (fMRI) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' It is well known that clusters of anatomical brain regions, known as functional connectivity networks (FCNs), encode temporal relationships which can serve as useful biomarkers for understanding brain function and dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Previous works, however, ignore the temporal dynamics of the brain and focus on static graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In this paper, we propose a dynamic brain graph deep generative model (DBGDGM) which simul- taneously clusters brain regions into temporally evolving communities and learns dynamic unsupervised node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM represents brain graph nodes as embeddings sampled from a distribution over communities that evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We parameterise this community distribution using neural networks that learn from subject and node embeddings as well as past community assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Experiments demonstrate DBGDGM outperforms baselines in graph generation, dynamic link prediction, and is com- parable for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, an analysis of the learnt community distributions reveals overlap with known FCNs reported in neuroscience literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Keywords: Dynamic graph, generative model, functional magnetic resonance imaging 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Introduction Functional magnetic resonance imaging (fMRI) is a non-invasive imaging technique pri- marily used to measure blood-oxygen level dependent (BOLD) signal in the brain (Huettel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' A natural representation of fMRI data is as a discrete-time graph, henceforth referred to as a dynamic brain graph (DBG), consisting of a set of fixed nodes correspond- ing to anatomically separated brain regions and a set of time-varying edges determined by a measure of dynamic functional connectivity (dFC) (Calhoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' DBGs have been widely used in graph-based network analysis for understanding brain function (Hirsch and Wohlschlaeger, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Raz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016) and dysfunction (Alonso Mart´ınez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dautricourt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ∗ Contributed equally © 2023 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Campbell, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Spasov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Toschi & P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Li`o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='11408v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='LG] 26 Jan 2023 Campbell Spasov Toschi Li`o Recently, there is growing interest in using deep learning-based methods for learning representations of graph-structured data (Goyal and Ferrara, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hamilton, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' A graph representation typically consists of a low-dimensional vector embedding of either the entire graph (Narayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017) or a part of it’s structure such as nodes (Grover and Leskovec, 2016), edges (Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019), or sub-graphs (Adhikari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Although originally formulated for static graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' not time-varying), several existing methods have been extended (Mahdavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020), and new ones proposed (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Sankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020), for dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The embeddings are usually learnt in either a supervised or unsupervised fashion and typically used in tasks such as node classification (Pareja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020) and dynamic link prediction (Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To date, very few deep learning-based methods have been designed for, or existing methods applied to, representation learning of DBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Those that do, tend to use graph neural networks (GNNs) that are designed for learning node- and graph-level embeddings for use in graph classification (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Although node/graph- level embeddings are effective at representing local/global graph structure, they are less adept at representing topological structures in-between these two extremes such a clusters of nodes or communities (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Recent methods that explicitly incorporate community embeddings alongside node embeddings have shown improved performance for static graph representation learning tasks (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Cavallari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' How to leverage the relatedness of graph, node, and community embeddings in a unified framework for DBG representation learning remains under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We refer to Appendix A for a summary of related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Contributions To address these shortcomings, we propose DBGDGM, a hierarchical deep generative model (DGM) designed for unsupervised representing learning on DBGs derived from multi-subject fMRI data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM represents nodes as embed- dings sampled from a distribution over communities that evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The community distribution is parameterized using neural networks (NNs) that learn from graph and node embeddings as well as past community assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We evaluate DBGDGM on multi- ple real-world fMRI datasets and show that it outperforms state-of-the-art baselines for graph reconstruction, dynamic link prediction, and achieves comparable results for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Problem formulation We consider a dataset of multi-subject DBGs derived from fMRI data D ≡ G(1:S, 1:T) = {G(s, t)}S, T s, t=1 that share a common set of nodes V = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' , vN} over T ∈ N timepoints for S ∈ N subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Each G(s, t) ∈ G(1:S, 1:T) denotes a non-attributed, unweighted, and undirected brain graph snapshot for the s-th subject at the t-th timepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We define a brain graph snapshot as a tuple G(s, t) = (V, E(s, t)) where E(s, t) ⊆ V × V denotes an edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The i-th edge for the s-th subject at the t-th timepoint e(s, t) i ∈ E(s, t) is defined e(s, t) i = (w(s,t) i , c(s,t) i ) where w(s,t) i is a source node and c(s,t) i is a target node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We assume each node corresponds to a brain region making the number of nodes |V| = V ∈ N fixed over subjects and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We also assume edges correspond to a measure of dFC allowing the number of edges |E(s, t)| = E(s, t) ∈ N vary over subjects and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We further assume there exists 2 DBGDGM: Dynamic Brain Graph Deep Generative Model K ∈ N clusters of nodes, or communities, the membership of which dynamically changes over time for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Let z(s, t) i ∈ [1 : K] denote the latent community assignment of the i-th edge for the s-th subject at the t-th timepoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For each subject’s DBG our aim is to learn, in an unsupervised fashion, graph α(s) ∈ RHα, node φ(s, t) 1:N = [φ(s, t) n ] ∈ RN×Hφ, and community ψ(s, t) 1:K = [ψ(s, t) k ] ∈ RK×Hψ representations of dimensions Hα, Hφ, Hψ ∈ N, respectively, for use in a variety of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Method Figure 1: Plate diagram for DBGDGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' La- tent and observed variables are denoted by white-and gray-shaded circles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Solid black squares denote non-linear map- pings parameterized by NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' DBGDGM defines a hierarchical deep gen- erative model and inference network for the end-to-end learning of graph, node, and community embeddings from multi- subject DBG data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM treats the embeddings and edge commu- nity assignments as latent random vari- ables collectively denoted Ω(s, t) = {α(s), φ(s, t) 1:N , ψ(s, t) 1:K , {z(s, t) i }E(s, t) i=1 }, which along with the observed DBGs, defines a proba- bilistic latent variable model with joint den- sity pθ(G1:S, 1:T , Ω1:S, 1:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Generative model Graph embeddings We begin the gen- erative process by sampling graph embed- dings from a prior α(s) ∼ pθα(α(s)) imple- mented as a normal distribution following pθα(α(s)) = Normal(0Hα, IHα) (1) where 0Hα is a matrix of zeros and IHα is a identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Each embedding is a vector α(s) ∈ RHα representing subject-specific information that remains fixed over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Node and community embeddings Next, let φ(s, t) n ∈ RHφ and ψ(s, t) k ∈ RHψ denote the n-th node and the k-th community embedding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To incorporate tempo- ral dynamics, we assume node and community embeddings are related through Markov chains with prior transition distributions φ(s, t) n ∼ pθφ(φ(s, t) n |φ(s, t−1) n , α(s)) and ψ(s, t) k ∼ pθψ(ψ(s, t) k |ψ(s, t−1) k , α(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We specify each prior to be a normal distribution following pθφ(φ(s, t) n |φ(s, t−1) n , α(s)) = Normal(φ(s, t−1) n , σφIHφ) (2) pθψ(ψ(s, t) k |ψ(s, t−1) k , α(s)) = Normal(ψ(s, t−1) k , σψIHψ) (3) where the graph embeddings are used for initializing the means, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', φ(s, 0) n = α(s), ψ(s, 0) k = α(s) and the standard deviations σφ, σψ ∈ R>0 are hyperparameters controlling how smoothly each embedding changes between consecutive timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3 Campbell Spasov Toschi Li`o Edge generation We next describe the edge generative process of a graph snapshot G(s, t) ∈ G(1:S, 1:T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Similar to Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' for each edge e(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i = (w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) ∈ E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) we first sample a latent community assignment z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∈ [1 : K] from a conditional prior z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∼ pθz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) implemented as a categorical distribution pθz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) = Categorical(π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) θz ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) θz = MLPθz(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ) (4) where MLPθz : RHφ → RK is a Lz-layered multilayered perception (MLP) that parame- terizes community probabilities using node embeddings indexed by w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In other words, each source node w(s, t) i is represented as a mixture of communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' A linked target node c(s, t) i ∈ [1 : N] is then sampled from the conditional likelihood c(s, t) i ∼ pθc(c(s, t) i |z(s, t) i ) which is also implemented as a categorical distribution pθc(c(s, t) i |z(s, t) i ) = Categorical(π(s, t) θc ), π(s, t) θc = MLPθc(ψ(s, t) zi ) (5) where MLPθc : RHψ → RN is a Lc-layered MLP that parameterizes node probabilities using community embeddings indexed by z(s, t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' That is, each community assignment z(s, t) i is represented as a mixture of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' By integrating out the latent community assignment variable p(c(s, t) i |w(s, t) i ) = � z(s, t) i ∈[1:K] pθc(c(s, t) i |z(s, t) i )pθz(z(s, t) i |w(s, t) i ) (6) we define the likelihood of node c(s, t) i being a linked neighbor of node w(s, t) i , in a given graph snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Factorized generative model Given this model specification, the joint probability of the observed data and the latent variables can be factorized following pθ(G1:S 1:T , Ω1:S,1:T ) = S � s=1 � pθα(α(s)) T � t=1 � V� n=1 pθφ(φ(s, t) n |φ(s, t−1) n ) K � k=1 pθψ(ψ(s,t) k |ψ(s,t−1) k ) E(s, t) � i=1 pθz(z(s, t) i |φ(s, t) wi )pθc(c(s, t) i |ψ(s, t) zi ) �� (7) where θ = {θc , θz} is the set of generative model parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', NN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The gener- ative model of DBGDGM summarized in Appendix B 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Inference network To learn the embeddings, we must infer the posterior distribution over all latent variables conditioned on the observed data pθ(Ω(1:S, 1:T)|G(1:S, 1:T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' However, exact inference is in- tractable due the log marginal likelihood requiring integrals that are hard to evaluate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 4 DBGDGM: Dynamic Brain Graph Deep Generative Model log pθ(G(1:S, 1:T)) = � Ω log pθ(G(1:S, 1:T), Ω(1:S, 1:T))dΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' As a result, we use variational infer- ence (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 1999) to approximate the true posterior with a variational distribution qλ(Ω(1:S,1:T)) with parameters λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To do this, we maximize a lower bound on the log marginal likelihood of the DBGs, referred to as the ELBO (evidence lower bound), defined as LELBO(θ, λ) = Eqλ � log pθ(G1:S, 1:T , Ω1:S, 1:T ) qλ(Ω(1:S, 1:T)) � ≤ log pθ(G(1:S, 1:T)) (8) where Eqλ[·] denotes the expectation taken with respect to the variational distribution qλ(Ω(1:S, 1:T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' By maximizing the ELBO with respect to the generative and variational parameters θ and λ we train our generative model and perform Bayesian inference, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Structured variational distribution To ensure a good approximation to true posterior, we retain the Markov properties of the node and community embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This results in a structured variational distribution (Hoffman and Blei, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Saul and Jordan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 1995) which factorizes following qλ(Ω(1:S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 1:T)) = S � s=1 � qλα(α(s)) T � t=1 � V� n=1 qλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) K � k=1 qλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k | ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ) E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) � i=1 qλz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci ) �� (9) where each distribution is specified to mimic the structure of the generative model so that qλα(α(s)) = Normal(µ(s) λα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s) λα ) (10) qλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n |φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) = Normal(µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ ) {µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λφ } = GRUλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) (11) qλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k |ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) = Normal(µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ ) {µ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λψ } = GRUλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ) (12) qλz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci ) = Categorical(π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λz ) π(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) λz = MLPλz(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ⊙ φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci ) (13) where GRUλj : RHj → RHj is a Lj-layered GRU for each j ∈ {φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ψ} and MLPλz : RHφ → RK is Lz-layered MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Furthermore, we use MLPs to initialize the GRUs with the graph embeddings such that φ(s, 0) n = MLPλφ(α(s)) and ψ(s, 0) k = MLPλψ(α(s)) where MLPλj : RNα → RNj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This allows for subject-specific variation to be incorporated in the temporal dynamics of the node and community embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Another difference with the generative model is now the variational distribution of the community assignment qλz(·) in- cludes information from neighboring nodes via c(s, t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, we use the same NN from the generative model to parameterize the variational distribution of the community assignment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', λz = θz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This not only spares additional trainable parameters for the variational dis- tribution but also further links the variational parameters of qλ(·) to generative parameters of pθ(·) resulting in more robust learning (Farnoosh and Ostadabbas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The set of pa- rameters for the inference network is therefore λ = {λα = {µ(s) λα, σ(s) λα }S s=1, λφ, λψ, λz = θz}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 5 Campbell Spasov Toschi Li`o Model HCP UKB NLL (↓) MSE (↓) NLL (↓) MSE (↓) CMN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='029 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='005 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='861 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='017 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='050 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * VGAE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='857 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='017 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='051 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='851 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='027 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='061 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * OSBM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='808 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='026 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='051 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='726 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='039 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * VGRAPH 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='569 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='046 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='022 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='716 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='037 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * VGRNN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='674 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='034 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='011 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='649 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='035 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * ELSM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='924 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='040 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='809 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='024 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='115 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 * DBGDGM 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='587 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='586 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='003 AUROC (↑) AP (↑) AUROC (↑) AP (↑) CMN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='665 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='007 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='654 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='006 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='678 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='668 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='005 * VGAE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='661 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='010 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='674 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='008 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='688 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='010 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='607 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='009 * OSBM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='655 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='027 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='675 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='024 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='678 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='032 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='033 * VGRAPH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='664 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='621 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='001 * VGRNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='689 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='007 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='698 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='006 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='698 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='009 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='696 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='007 * ELSM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='669 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='004 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='662 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='661 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='001 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='662 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='002 * DBGDGM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='768 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='732 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='786 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='762 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='038 Table 1: Graph reconstruction (top) and dynamic link prediction (bottom) results (mean ± standard deviation over 5 runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' First and second-best results shown in bold and underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Statistically significant difference from DBGDGM marked *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Training objective Substituting the variational distribution from (9) and the joint dis- tribution from (7) into the ELBO (8) gives the full training objective which can be optimized using stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We estimate all gradients using the reparameterization trick (Kingma and Welling, 2013) and the Gumbel-softmax trick (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Mad- dison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We refer to Appendix B further details on the ELBO and learning the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Experiments We evaluate DBGDGM against baseline models on the tasks of graph reconstruction, dy- namic link prediction, and graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Each task is designed to evaluate the use- fulness of the learnt embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Datasets We construct two multi-subject DBG datasets using publicly available fMRI scans from the Human Connectome Project (HCP) (Van Essen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2013) and UK Biobank (UKB) (Sudlow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We randomly sample S = 300 subjects ensuring an even male/female split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To create DBGs, we parcellate each scan into V = 360 region-wise BOLD signals using the Glasser atlas (Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016), apply sliding-window Pearson correlation (Calhoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014) with a non-overlapping window of size and stride of 30, and threshold the top 5% values of the lower triangle of each correlation matrix as connected following Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The described procedure gives T = 16 graph snapshots for each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Biological sex is taken as graph-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We refer to Appendix C for further details on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 6 DBGDGM: Dynamic Brain Graph Deep Generative Model Baselines We compare DBGDGM against a range of different unsupervised probabilistic baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For static baselines, we include variational graph autoencoder (VGAE) (Kipf and Welling, 2016b), a deep generative version of the overlapping stochastic block model (OSBM) (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019), and vGraph (VGRAPH) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For dynamic baselines we include variational graph recurrent neural network (VGRNN) (Hajiramezanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) and evolving latent space model (ELSM) (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For the graph re- construction and link prediction tasks, we also include a heuristic baseline based on common neighbors between nodes at previous snapshots (CMN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, for graph classification we include a support vector machine which takes as import static FC matrices (FCM) (Abra- ham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Further details about baseline model can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Implementation We split both datasets into 80/10/10% training/validation/test data along the time dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We train all models using the Adam optimizer (Kingma and Ba, 2014) with decoupled weight decay (Loshchilov and Hutter, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' All baseline hy- perparameters are set following their original implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For DBGDGM, choose the number of communities K based on validation NLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, we train all models 5 times using different random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Implementation details can be found in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Evaluation metrics For graph reconstruction, we evaluate the probability of the edges in the test dataset using negative log-likelihood (NLL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We also compare the mean-squared error (MSE) between actual and reconstructed node degree over all test snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For dy- namic link prediction, we sample an equal number of positive and negative edges in the test dataset and measure performance using area under the receiver operator curve (AUROC) and average precision (AP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, for graph classification we predict the biological sex for each subjects’ DBG and evaluate on accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To predict graph labels, we average node embeddings per subject for the baselines and the community embeddings for DBGDGM before training a SVM using 10-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For comparing models, we use the al- most stochastic order (ASO) test (Dror et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) with significance level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='05 and correct for multiple comparisons (Bonferroni, 1936).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Results Dynamic graph reconstruction and link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We summarize the average test results of all models over 5 runs using optimally tuned hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' From Table 1, it is clear that DBGDGM outperforms baselines on both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For graph reconstruction, DBGDGM shows an 18% and 30% relative improvement in NLL on HCP and UKB, re- spectively, compared to the second-best baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For dynamic link prediction, the relative improvement is > 11% in AUCROC and > 5% in AP compared to second-best baselines de- pending on dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We attribute these statistically significant gains to DBGDGM’s ability to learn dynamic brain connectivity more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Graph classification For graph classification, DBGDGM achieves ∼ 75% accuracy for HCP and ∼ 73% for UKB (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We outperform 4 baselines and show indiscernible performance to VGAE and OSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To show the interpretative power of DBGDGM, we re- run the graph classification experiment for HCP with the embeddings of each community separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We find a community which comprises brain regions in the Cingulo-opercular 7 Campbell Spasov Toschi Li`o Figure 2: Graph classification results (5 runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Statistical significance from DBGDGM marked *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Figure 3: Overlap between communities learned by DBGDGM and FCNs from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (CON) and the Somatomotor (SMN) networks, which achieves 68% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' This finding is in agreement with studies that show SMN is predictive of gender (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Interpretability analysis We use the learnt distributions over the nodes to calculate overlap between each community and known functional connectivity networks (FCNs) from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019) (see Appendix F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Figure 3 shows that DBGDGM finds communities that sig- nificantly overlap with existing FCNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In particular, nodes in community 1 almost fully corresponds to the visual network (VIS1 + VIS2), which is in keeping with the nature of the experiment (the resting state data was acquired with eyes open and cross-hair fixation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Remarkably, the second and third most homogeneous communities correspond to a large degree to the DMN, which is well known to dominate resting state activity as a whole (Yeshurun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The inspection of additional communities and respective predictive power, along with their evolution in time at the region-of-interest granularity, has the poten- tial to unveil the yet largely unexplored relationships between dynamic brain connectivity changes and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' psychiatric or neurological disorders (Heitmann and Breakspear, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='HCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='100% ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='OSBM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='VGRNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ELSM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='DBGDGM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='VGRAPH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='DBGDGM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='VGRNN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='FCMDBGDGM: Dynamic Brain Graph Deep Generative Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Conclusion We propose DBGDGM, a hierarchical DGM designed for unsupervised representing learning of DBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, DBGDGM jointly learns graph-, community-, and node-level embed- dings that outperform baselines on classification, interpretability, and dynamic link predic- tion with statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Moreover, an analysis of the learnt dynamic community- node distributions shows significant overlap with existing FCNs from neuroscience literature further validating our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Acknowledgments This work is supported by The Alan Turing Institute under the EPSRC grant EP/N510129/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 1U54MH091657) funded by the 16 NIH Institutes and Centers 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Calhoun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Assessing dynamic brain graphs of time-varying connectivity in fmri data: Application to healthy controls and patients with schizophrenia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' NeuroImage, 107:345–355, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ISSN 1053-8119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='neuroimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' URL https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/science/article/pii/S105381191401012X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Chao Zhang, Chase C Dougherty, Stefi A Baum, Tonya White, and Andrew M Michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Functional connectivity predicts gender: Evidence for gender differences in resting brain connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Human brain mapping, 39(4):1765–1776, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Wenbin Zhang, Liming Zhang, Dieter Pfoser, and Liang Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Disentangled dynamic graph deep generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In Proceedings of the 2021 SIAM International Conference on Data Mining (SDM), pages 738–746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' SIAM, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lekui Zhou, Yang Yang, Xiang Ren, Fei Wu, and Yueting Zhuang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dynamic network embedding by modeling triadic closure process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In Proceedings of the AAAI conference on artificial intelligence, volume 32, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Related work Dynamic graph generative models Classic generative models for graph-structured data are designed for capturing a small set of specific properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', degree distribution, eigenvalues, modularity) of static graphs (Erdos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 1960;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Barab´asi and Albert, 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Nowicki and Snijders, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' DGMs that exploit the learning capacity of NNs are able to learn more expressive graph distributions (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Kipf and Welling, 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Sarkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Recent DGMs for dynamic graphs are majority VAE-based (Kingma and Welling, 2013) and cannot learn community representations (Hajiramezanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Gracious et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The few that do, are designed for static graphs (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Cavallari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Learning representations of dynamic brain graphs Unsupervised representation learning methods for DBGs tend to focus on clustering DBGs into a finite number of con- nectivity patterns that recur over time (Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Spencer and Goodfellow, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Community detection is another commonly used method but mainly applied to static brain graphs (Pavlovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Esfahlani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Extensions to DBGs are typically not end-to-end trainable and do not scale to multi-subject datasets (Ting et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Martinet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Recent deep learning-based methods are predominately GNN-based (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Dahan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Unlike DBGDGM, these methods are supervised and focus on learning deterministic node- and graph-level representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Method B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Generative model Algorithm 1 summarizes the generative model for DBGDGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 15 Campbell Spasov Toschi Li`o Algorithm 1: DBGDGM generative model Input: {E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)}S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' T s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t=1 Hyperparameters: K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Lz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ2 ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σ2 φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Initialize: D ← ∅ for s ← 1 to S do α(s) ∼ p(α(s)) = Normal(0Hα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' IHα) for t ← 1 to T do for k ← 1 to K do ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='t) k ∼ p(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k |ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ) = Normal(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σψIHψ) end for n ← 1 to V do φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='t) n ∼ p(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n |φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n ) = Normal(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' σφIHφ) end ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ← ∅ for i ← 1 to |E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)| do z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∼ p(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) = Categorical(fθπ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi )) c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ∼ p(c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ) = Categorical(fθπ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) zi )) ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ← ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ∪ {(w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i )} end G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ← (V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ˜E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)) D ← D ∪ {G(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t)} end end B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Training objective and learning the parameters Substituting the variational distribution from (9) and the joint distribution from (7) into the ELBO (8) gives the full training objective defined as LELBO(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' λ) = S � s=1 T � t=1 E(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) � i=1 � Eqλz qλψ � log pθ(c(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i |w(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) zi ) � − Eqλφ � DKL[qλz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) ci )||pθz(z(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) i | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) wi )] �� − S � s=1 � DKL[qλα(α(s))||pθα(α(s))] T � t=1 � (14) − V � n=1 Eqλφ � DKL[qλφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n )||pθφ(φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) n | φ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) n )] � − K � k=1 Eqλψ � DKL[qλψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k | ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k )||pθψ(ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t) k | ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' t−1) k )] ��� 16 DBGDGM: Dynamic Brain Graph Deep Generative Model where DKL[·||·] denotes the Kullback-Leibler (KL) divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' By maximizing (14), the parameters (θ, λ) of the generative model and inference network can be jointly learnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Learning the parameters In order to use efficient stochastic gradient-based optimiza- tion techniques (Robbins and Monro, 1951) for learning (θ, λ), the gradient of the ELBO has to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The main challenge of this is obtaining gradients of the variables under expectation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', Eq∗[·], since they are sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' To allow gradients to flow through these sampling steps, we use the reparameterization trick (Kingma and Welling, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Rezende et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014) for the normal distributions and the Gumbel-softmax trick (Jang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Maddison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016) for the categorical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' All gradients are now easily com- puted via back-propagation (Rumelhart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 1986) making DBGDGM end-to-end train- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In addition, we analytically calculate the KL terms for both normal and categorical distributions, which leads to lower variance gradient estimates and faster training as com- pared to noisy Monte Carlo estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Datasets To create multi-subject DBG datasets, we use real fMRI scans from the UK Biobank (Sud- low et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2015) and Human Connectome Project (Van Essen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Both data sources represent well-characterized population cohorts that have undergone standardized neuroimaging and clinical assessments to ensure high quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' UK Biobank1 (UKB) The UKB dataset consists of S = 300 resting-rate fMRI scans (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' 3D image of the brain taken over consecutive timepoints) randomly sampled from the v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='3 January 2017 release ensuring an equal male/female split (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' sex balanced) with an age range of 44 − 57 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The total number of images for each scan is 490 timepoints (6 minutes duration with a repetition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='74s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The dataset is minimally preprocessed following the pipeline described in Alfaro-Almagro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Human Connectome Project2 (HCP) The HCP dataset similarly consists of S = 300 sex balanced resting-state fMRI scans randomly sampled from the S1200 release with an age range of 22 − 35 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Only images from the first scanning-session using left-right phase encoding are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The total number of images for each scan is 1, 200 timepoints (15 minutes duration with a repetition time of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='72s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The dataset is minimally preprocessed following the pipeline described in Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2013) Further preprocessing The fMRI scans from each dataset are further preprocessed to create DBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Firstly, each scan is transformed into a multivariate timeseries of BOLD signals using the Glasser atlas (Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2016) to average voxels within V = 360 brain regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Next, to ensure comparability with UKB, we truncate the length of HCP timeseries to 490 timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Following the commonly used sliding-window method (Calhoun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2014), we use Pearson correlation to calculate FC matrices within non-overlapping windows of length 1 < W ≤ 490 along the temporal dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' At every window, we create an edge set of a unweighted and undirected graph with no self-edges by thresholding the top 1 ≤ ϵ < 100 percentile values of the lower triangle of the FC matrix (excluding the principal 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ukbiobank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='uk 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='humanconnectome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='org 17 Campbell Spasov Toschi Li`o diagonal) as connected following Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For both datasets, we choose W = 30 and ϵ = 5 resulting in T = ⌊490/30⌋ = 16 graph snapshots each with E(s, t) = ⌊(360(360 − 1)/2)(5/100)⌋ = 3, 231 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Baselines We compare DBGDGM against a range of static and dynamic unsupervised graph repre- sentation learning baseline models, all with publicly available code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In particular, we focus on baselines that are generative and can quantify uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We leave comparisons to popular deterministic baselines such as DynamicTriad (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018), DySAT (Sankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020), and DynNode2Vec (Mahdavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2018) for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Furthermore, since all of the baselines were originally designed to model large single-graph datasets, we had to adapt each implementation to work with smaller multi-graph datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Variational graph auto encoder3 (VGAE) (Kipf and Welling, 2016b) An extension of the variational autoencoder (Kingma and Welling, 2013) (VAE) for graph structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Specifically, VGAE uses a graph convolutional network (GCN) (Kipf and Welling, 2016a) to learn a distribution over node embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Originally designed for static graphs, we train VGAE on each dynamic graph snapshot independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Overlapping stochastic block model4 (OSBM) (Mehta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) A deep gener- ative version of the overlapping stochastic block model (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In particular, OSBM places a stick-breaking prior over the number of communities which allows the model to automatically infer the optimal number of communities from the data during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Similar to VGAE, OSBM uses a GCN to parameterize the distribution over node embed- dings and is designed for static graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Variational graph RNN5 (VGRNN) (Hajiramezanali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) An extension of VGAE for dynamic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Using a modified graph RNN architecture, VGRNN is able to learn dependencies between and within changing graph topology over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Similar to DBGDGM, the prior distribution over node embeddings is parameterized using hidden states from previous timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Evolving latent space model6 (ELSM) (Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) A generative model for dynamic graphs that learns node embeddings and performs community detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In par- ticular, node embeddings are initially sampled from a Gaussian mixture model over com- munities and then evolved over time using an LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Unlike the previous baselines, ELSM does not use a GNNs to parameterize model distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' vGraph7 (VGRAPH) (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019) Similar to DBGDGM, VGRAPH simultane- ously learns node embeddings and community assignments by modeling nodes as being 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/tkipf/gae 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/nikhil-dce/SBM-meet-GNN 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/VGraphRNN/VGRNN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/sh-gupta/ELSM 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='com/fanyun-sun/vGraph 18 DBGDGM: Dynamic Brain Graph Deep Generative Model generated from a mixture of communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The generative process of VGRAPH also re- lies on edge information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Since VGRAPH only models static graphs, we train it on each dynamic graph snapshot independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Common neighbors (CMN) In light of recent work demonstrating that heuristic meth- ods are able to outperform deep-learning based models on dynamic link prediction tasks (Skard- ing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Poursafaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022), we include our own heuristic-based generative model baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' More formally, let π(t) vi ∈ [0, 1]V denote a vector of Jaccard index scores for node v(t) i ∈ V with all other nodes v(t) j ∈ V for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' The Jaccard index between two nodes v(t) i , v(t) j ∈ V is defined |Γ(v(t) i ) ∩ Γ(v(t) j )|/|Γ(v(t) i ) ∪ Γ(v(t) j )| where Γ(v(t) i ) denotes the set of neighbors of node v(t) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We define the probability of node v(t) i having a linked neighbor v(t) j at snapshot t as p(v(t) j |v(t) i ) = Categorical(π(t−1) vi ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (15) This simple generative model captures the intuition that nodes are more likely to form links if they had common neighbors in a previous snapshot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Implementation details Software and hardware All models are developed in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='7 (Python Core Team, 2019) using scikit-learn 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1 (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2011), PyTorch(Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2019), and numpy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1 (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Statistical significance tests are carried out using deep- significance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='1 (Ulmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Experiments are performed on a Linux server (Debian 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='113-1) with a NVIDIA RTX A6000 GPU with 48 GB memory and 16 CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Training and testing All baselines are implemented as per the original paper and/or code repository given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For the static graph baselines VGAE, OSBM, VGRAPH we train on each snapshot independently and use the node and/or community embeddings at the last training snapshot to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Hyperparameter optimization We use model and training hyperparameter values de- scribed in the original implementation of each baseline as a starting point for tuning on the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Since searching for optional values for each hyperparameter con- figuration was outside the scope of this paper, we focus mainly on tuning the dimensions of hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For DBGDGM, we use a learning rate of 1e-4 with a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We choose the number of communities K ∈ {3, 6, 8, 12, 16, 24} based on lowest average validation NLL (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In the generative model, we fix the temporal smoothness hyperparameters σφ = σψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' In the inference network, we fix the number of layers for all NNs to Lφ = Lψ = Lz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' For the Gumbel-softmax reparameterization trick we anneal the softmax temperature parameter starting from a maximum of 1 to a minimum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='05 at a rate of 3e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Finally, we train all models for 1, 000 epochs using early-stopping with a patience of 15 based on the lowest validation NLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Interpretability analysis Using DBGDGM, for each community we average the node distributions across subjects and timepoints and take the top 10% most probable nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' We use these high probability 19 Campbell Spasov Toschi Li`o 3 6 8 12 16 24 Number of communities 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='80 Validation nll hcp 3 6 8 12 16 24 Number of communities 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='40 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='45 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content='75 Validation nll ukb Figure 4: Elbow plot for finding the optimal number of communities K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' nodes to calculate overlap between each community and the brain regions that comprise each functional network from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' More specifically, the coloured proportions in Figure 3 represent the proportion of top nodes in each community, which belong to a given functional network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' Abbreviation Functional network AUD Auditory network CON Cingulo-opercular network DAN Dorsal-attention network DMN Default mode network FPN Frontoparietal network LAN Language network ORA Orbito-affective network PMM Posterior-multimodal network SMN Somatomotor network VIS1 Visual network 1 VIS2 Visual network 2 VMM Ventral-multimodal network Table 2: Functional connectivity networks (FCNs) from Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} +page_content=' (2019) 20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dFIT4oBgHgl3EQf9yvA/content/2301.11408v1.pdf'} diff --git a/a9FAT4oBgHgl3EQfXR0t/content/tmp_files/2301.08532v1.pdf.txt b/a9FAT4oBgHgl3EQfXR0t/content/tmp_files/2301.08532v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f65e2ffb2226e84a4b7e6590c5adcf0092aa341e --- /dev/null +++ b/a9FAT4oBgHgl3EQfXR0t/content/tmp_files/2301.08532v1.pdf.txt @@ -0,0 +1,419 @@ +Repercussion of the a0(1710) [a0(1817)] resonance and future developments +E. Oset,1, ∗ L. R. Dai,2, † and L. S. Geng3, ‡ +1Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC +Institutos de Investigaci´on de Paterna, Aptdo.22085, 46071 Valencia, Spain +2School of Science, Huzhou University, Huzhou 313000, Zhejiang, China +3School of Physics, Beihang University, Beijing 102206, China +(Dated: January 23, 2023) +In a recent paper the BESIII Collaboration reported the observation of a scalar meson of spin-parity JP = 0+ with +isospin I = 1, branded as a0(1817). The state is seen as a peak in the K0 +SK+ mass distribution in the D+ +s → K0 +SK+π0 +decay [1]. Its mass and width are reported as +Ma0 = 1817 ± 8stat ± 20sys MeV; +Γa0 = 97 ± 22stat ± 15sys MeV +(1) +Prior to this finding, a BABAR experiment reported an a0(1710) state from the π+η, π−η mass distributions in +the ηc → π+π−η decay [2]. +The mass and width of this state are obtained as M = 1709 ± 5stat ± 2sys MeV, +Γ = 110 ± 152stat ± 11sys MeV. +The new a0 resonance in the sector of light quarks comes as a real surprise at a moment when, however, many +new mesonic states are obtained in the heavy quark sector [3]. An isospin I = 0, f0(1710) resonance has, however, +been known for quite some time [4]. Another BESIII experiment reporting on D+ +s → π+f0(1710) has brought new +elements to this discussion. Indeed in Ref. [5] it was found the branching fraction +Br[D+ +s → π+“f0(1710)”; “f0(1710)” → K+K−] = (1.0 ± 0.2 ± 0.3) × 10−3 , +(2) +while from Ref. [6] it was found that +Br[D+ +s → π+“f0(1710)”; “f0(1710)” → K0 +SK0 +S] = (3.1 ± 0.3 ± 0.1) × 10−3 , +(3) +where “f0(1710)” was supposed to be the f0(1710) resonance. Yet, it was concluded that it could not be, because +from Eqs. (2), (3) one finds +R1 = Γ(D+ +s → π+“f0(1710)” → π+K0 ¯K0) +Γ(D+ +s → π+“f0(1710)” → π+K+K−) = 6.20 ± 0.67 . +(4) +If “f0(1710)” was the f0(1710) resonance this latter ratio should be 1. It was concluded that hidden below, or around +the f0(1710), there should be an I = 1 resonance responsible for this surprising large ratio. Indeed, since +|K ¯K, I = 0⟩ = − 1 +√ +2 +� +K0 ¯K0 + K+K−� +, +|K ¯K, I = 1, I3 = 0⟩ = +1 +√ +2 +� +K0 ¯K0 − K+K−� +, +(5) +there could be a mixture of the two resonances and their interference would be responsible for a different K+K− or +K0 ¯K0 production. +It is clear that the three experiments are seeing an I = 1 resonance in the region [1700 − 1800] MeV and it is +unlikely that they correspond to three different resonances, but this is something to be settled by further experiments +to which we will come back. Below this energy there is one a0 resonance very well known, the a0(980) resonance, +which decays to πη, with an apparent width of about [50 − 100] MeV [4]. +∗Electronic address: oset@ific.uv.es +†Electronic address: dailianrong@zjhu.edu.cn +‡Electronic address: lisheng.geng@buaa.edu.cn +arXiv:2301.08532v1 [hep-ph] 20 Jan 2023 + +2 +From the standard q¯q quark model point of view the a+ +0 (980) would be u ¯d. It might be surprising that a state like +that with no strange quarks decays to K ¯K. The answer to this lies in the hadronization of the u ¯d which gets attached +to a ¯qq state with the quantum numbers of the vacuum via +u ¯d → +� +i +u ¯qiqi ¯d = u(¯uu + ¯dd + ¯ss) ¯d , +(6) +as shown in Fig. 1. +¯d +u +¯qi +qi +FIG. 1: Hadronization of a u ¯d component into two mesons. +It is most convenient to write the q¯q matrix in SU(3) in terms of mesons given by Eq. (7) for pseudoscalar mesons, +P, and Eq. (8) for vector mesons, V , +P = +� +� +� +� +� +π0 +√ +2 + +η +√ +3 + η′ +√ +6 +π+ +K+ +π− +− π0 +√ +2 + +η +√ +3 + η′ +√ +6 +K0 +K− +¯K0 +− η +√ +3 + +� +2 +3η′ +� +� +� +� +� , +(7) +V = +� +� +� +� +ρ0 +√ +2 + +ω +√ +2 +ρ+ +K∗+ +ρ− +− ρ0 +√ +2 + +ω +√ +2 +K∗0 +K∗− +¯K∗0 +φ +� +� +� +� . +(8) +We find then for Eq. (6) written in terms of pseudoscalar mesons +u ¯d → +� +i +P1iPi2 = (P 2)12 , +(9) +which gives +u ¯d → +2 +√ +3ηπ+ + K+ ¯K0 +(10) +A rough estimation of the decay width of the new a0 with a mass around 1780 MeV and the a0(980), based solely on +the phase space and decay in S-wave (Γ ∼ pi, the momentum of each meson pair in the decay) gives a ratio of 4.27, +hence we should expect a width for the new a0 of the order of [213 − 427] MeV. Yet, the width of the new a0 is of +the order of 100 MeV, see Eq. (1). +There seems to be something special in these a0 resonances. Indeed, it was already long ago that the a0(980) was +advocated as being a K ¯K molecule [7]. The advent of the chiral unitary approach combining dynamics of chiral +Lagrangians with unitarity in coupled channels [8] made this idea more quantitative, since the a0(980) emerges as a +consequence of the interaction of the coupled channels πη and K ¯K. +The success of the chiral unitary approach describing the low-lying scalar mesons f0(500), a0(980), f0(980), K∗ +0(700), +prompted the extension of these ideas to the interaction of vector mesons in [9], using as a source of interaction of +the vector mesons the local hidden gauge approach of [10]. Interestingly, two resonances were found in the region of +energies discussed here, one with I = 0, f0(1721) with Γ ≃ 133 MeV, which was associated to the f0(1710) and one +with I = 1, a0(1777) with Γ ≃ 148 MeV (we will call it a0(1780) in what follows), for which there was no experimental + +3 +information at the time of its prediction. The f0(1710) couples in that approach to K∗ ¯K∗, ρρ, ωω, ωφ, φφ, while the +a0(1780) couples to the channels K∗ ¯K∗, ρω, ρφ, but the largest coupling in both cases is to the K∗ ¯K∗ component, +which makes the two resonances qualify roughly as K∗ ¯K∗ molecules in analogy to the K ¯K approximate nature of the +a0(980) [8]. Similar conclusions have been reached more recently in [11]. The smaller binding of the a0(1780) comes +as a natural consequence of a weaker potential in I = 1 than in I = 0 [9]. +While we expect that the new experimental finding of the a0(1817) will trigger much theoretical work trying to +understand the nature of the resonance, it is certainly most appealing to consider that this resonance corresponds to +the a0(1780) that was predicted in [9]. The success in other resonances obtained in that approach, which could be +associated to known states, as well as the success in the predictions of the chiral unitary approach in other sectors, +make us to consider this as a most likely option. Next step is to test the theory in experiment. In this direction, the +work done in [12] is illuminating. The mechanisms for D+ +s decay at the quark level corresponding to external emission +were considered as shown in Fig. 2. +¯s +¯s +c +s +W + +¯d +u +π+ +D+ +s +(a) +¯s +¯s +c +s +W + +¯d +u +π+ +D+ +s +(b) +¯s +¯s +c +s +W + +¯d +u +D+ +s +(c) +FIG. 2: Cabibbo-favored decay mode of D+ +s at the quark level with external emission (a); Hadronization of the s¯s component +(b); Hadronization of the ¯du component (c). +The hadronization of s¯s or ¯du components was allowed, following the method of Eqs. (9),(10), producing pairs of +vector mesons. Subdominant internal emission mechanisms were also considered and the different mechanisms exciting +f0(1710) and a0(1780) were identified, and their interference produced a ratio R1 different than unity. One should +mention that while the f0(1710) and a0(1780) resonances are made from vector-vector components, the experiment +measures K ¯K for which the transition of vector-vector to K ¯K must be implemented as depicted in Fig. 3. Fine +K∗ +π +K +¯K∗ +¯K +(a) +φ +¯K +K +ρ, ω, φ +¯K +(b) +FIG. 3: K∗ ¯K∗ → K ¯K transitions driven by π exchange (a) and φ(ρ, ω, φ) → K ¯K transitions driven by K exchange (b). +tunning two free parameters of the theory, the ratio of Eq. (4) could be reproduced in [12]. Yet, the challenge was +then to predict the ratio +R2 = Γ(D+ +s → π0a0(1710)+ → π0K+K0 +S) +Γ(D+ +s → π0“f0(1710)” → π+K+K−) , +(11) +where the numerator corresponds to the π0a+ +0 (1780) production, and a value +Rtheo +2 +≃ 1.31 ± 0.12 + +4 +was obtained, from which the branching fraction +Br[D+ +s → π0a0(1710)+; a0(1710)+ → K+K0 +S] ≃ (1.3 ± 0.4) × 10−3 +(12) +resulted. This was a prediction before this ratio was measured in [1], a fair prediction compared with the branching +fraction reported in [1] of (3.44 ± 0.52 ± 0.32) × 10−3 considering the smallness of the number for a Ds decay and the +amount of modeling required in [1] to extract the a0(1817) signal. Indeed, in the experiment of [1] (see Fig.2 (a) of that +paper), a prominent peak around 1770 MeV is seen in the K0 +SK+ mass distribution, to which there are five channels +contributing, K+ ¯K∗0(892), K0 +SK∗(892), K∗K∗(1410)0, a0(980)+π0, a0(1817)+π0, out of which the a0(1817)π0 is only +a small fraction. On the other hand further developments around the idea of [12] have been carried out very recently +including the D+ +s → K∗+ ¯K0 → π+K0 +SK0 +S mechanism in Ref. [13] and the extra one D+ +s → π0a0(980) → π0K+K0 +S in +[14], by means of which all the mass distributions in the D+ +s → π+K0 +SK0 +S and D+ +s → π0K+K0 +S reactions are all well +reproduced, showing the relevance of the a0(1780) state in the process. +It would be most interesting to devise different reactions where the a0(1817) can be observed and we shall discuss +this issue providing some perspective for future experiments. +1) In order to clean up the spectrum from unwanted contributions of other resonances, one can make cuts in the +mass distributions to eliminate the K∗(892) contributions. This idea has already been used with success in [15] +in the study of the D+ +s → π+π0η decay, looking at the a0(980) contribution, in which a cut was performed to +eliminate the ηρ+ contribution by taking Mπ+π0 > 1 GeV. +2) One can investigate strong decays which are more selective concerning isospin conservation than weak decays, +where generally isospin is not conserved. In this case one can look at different cases. Let us begin by one +reaction where one would have the violation, J/ψ → φK+K−(K0 ¯K0). One can also have ω instead of φ in this +decay. From the perspective of molecular structure for f0(1710) and f0(1780) we have to look for combinations +of one φ and the vector-vector components. This can be done looking for SU(3) invariants of three vectors, given +the fact that J/ψ is an SU(3) singlet. This can be done with the matrix V of Eq. (8), and the invariants are +⟨V V V ⟩, ⟨V ⟩⟨V V ⟩, ⟨V ⟩⟨V ⟩⟨V ⟩, where ⟨ ⟩ indicates the trace in SU(3), but it was shown in [16] and other works +that the ⟨V V V ⟩ structure was prefered by experiments. One finds immediately that the structure coming from +⟨V V V ⟩ containing at least one φ field is +3 (K∗+K∗− + K∗0 ¯K∗0)φ + φφφ , +which leaves the K∗ ¯K∗ and φφ channels to produce the f0(1710). This is done through a mechanism shown in +Fig. 4. The interesting thing is that given the different masses of K∗+, K∗− in the loops of Fig. 4 and in the +J/ψ +φ +K∗, φ +¯K∗, φ +V +V +P +K +¯K +FIG. 4: Mechanism to produce the f0, a0 resonances decaying to K ¯K. +The thick dot indicates the transition T matrix +V V → V V . +construction of the V V → V V scattering matrix, isospin will be slightly broken, but sufficiently to allow the +production of a0(1780) and produce a difference in the K+K−, K0 ¯K0 production in the region of 1780 MeV. +This kind of isospin violation due to different masses of K+, K0 was already emphasized in [17]. Once again, +because of the production of the a0(1780), we expect to have different K+K−, K0 ¯K0 production rates. +We should note that because of the width of the K∗, the effect of the K∗+, K∗0 mass difference should be much +reduced with respect to that in reactions driven by K+K−,K0 ¯K0 intermediate states. Yet, through interference +effect, and provided one has good statistics, observable effects are foreseen. +3) Next one can look at reactions that select the I = 1 component in the decay. Let us take for instance +J/ψ → ρ+K0K− ; +J/ψ → π+K0K− + +5 +This will select necessarily the a0(1780)−. Here we do not have to worry about isospin violation since there is no +f0(1710) with negative charge. The only problem could stem from a contribution of the ρK or πK interaction. +The πK should appear as a strong signal of K∗, but once again this can be removed eliminating this contribution +with cuts in the πK invariant mass. The ρK would show as a K1(1270) resonance peak, but once again one +could remove this contribution with cuts. Alternatively, one can also make the analysis of the reaction with a +partial wave analysis, as done in [1]. +4) Finally let us give another suggestion to further pin down the a0 resonance. As discussed above and in [9], the +a0(1780) resonance couples to K∗ ¯K∗, ρω, ρφ. So far the investigations have been done by looking at K ¯K, which +comes from the K∗ ¯K∗ component. One can look instead to the channels ρω, ρφ. Those channels already filter +an isospin I = 1 and would provide a new look at the a0(1780)(a0(1817)) resonance. The ρω channel is open. +The ρφ channel is closed using the nominal masses of the particles, but allowed considering the ρ width. +5) Apart from the former discussion, with suggestion of new experiments, it is worth mentioning here some of the +relevant ideas that the discovery of the new a0 state has spurred within different theoretical groups. In Ref. +[18] the new resonance has served to classify existing a0 states into a Regge trajectory, predicting that there +should be a new a0 resonance at 2115 MeV. In Ref. [19] the idea of [9] is retaken, showing that the addition of +pseudoscalar-pseudoscalar channels does not significantly alter the results obtained with just the vector-vector +channels. In that work, a discussion is made about how the partnership of the known f0 and a0 resonances is +affected by the discovery of the new resonance, which has repercussion on the on-going discussion about the +possibility of the f0(1500) or f0(1710) states corresponding to glueball states, formed purely from gluons and +their interaction [20]. +As we have shown, the new a0(1817) resonance observed in [1] is an important state that sheds light into the structure +of scalar mesons in the light quark sector and other relevant issues currently under debate in hadron physics. Given +its relevance, the observation of that state in different reactions is very important, as well as the investigation of its +decay channels. In this letter we have offered a perspective to make progress in this direction. +ACKNOWLEDGEMENT +This work is partly supported by the National Natural Science Foundation of China under Grants Nos. 12175066, +11975009, 12147219 and Nos. 11975041, 11735003. This work is also partly supported by the Spanish Ministerio de +Economia y Competitividad (MINECO) and European FEDER funds under Contracts No. FIS2017-84038-C2-1-P +B, PID2020-112777GB-I00, and by Generalitat Valenciana under contract PROMETEO/2020/023. This project has +received funding from the European Union Horizon 2020 research and innovation programme under the program +H2020-INFRAIA-2018-1, grant agreement No. 824093 of the STRONG-2020 project. +[1] BESIII Collaboration, Ablikim M, et al. Observation of an a0-like state with mass of 1.817 GeV in the study of D+ +s → +K0 +SK+π0 decays. Phys Rev Lett 2022; 129:182001 +[2] BABAR Collaboration, Lees JP, et al. Light meson spectroscopy from Dalitz plot analyses of ηc decays to η′K+K−, +η′π+π−, and ηπ+π− produced in two-photon interactions. Phys Rev D 2021; 104: 072002 +[3] Chen HX, Chen W, Liu X, Liu YR, Zhu SL. An updated review of the new hadron states. Rept Prog Phys 2023; 86: +026201 +[4] Particle Data Group, Workman RL, et al. The review of particle physics. Prog Theor Exp Phys 2022;2022:083C01 +[5] BESIII Collaboration, Ablikim M, et al. Amplitude analysis and branching fraction measurement of D+ +s → K+K−π+. +Phys Rev D 2021;104:012016 +[6] BESIII Collaboration, Ablikim M, et al. Study of the decay D+ +s → K0 +SK0 +Sπ+ and observation of an isovector partner to +f0(1710). Phys Rev D 2022; 105: L051103 +[7] Weinstein J and Isgur N. K ¯K molecules. Phys Rev D 1990;41:2236 +[8] Oller JA and Oset E. Chiral symmetry amplitudes in the S wave isoscalar and isovector channels and the σ, f0(980), +a0(980) scalar mesons. Nucl Phys A 1997;620:438 +[9] Geng LS and Oset E. Vector meson-vector meson interaction in a hidden gauge unitary approach. Phys Rev D +2009;79:074009 +[10] Bando M, Kugo T, Yamawaki K. Nonlinear realization and hidden local symmetries. Phys Rept 1988;164:217 +[11] Du ML, G¨ulmez D, Guo FK, Meißner UG, Wang Q. Interactions between vector mesons and dynamically generated +resonances. Eur Phys J C 2018; 78: 988 + +6 +[12] Dai LR, Oset E, Geng LS. The D+ +s → π+K0 +SK0 +S reaction and the I = partner of the f0(1710) state. Eur Phys J C 2022; +82: 225 +[13] Zhu X, Li DM, Wang E, Geng LS, and Xie JJ. Theoretical study of the process D+ +s → π+K0 +SK0 +S and the isovector partner +of f0(1710). Phys Rev D 2022;105:116010 +[14] Zhu X, Wang HN, Li DM, Wang E, Geng LS, Xie JJ. Further understanding the nature of a0(1710) in the D+ +s → π0K+K0 +S +decay. arXiv:2210.12992 [hep-ph] +[15] BESIII Collaboration, Ablikim M, et al. Amplitude analysis of D+ +s → π+π0η and first observation of the W-annihilation +dominant decays D+ +s → a0(980)+π0 and D+ +s → a0(980)0π+. Phys Rev Lett 2019; 123: 112001 +[16] Debastiani VR, Liang WH, Xie JJ, Oset E. Predictions for ηc → ηπ+π− producing f0(500), f0(980) and a0(980). Phys +Lett B 2017; 766: 59 +[17] Achasov NN, Devyanin SA, Shestakov GN. S∗-δ0 mixing as a threshold phenomenon. Phys Lett B 1979; 88: 367 +[18] Guo D, Chen W, Chen HX, Liu X, and Zhu SL. Newly observed a0(1817) as the scaling point of constructing the scalar +meson spectroscopy. Phys Rev D 2022;105:114014 +[19] Wang ZL and Zou BS. Two dynamical generated a0 resonances by interactions between vector mesons. Eur Phys J C 2022; +82: 509 +[20] Amsler C and Close FE. Is f0(1500) a scalar glueball? Phys Rev D 1996; 53: 295 + diff --git a/a9FAT4oBgHgl3EQfXR0t/content/tmp_files/load_file.txt b/a9FAT4oBgHgl3EQfXR0t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f04d0679fb74b239ac526506f919629082f07968 --- /dev/null +++ b/a9FAT4oBgHgl3EQfXR0t/content/tmp_files/load_file.txt @@ -0,0 +1,214 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf,len=213 +page_content='Repercussion of the a0(1710) [a0(1817)] resonance and future developments E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Oset,1, ∗ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Dai,2, † and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Geng3, ‡ 1Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigaci´on de Paterna, Aptdo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='22085, 46071 Valencia, Spain 2School of Science, Huzhou University, Huzhou 313000, Zhejiang, China 3School of Physics, Beihang University, Beijing 102206, China (Dated: January 23, 2023) In a recent paper the BESIII Collaboration reported the observation of a scalar meson of spin-parity JP = 0+ with isospin I = 1, branded as a0(1817).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The state is seen as a peak in the K0 SK+ mass distribution in the D+ s → K0 SK+π0 decay [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Its mass and width are reported as Ma0 = 1817 ± 8stat ± 20sys MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Γa0 = 97 ± 22stat ± 15sys MeV (1) Prior to this finding, a BABAR experiment reported an a0(1710) state from the π+η, π−η mass distributions in the ηc → π+π−η decay [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The mass and width of this state are obtained as M = 1709 ± 5stat ± 2sys MeV, Γ = 110 ± 152stat ± 11sys MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The new a0 resonance in the sector of light quarks comes as a real surprise at a moment when, however, many new mesonic states are obtained in the heavy quark sector [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' An isospin I = 0, f0(1710) resonance has, however, been known for quite some time [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Another BESIII experiment reporting on D+ s → π+f0(1710) has brought new elements to this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Indeed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' [5] it was found the branching fraction Br[D+ s → π+“f0(1710)”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' “f0(1710)” → K+K−] = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='3) × 10−3 , (2) while from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' [6] it was found that Br[D+ s → π+“f0(1710)”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' “f0(1710)” → K0 SK0 S] = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='1) × 10−3 , (3) where “f0(1710)” was supposed to be the f0(1710) resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Yet, it was concluded that it could not be, because from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (2), (3) one finds R1 = Γ(D+ s → π+“f0(1710)” → π+K0 ¯K0) Γ(D+ s → π+“f0(1710)” → π+K+K−) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='67 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (4) If “f0(1710)” was the f0(1710) resonance this latter ratio should be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' It was concluded that hidden below, or around the f0(1710), there should be an I = 1 resonance responsible for this surprising large ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Indeed, since |K ¯K, I = 0⟩ = − 1 √ 2 � K0 ¯K0 + K+K−� , |K ¯K, I = 1, I3 = 0⟩ = 1 √ 2 � K0 ¯K0 − K+K−� , (5) there could be a mixture of the two resonances and their interference would be responsible for a different K+K− or K0 ¯K0 production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' It is clear that the three experiments are seeing an I = 1 resonance in the region [1700 − 1800] MeV and it is unlikely that they correspond to three different resonances, but this is something to be settled by further experiments to which we will come back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Below this energy there is one a0 resonance very well known, the a0(980) resonance, which decays to πη, with an apparent width of about [50 − 100] MeV [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' ∗Electronic address: oset@ific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='uv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='es †Electronic address: dailianrong@zjhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='cn ‡Electronic address: lisheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='geng@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='cn arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='08532v1 [hep-ph] 20 Jan 2023 2 From the standard q¯q quark model point of view the a+ 0 (980) would be u ¯d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' It might be surprising that a state like that with no strange quarks decays to K ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The answer to this lies in the hadronization of the u ¯d which gets attached to a ¯qq state with the quantum numbers of the vacuum via u ¯d → � i u ¯qiqi ¯d = u(¯uu + ¯dd + ¯ss) ¯d , (6) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' ¯d u ¯qi qi FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 1: Hadronization of a u ¯d component into two mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' It is most convenient to write the q¯q matrix in SU(3) in terms of mesons given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (7) for pseudoscalar mesons, P, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (8) for vector mesons, V , P = � � � � � π0 √ 2 + η √ 3 + η′ √ 6 π+ K+ π− − π0 √ 2 + η √ 3 + η′ √ 6 K0 K− ¯K0 − η √ 3 + � 2 3η′ � � � � � , (7) V = � � � � ρ0 √ 2 + ω √ 2 ρ+ K∗+ ρ− − ρ0 √ 2 + ω √ 2 K∗0 K∗− ¯K∗0 φ � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (8) We find then for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (6) written in terms of pseudoscalar mesons u ¯d → � i P1iPi2 = (P 2)12 , (9) which gives u ¯d → 2 √ 3ηπ+ + K+ ¯K0 (10) A rough estimation of the decay width of the new a0 with a mass around 1780 MeV and the a0(980), based solely on the phase space and decay in S-wave (Γ ∼ pi, the momentum of each meson pair in the decay) gives a ratio of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='27, hence we should expect a width for the new a0 of the order of [213 − 427] MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Yet, the width of the new a0 is of the order of 100 MeV, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' There seems to be something special in these a0 resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Indeed, it was already long ago that the a0(980) was advocated as being a K ¯K molecule [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The advent of the chiral unitary approach combining dynamics of chiral Lagrangians with unitarity in coupled channels [8] made this idea more quantitative, since the a0(980) emerges as a consequence of the interaction of the coupled channels πη and K ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The success of the chiral unitary approach describing the low-lying scalar mesons f0(500), a0(980), f0(980), K∗ 0(700), prompted the extension of these ideas to the interaction of vector mesons in [9], using as a source of interaction of the vector mesons the local hidden gauge approach of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Interestingly, two resonances were found in the region of energies discussed here, one with I = 0, f0(1721) with Γ ≃ 133 MeV, which was associated to the f0(1710) and one with I = 1, a0(1777) with Γ ≃ 148 MeV (we will call it a0(1780) in what follows), for which there was no experimental 3 information at the time of its prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The f0(1710) couples in that approach to K∗ ¯K∗, ρρ, ωω, ωφ, φφ, while the a0(1780) couples to the channels K∗ ¯K∗, ρω, ρφ, but the largest coupling in both cases is to the K∗ ¯K∗ component, which makes the two resonances qualify roughly as K∗ ¯K∗ molecules in analogy to the K ¯K approximate nature of the a0(980) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Similar conclusions have been reached more recently in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The smaller binding of the a0(1780) comes as a natural consequence of a weaker potential in I = 1 than in I = 0 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' While we expect that the new experimental finding of the a0(1817) will trigger much theoretical work trying to understand the nature of the resonance, it is certainly most appealing to consider that this resonance corresponds to the a0(1780) that was predicted in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The success in other resonances obtained in that approach, which could be associated to known states, as well as the success in the predictions of the chiral unitary approach in other sectors, make us to consider this as a most likely option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Next step is to test the theory in experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' In this direction, the work done in [12] is illuminating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The mechanisms for D+ s decay at the quark level corresponding to external emission were considered as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' ¯s ¯s c s W + ¯d u π+ D+ s (a) ¯s ¯s c s W + ¯d u π+ D+ s (b) ¯s ¯s c s W + ¯d u D+ s (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 2: Cabibbo-favored decay mode of D+ s at the quark level with external emission (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Hadronization of the s¯s component (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Hadronization of the ¯du component (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The hadronization of s¯s or ¯du components was allowed, following the method of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (9),(10), producing pairs of vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Subdominant internal emission mechanisms were also considered and the different mechanisms exciting f0(1710) and a0(1780) were identified, and their interference produced a ratio R1 different than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' One should mention that while the f0(1710) and a0(1780) resonances are made from vector-vector components, the experiment measures K ¯K for which the transition of vector-vector to K ¯K must be implemented as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Fine K∗ π K ¯K∗ ¯K (a) φ ¯K K ρ, ω, φ ¯K (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 3: K∗ ¯K∗ → K ¯K transitions driven by π exchange (a) and φ(ρ, ω, φ) → K ¯K transitions driven by K exchange (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' tunning two free parameters of the theory, the ratio of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (4) could be reproduced in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Yet, the challenge was then to predict the ratio R2 = Γ(D+ s → π0a0(1710)+ → π0K+K0 S) Γ(D+ s → π0“f0(1710)” → π+K+K−) , (11) where the numerator corresponds to the π0a+ 0 (1780) production, and a value Rtheo 2 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='12 4 was obtained, from which the branching fraction Br[D+ s → π0a0(1710)+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' a0(1710)+ → K+K0 S] ≃ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='4) × 10−3 (12) resulted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This was a prediction before this ratio was measured in [1], a fair prediction compared with the branching fraction reported in [1] of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='32) × 10−3 considering the smallness of the number for a Ds decay and the amount of modeling required in [1] to extract the a0(1817) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Indeed, in the experiment of [1] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='2 (a) of that paper), a prominent peak around 1770 MeV is seen in the K0 SK+ mass distribution, to which there are five channels contributing, K+ ¯K∗0(892), K0 SK∗(892), K∗K∗(1410)0, a0(980)+π0, a0(1817)+π0, out of which the a0(1817)π0 is only a small fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' On the other hand further developments around the idea of [12] have been carried out very recently including the D+ s → K∗+ ¯K0 → π+K0 SK0 S mechanism in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' [13] and the extra one D+ s → π0a0(980) → π0K+K0 S in [14], by means of which all the mass distributions in the D+ s → π+K0 SK0 S and D+ s → π0K+K0 S reactions are all well reproduced, showing the relevance of the a0(1780) state in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' It would be most interesting to devise different reactions where the a0(1817) can be observed and we shall discuss this issue providing some perspective for future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 1) In order to clean up the spectrum from unwanted contributions of other resonances, one can make cuts in the mass distributions to eliminate the K∗(892) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This idea has already been used with success in [15] in the study of the D+ s → π+π0η decay, looking at the a0(980) contribution, in which a cut was performed to eliminate the ηρ+ contribution by taking Mπ+π0 > 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 2) One can investigate strong decays which are more selective concerning isospin conservation than weak decays, where generally isospin is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' In this case one can look at different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Let us begin by one reaction where one would have the violation, J/ψ → φK+K−(K0 ¯K0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' One can also have ω instead of φ in this decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' From the perspective of molecular structure for f0(1710) and f0(1780) we have to look for combinations of one φ and the vector-vector components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This can be done looking for SU(3) invariants of three vectors, given the fact that J/ψ is an SU(3) singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This can be done with the matrix V of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' (8), and the invariants are ⟨V V V ⟩, ⟨V ⟩⟨V V ⟩, ⟨V ⟩⟨V ⟩⟨V ⟩, where ⟨ ⟩ indicates the trace in SU(3), but it was shown in [16] and other works that the ⟨V V V ⟩ structure was prefered by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' One finds immediately that the structure coming from ⟨V V V ⟩ containing at least one φ field is 3 (K∗+K∗− + K∗0 ¯K∗0)φ + φφφ , which leaves the K∗ ¯K∗ and φφ channels to produce the f0(1710).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This is done through a mechanism shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The interesting thing is that given the different masses of K∗+, K∗− in the loops of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 4 and in the J/ψ φ K∗, φ ¯K∗, φ V V P K ¯K FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 4: Mechanism to produce the f0, a0 resonances decaying to K ¯K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The thick dot indicates the transition T matrix V V → V V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' construction of the V V → V V scattering matrix, isospin will be slightly broken, but sufficiently to allow the production of a0(1780) and produce a difference in the K+K−, K0 ¯K0 production in the region of 1780 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This kind of isospin violation due to different masses of K+, K0 was already emphasized in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Once again, because of the production of the a0(1780), we expect to have different K+K−, K0 ¯K0 production rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' We should note that because of the width of the K∗, the effect of the K∗+, K∗0 mass difference should be much reduced with respect to that in reactions driven by K+K−,K0 ¯K0 intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Yet, through interference effect, and provided one has good statistics, observable effects are foreseen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 3) Next one can look at reactions that select the I = 1 component in the decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Let us take for instance J/ψ → ρ+K0K− ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' J/ψ → π+K0K− 5 This will select necessarily the a0(1780)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Here we do not have to worry about isospin violation since there is no f0(1710) with negative charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The only problem could stem from a contribution of the ρK or πK interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The πK should appear as a strong signal of K∗, but once again this can be removed eliminating this contribution with cuts in the πK invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The ρK would show as a K1(1270) resonance peak, but once again one could remove this contribution with cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Alternatively, one can also make the analysis of the reaction with a partial wave analysis, as done in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 4) Finally let us give another suggestion to further pin down the a0 resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' As discussed above and in [9], the a0(1780) resonance couples to K∗ ¯K∗, ρω, ρφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' So far the investigations have been done by looking at K ¯K, which comes from the K∗ ¯K∗ component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' One can look instead to the channels ρω, ρφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Those channels already filter an isospin I = 1 and would provide a new look at the a0(1780)(a0(1817)) resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The ρω channel is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' The ρφ channel is closed using the nominal masses of the particles, but allowed considering the ρ width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 5) Apart from the former discussion, with suggestion of new experiments, it is worth mentioning here some of the relevant ideas that the discovery of the new a0 state has spurred within different theoretical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' [18] the new resonance has served to classify existing a0 states into a Regge trajectory, predicting that there should be a new a0 resonance at 2115 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' [19] the idea of [9] is retaken, showing that the addition of pseudoscalar-pseudoscalar channels does not significantly alter the results obtained with just the vector-vector channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' In that work, a discussion is made about how the partnership of the known f0 and a0 resonances is affected by the discovery of the new resonance, which has repercussion on the on-going discussion about the possibility of the f0(1500) or f0(1710) states corresponding to glueball states, formed purely from gluons and their interaction [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' As we have shown, the new a0(1817) resonance observed in [1] is an important state that sheds light into the structure of scalar mesons in the light quark sector and other relevant issues currently under debate in hadron physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Given its relevance, the observation of that state in different reactions is very important, as well as the investigation of its decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' In this letter we have offered a perspective to make progress in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work is partly supported by the National Natural Science Foundation of China under Grants Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 12175066, 11975009, 12147219 and Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 11975041, 11735003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This work is also partly supported by the Spanish Ministerio de Economia y Competitividad (MINECO) and European FEDER funds under Contracts No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' FIS2017-84038-C2-1-P B, PID2020-112777GB-I00, and by Generalitat Valenciana under contract PROMETEO/2020/023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' This project has received funding from the European Union Horizon 2020 research and innovation programme under the program H2020-INFRAIA-2018-1, grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 824093 of the STRONG-2020 project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' [1] BESIII Collaboration, Ablikim M, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Observation of an a0-like state with mass of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='817 GeV in the study of D+ s → K0 SK+π0 decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Phys Rev Lett 2022;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Phys Lett B 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 88: 367 [18] Guo D, Chen W, Chen HX, Liu X, and Zhu SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Newly observed a0(1817) as the scaling point of constructing the scalar meson spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Phys Rev D 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content='105:114014 [19] Wang ZL and Zou BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Two dynamical generated a0 resonances by interactions between vector mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Eur Phys J C 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 82: 509 [20] Amsler C and Close FE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Is f0(1500) a scalar glueball?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' Phys Rev D 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} +page_content=' 53: 295' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9FAT4oBgHgl3EQfXR0t/content/2301.08532v1.pdf'} diff --git a/ddAzT4oBgHgl3EQfLvvB/content/tmp_files/2301.01121v1.pdf.txt b/ddAzT4oBgHgl3EQfLvvB/content/tmp_files/2301.01121v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..07db695af2572d5e78e0ef4c56e7c351397f78d9 --- /dev/null +++ b/ddAzT4oBgHgl3EQfLvvB/content/tmp_files/2301.01121v1.pdf.txt @@ -0,0 +1,4425 @@ +arXiv:2301.01121v1 [math.AT] 3 Jan 2023 +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +MICHAEL BORINSKY AND KAREN VOGTMANN +Abstract. The moduli space of rank n graphs, the outer automorphism group of the free +group of rank n and Kontsevich’s Lie graph complex have the same rational cohomology. We +show that the associated Euler characteristic grows like −e−1/4 (n/e)n/(n log n)2 as n goes to +infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n. +1. Introduction +The moduli space MGn of finite metric graphs with fundamental group Fn was introduced in +[17] as a tool for studying the group Out(Fn) of outer automorphisms of a free group. By the +main result in that paper MGn is the quotient of a contractible space, called Outer space, on +which Out(Fn) acts with finite stabilizers. Thus the homology of Out(Fn) with trivial rational +coefficients is equal to the homology of MGn. +Kontsevich showed in [22, 23] that the homology of MGn can also be identified with the +cohomology of his Lie graph complex, which can in turn be identified with the primitive part of +the cohomology of the Lie algebra of symplectic derivations of a free Lie algebra (see [15] for a +detailed exposition of Kontsevich’s results). In [5] Berglund and Madsen found this Lie algebra +in a very different context, and proved that its cohomology is a sub-algebra of the cohomology of +the block diffeomorphism group of even-dimensional products of spheres. In more recent years, +algebraic geometers have studied MGn as a tropical analog of the classical moduli space Mn +of smooth complex curves of genus n. The simplicial completion of Outer space descends to a +natural compactification of MGn, which the tropical geometers have dubbed the moduli space +of tropical curves, by analogy with the Deligne-Mumford compactification of Mn (see, e.g., +[13]). In yet another context, MGn may be considered a natural parameter space for the n-loop +contribution to certain Feynman amplitudes. This direction has been explored, for example, by +Bloch, Berghoff and Kreimer [7, 4]. +In this paper we prove a formula for the Euler characteristic of MGn, and then determine +its asymptotic growth rate. The asymptotic result depends on our results in [10], where we +determined the asymptotic growth rate of the rational or virtual Euler characteristic χ(Out(Fn)). +This is a rational number closely related to the alternation sum of the Betti numbers, but +which has better group-theoretic properties, making it easier to compute. The rational Euler +characteristic of Out(Fn) coincides with the number Kontsevich referred to as the orbifold Euler +characteristic of his Lie graph complex. The actual alternating sum of the Betti numbers is +denoted e(Out(Fn)) in this paper to distinguish it from χ(Out(Fn)), and is called the integral +Euler characteristic to conform with other terminology in the literature. +If we are primarily interested in the cohomology of the space MGn or (equivalently) the group +Out(Fn), then the number e(Out(Fn)) = e(MGn) is clearly more relevant. Brown [12] showed +that the rational and integral Euler characteristics of a group Γ are closely related, namely e(Γ) +can be calculated from the rational Euler characteristics of centralizers of finite-order elements +by the formula +e(Γ) = +� +[α] +χ(C(α)). +1 + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +2 +Here the sum is over representatives α for the conjugacy classes of finite-order elements (including +the identity), and C(α) is the centralizer of α. +The number e(Out(Fn)) was calculated for n ≤ 11 by Morita, Sakasai and Suzuki [25], using +methods from symplectic representation theory. In the present paper we use Brown’s formula, +results on centralizers from [24], an adaptation of Joyal’s theory of species [20] and further +development of the asymptotic methods of [10] to first give an effective formula for e(Out(Fn)) +and then to determine its asymptotic growth rate. +The effective formula is developed in Sections 2-3 and summarized in Theorem 3.12. Based +on it, we wrote a computer program to compute e(Out(Fn)) for n ≤ 15. The results are listed in +Appendix A. Further optimizations of this program by Jos Vermaseren enabled the computation +of the numbers e(Out(Fn)) for all n ≤ 100 [9]. +Section 4 is devoted to proving the following asymptotic result. +Theorem 1.1. The integral Euler characteristic e(Out(Fn)) has the asymptotic behaviour +e(Out(Fn)) ∼ −e− 1 +4 +�n +e +�n +1 +(n log n)2 as n → ∞. +Here the notation an ∼ bn means that limn→∞ an/bn = 1. In particular this verifies the +fact, suggested by our results on the rational Euler characteristic in [10], that there is a huge +amount of cohomology in odd dimensions. Since the cohomology of Out(Fn) is a direct summand +of the cohomology of Aut(Fn), we reach the same conclusion for Aut(Fn). +The cohomology +Hk(Out(Fn); Q) is known to vanish for both k < 4n/5 and k > 2n−3 (see [14] and the references +there), thus all of this cohomology must be concentrated in dimensions 4n/5 ≤ k ≤ 2n − 3. The +only odd-dimensional class known to date occurs in H11(Out(F8)) [2]. It is interesting to note +that this is the largest possible dimension, the virtual cohomological dimension of Out(F8). +This is in contrast to the fact that the groups GLn(Z) and mapping class groups of punctured +surfaces, both of which are often considered analogs of Out(Fn), have no rational cohomology in +their virtual cohomological dimension. +Comparing Theorem 1.1 with our results in [10] on χ(Out(Fn)) gives +Corollary 1.2. The ratio limn→∞ e(Out(Fn))/χ(Out(Fn)) = e− 1 +4 ≈ 0.78. +Proof. Use Theorem 1.1, [10, Thm. A] and Stirling’s formula (see, e.g., [1, Eq. (3.9)]). +□ +This solves Problem 6.5 of the paper [25] by Morita, Sakasai and Suzuki. +More precise +asymptotic statements and comments about the rate of convergence are given in Section 4.1. +In his original paper [22] Kontsevich introduced commutative and associative graph complexes +in addition to the Lie graph complex. The methods of the present paper can be modified to +compute the Euler characteristics for both of these other graph complexes as well as to determine +their asymptotic behavior (see [8]); they can also be used to do the same for other moduli spaces +of graphs, such as colored graphs or graphs with leaves. As Kontsevich noted, the associative +graph complex computes the homology of mapping class groups of punctured surfaces. Both +the rational and integral Euler characteristics of these groups for a once-punctured surface were +originally computed by Harer and Zagier [19]. They also computed the asymptotics and deduced +the existence of lots of cohomology. Getzler and Kapranov partially extended this to surfaces +with more than one puncture (without determining the asymptotics) as an application of their +general theory of modular operads [18], and we remark that our method of finding the generating +function is similar to theirs. +In [22] Kontsevich also defined odd versions of his graph complexes, and noted that in the +Lie case the primitive part of the homology computes the cohomology of Out(Fn) with twisted +coefficients �Q, where the twisting is given by composing the natural map from Out(Fn) to GLn(Z) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +3 +with the determinant map. This odd version of Lie graph homology occurs, for example, in the +study of diffeomorphism groups of odd-dimensional products of spheres [6]. In a final section, +Section 5, we explain how to modify our results to compute the Euler characteristic of this odd +Lie graph complex, which we denote eodd(Out(Fn)). The results also extend to the analysis of +the asymptotics, and we find +Theorem 1.3. The ratio limn→∞ eodd(Out(Fn))/χ(Out(Fn)) = e +1 +4 ≈ 1.28. +Acknowledgments +We are grateful to Jos Vermaseren for generous FORM programming help and to Thomas +Willwacher for illuminating discussions. MB was supported by Dr. Max Rössler, the Walter +Haefner Foundation and the ETH Zürich Foundation. +2. A first formula for e(Out(Fn)) +2.1. The rational Euler characteristic. As noted in the introduction, Brown’s theorem says +we can compute e(Out(Fn)) by adding up the rational Euler characteristics of centralizers of +finite-order elements. One way to compute the rational Euler characteristic of a group Γ is to +find a contractible cell complex Y on which Γ acts properly and cocompactly; the rational Euler +characteristic χ(Γ) is then given by the formula +χ(Γ) = +� +[σ] +(−1)dim(σ) +|Stab(σ)| , +where we sum over all orbits [σ] of cells, σ is a representative from the orbit and Stab(σ) is +its stabilizer under the group action. Fortunately, we have such complexes Y for centralizers of +finite-order elements of Out(Fn). +The entire group Out(Fn) centralizes the identity. Recall from [17] that Out(Fn) acts properly +and cocompactly on the spine Kn of Outer space. Kn is a contractible cube complex with one +k-dimensional cube for each equivalence class [G, Φ, g] of triples (G, Φ, g), where +• (G, Φ) is a connected forested graph, and +• g: Fn → π1(G) is an isomorphism, called a marking. +Here by a graph we mean a CW-complex of dimension 0 or 1, a forest is a graph without cycles, +and a subforest of G is a subcomplex that is a forest and contains all of the vertices of G. We say a +graph is of rank n if its fundamental group is of rank n. A graph is admissible if it has no isolated, +univalent or bivalent vertices, and a forested graph is a pair (G, Φ) consisting of an admissible +graph G and a subforest Φ. Two triples (G, Φ, g) and (G′, Φ′, g′) are equivalent if there is a graph +isomorphism h: G → G′ sending Φ to Φ′ and inducing an isomorphism h∗ : π(G) → π(G′), such +that g−1h∗g = id . +The spine K = Kn is contractible, and the action of Out(Fn) on K simply changes the marking +g. Thus there is one orbit for each isomorphism class [G, Φ] of forested graphs. The stabilizer of +a cube [G, Φ, g] is isomorphic to Aut(G, Φ), the automorphisms of G that preserve Φ. Thus, +χ(Out(Fn)) = χ(C(id)) = +� +[G,Φ] +(−1)e(Φ) +|Aut(G, Φ)|, +where e(Φ) is the number of edges in Φ. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +4 +2.2. The equivariant spine. In [24] an equivariant version of K was introduced, which can be +used to study the centralizer of any finite order element of Out(Fn) (in fact the centralizer of +any finite-order subgroup). We briefly summarize the construction. A graph G is said to realize +a finite-order automorphism α if there is some marking g: Fn → π1(G) and automorphism fα +of G such that g−1(fα)∗g = α. Every finite-order element of Out(Fn) can be realized on some +admissible graph G, [16, 21]. This translates to the statement that the action of α on K has +at least one fixed point. The centralizer C(α) acts on the entire fixed-point set Kα of α. It +follows from [24] that this fixed point set is contractible, cocompact and has the structure of a +cube complex. Specifically, Kα has one cube for each equivalence class of triples (G, Φ, g)α where +(G, g) realizes α and Φ is a (possibly empty) forest in G that is invariant under the action of α. +Here (G, Φ, g)α is equivalent to (G′, Φ′, g′)α if there is an α-invariant automorphism h: G → G′ +sending Φ to Φ′ such that g−1h∗g = id, and again we write [G, Φ, g]α for the equivalence class. +The dimension of the cube [G, Φ, g]α is the number eα(Φ) of edge-orbits in Φ. +The stabilizer Stab[G, Φ, g]α is isomorphic to Autα(G, Φ), the automorphisms of G that com- +mute with the action of α and send Φ to itself, so +χ(C(α)) = +� +[G,Φ]α +(−1)eα(Φ) +|Autα(G, Φ)|, +where [G, Φ]α runs over isomorphism classes of pairs that realize α with a connected graph of +rank n. Brown’s theorem [12] then gives +e(Out(Fn)) = +� +[α] +� +[G,Φ]α +(−1)eα(Φ) +|Autα(G, Φ)|, +where [α] runs over conjugacy classes of finite-order elements α. +The group Aut(G, Φ) acts on itself by conjugation, so the orbit-stabilizer theorem gives +|Aut(G, Φ)| = |(orbit of α)| · |StabAut(G,Φ)(α)|. Since StabAut(G,Φ)(α) = Autα(G, Φ) this gives: +Theorem 2.1. +e(Out(Fn)) = +� +[G,Φ] +1 +|Aut(G, Φ)| +� +α∈Aut(G,Φ) +(−1)eα(Φ), +where we sum over all isomorphism classes [G, Φ] of connected forested graphs of rank n. +2.3. Disconnected graphs. Recall that the Euler characteristic of a connected graph of rank +n is χ(G) = 1 − n. The graph’s Euler characteristic is often better behaved than its rank. For +instance, the formula in Theorem 2.1 is easier to work with if we drop the requirement that G +be connected and shift the index by one, i.e. define +�en = +� +[G,Φ] +1 +|Aut(G, Φ)| +� +α∈Aut(G,Φ) +(−1)eα(Φ) +(1) +where we sum over all isomorphism classes [G, Φ] of (not necessarily connected) forested graphs +with χ(G) = −n. In this section we show how to recover e(Out(Fn+1)) once the numbers �en are +known. +We begin by deriving new formulas for e(Out(Fn+1)) and �en. Define a forested graph (G, Φ) +to be even if G has no automorphisms that induce an odd permutation of the edges of Φ. +Proposition 2.2. If we sum over all isomorphism classes [G, Φ] — +(1) —of even forested graphs with χ(G) = −n, then +� +[G,Φ] +(−1)e(Φ) = �en . + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +5 +(2) —of connected even forested graphs with χ(G) = −n, then +� +[G,Φ] +(−1)e(Φ) = e(Out(Fn+1)). +Remark 2.3. Conant and Vogtmann showed in [15] that Kontsevich’s Lie graph complex, as +defined in [22] is quasi-isomorphic to the complex spanned by even forested graphs. Thus this +propositions shows that e(Out(Fn)) is equal to the Euler characteristic of the Lie graph complex. +Proposition 2.2 is an immediate consequence of the following lemma. +Lemma 2.4. The sum � +α∈Aut(G,Φ)(−1)eα(Φ) vanishes if (G, Φ) has an automorphism that +induces an odd permutation on Φ and is equal to (−1)e(Φ)|Aut(G, Φ)| otherwise. +Proof. An element α ∈ Aut(G, Φ) induces a permutation αΦ ∈ Se(Φ) on the set of edges of Φ. +By definition, the number eα(Φ) is equal to the number of cycles of in the cycle decomposition +of αΦ. +The sign of a permutation is the parity of the number of its even cycles. Since the +parity of the number of odd cycles of a permutation on an n-element set is equal to the parity +of n, we have (−1)eα(Φ) = sign(αΦ)(−1)e(Φ). The sign function gives a homomorphism from +Aut(G, Φ) to the cyclic group of order 2, which is surjective if and only if Aut(G, Φ) contains an +odd permutation. If it is surjective, then half of the elements of Aut(G, Φ) have each sign, so +� +α∈Aut(G,Φ) sign(αΦ) = 0; otherwise � +α∈Aut(G,Φ) sign(αΦ) = (−1)e(Φ)|Aut(G, Φ)|. +□ +Proof of Proposition 2.2. Use Lemma 2.4, Eq. (1) and Theorem 2.1. +□ +Each pair consisting of a forested graph and an automorphism contributes to the sum in the +definition of �en, so evaluating this sum means doing a weighted count of forested graphs and +their automorphisms. We will do this counting by means of generating functions, i.e. formal +power series whose coefficients encode the counts we are interested in. +We can also use formal power series to describe the relation between �en and e(Out(Fn+1)). +By standard topological quantum field theory convention, we use the symbol ℏ as the formal +variable that marks the negative Euler characteristic of the graphs. +Theorem 2.5. +� +n≥0 +�en ℏn = +∞ +� +n=1 +� +1 +1 − ℏn +�e(Out(Fn+1)) +. +Proof. For any (possibly disconnected) admissible graph G and subforest Φ ⊂ G, set θ(G, Φ) = +(−1)e(Φ)ℏ−χ(G). Since an admissible graph is either trivial or has strictly negative Euler charac- +teristic, θ(G, Φ) ∈ Q[[ℏ]]. The function θ factors over connected components, i.e. +θ((G1, Φ1) ⊔ (G2, Φ2)) = θ(G1, Φ1)θ(G2, Φ2). +By Proposition 2.2(1), +� +n≥0 +�en ℏn = +� +[G,Φ] even +θ(G, Φ), +where we sum over all isomorphism classes of (possibly disconnected) even forested graphs [G, Φ]. +Each isomorphism class [G, Φ] can be described by giving a set of isomorphism classes of +connected graphs together with the multiplicity with which each connected class appears in the +disconnected class. +In the sum above, a component [g, ϕ] of [G, Φ] such that ϕ has an odd +number of edges can appear at most once, as otherwise [G, Φ] would have an odd automorphism. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +6 +Components with an even number of edges in the forest can appear with any multiplicity. The +sum of the function θ over all even admissible forested graphs hence satisfies the following identity: +� +n≥0 +�en ℏn = +� +[G,Φ] even +θ(G, Φ) = + + + + +� +[g,ϕ] +e(ϕ) odd +(1 + θ(g, ϕ)) + + + + + + + + +� +[g,ϕ] +e(ϕ) even +� +m≥0 +θ(g, ϕ)m + + + + +where the [g, ϕ] are isomorphism classes of connected forested graphs and e(ϕ) denotes the +number of edges of ϕ. +Using the fact that θ(g, ϕ) is −ℏ−χ(g) if [g, ϕ] is odd and ℏ−χ(g) if [g, ϕ] is even and evaluating +the sum over m, we obtain +� +n≥0 +�en ℏn = + + + + +� +[g,ϕ] +e(ϕ) odd +� +1 − ℏ−χ(g)� + + + + + + + + +� +[g,ϕ] +e(ϕ) even +1 +1 − ℏ−χ(g) + + + + = +� +n≥1 +(1 − ℏn)βn +odd +(1 − ℏn)βn +even , +where βn +odd and βn +even are the numbers of connected forested admissible graphs without odd edge- +automorphisms with Euler characteristic −n with an odd or even number of edges in the forest. +A connected graph with Euler characteristic −n has rank n + 1. Hence, by Proposition 2.2(2), +βn +even − βn +odd = e(Out(Fn+1)). +□ +We can now explain how to calculate the numbers e(Out(Fn)) recursively from the numbers +�en. Recall that the classical Möbius function µ is defined recursively for positive integers by +µ(1) = 1 and � +d|n µ(d) = 0 for all n ≥ 2. +Corollary 2.6. Let � +n≥1 en ℏn = log f(ℏ) be the logarithm of the series f(ℏ) = � +n≥0 �en ℏn. +The numbers en for n ≥ 1 are given recursively by +en = �en − 1 +n +n−1 +� +k=1 +k ek �en−k +and e(Out(Fn+1)) for n ≥ 1 by +e(Out(Fn+1)) = +� +d|n +µ(d) +d +en/d . +Proof. The recursive expression for en in terms of �en follows by taking the derivative of log f(x) +with respect to ℏ and using (log f(x))′ = f ′(x)/f(x). Taking the logarithm of the statement of +Theorem 2.5, using log(1/(1 − x)) = �∞ +n=1 xn/n and the definition of the Möbius function gives +the second formula. +□ +3. An effective formula for e(Out(Fn)) +3.1. Admissible trees. In the last section we reduced the problem of computing e(Out(Fn)) to +the problem of doing a weighted count of forested graphs. In order to count forested graphs we +will first count trees, then forests, then ways of matching the leaves of the forests to form forested +graphs. Throughout the rest of the paper we fix the following conventions and terminology. By +a tree we mean a connected graph with no cycles. A forest is a disjoint union of trees. We give +univalent vertices of trees and forests a special role and call them leaves. A tree or forest is +admissible if it has no isolated or bivalent vertices. We require admissible trees to have at least +one vertex, so a single edge which connects two leaves is not allowed. The edges of a tree or +forest are the 1-cells that are not attached to leaves (sometimes called internal edges). A rooted +tree is a tree where one distinguished univalent vertex takes the role of the root and the other + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +7 +univalent vertices are leaves. Our trees and rooted trees and forests will be labeled. That means +each leaf shall be decorated with a unique element from some given finite set U. +The set of all admissible labeled (or labeled rooted) trees forms a combinatorial species in the +sense of Joyal [20] (see also [3]). Since our counting problems fit neatly into the theory of species, +we review the relevant parts of this theory in the next section. +3.2. Species and generating functions. A combinatorial species is a functor from the group- +oid of finite sets to itself. More explicitly, a species S associates to every finite set U of labels, a +finite set S[U] of combinatorial objects such that every bijection U → V gives rise to a bijection +of sets S[U] → S[V ] in a way compatible with composition. In the case U = {1, . . ., n} we write +S[n] for S[U]. An element of S[U] for some U is an S-object, or more informally, an object of S. +A set partition of U with k blocks is a collection π = {B1, . . . , Bk} of k mutually disjoint subsets +of U such that � +B∈π B = U. From a given species S we can construct a new species Setk(S) by +associating to a set U all collections of objects {φ1, . . . , φk} such that φ1 ∈ S[B1], . . . , φk ∈ S[Bk] +for some partition of U into blocks B1, . . . , Bk. The functor Setk(S) is the species of sets of +size k which contain objects of S. To make the formulas more compact we agree that Set0(S)[0] +contains one element, the empty set, and that Setk(S)[0] is empty for all k ≥ 1. +Let A be an algebra and ω : S[U] → A a function that associates some element of A to each +object of S[U] in a way that is independent of the labeling. We can extend the weight function +to Setk(S) by setting the weight of the empty set to 1 and ω({φ1, . . . , φk}) = �k +ℓ=1 ω(φℓ). +When working with a formal power series F(x) = � aixi we will frequently use the coefficient +extraction operator [xn] to extract the coefficient of xn, so [xn]F(x) = an. We now define two +formal power series, in A[[x]] and A[[x, y]] respectively, by +S(x) = +� +n≥1 +xn +n! +� +φ∈S[n] +ω(φ) +and +SSet(x, y) = +� +n,k≥0 +xn +n! yk +� +Φ∈Setk(S)[n] +ω(Φ). +(2) +These are exponential generating functions for weighted counts: [xn]S(x) is 1/n! times the +number of objects φ ∈ S[n], counted with weight ω(φ), and [xnyk]S(x, y) is 1/n! times the +number of elements Φ ∈ Setk(S)[n], counted with weight ω(Φ). We will make frequent use of +the following exponential formula, a standard lemma that relates the power series S and SSet. +Lemma 3.1. +SSet(x, y) = exp (y S(x)) . +Proof. Use the expansion exp(X) = � +n≥0 Xn/n! and the fact that the number of set partitions +of a set of cardinality n into k blocks with sizes m1, . . . , mk is given by n!/(k!m1! . . . mk!). +□ +For every species S there is a natural action of Sn on S[n] that permutes the labels. The +orbit of an element φ ∈ S[n] under this action is an isomorphism class of combinatorial objects, +i.e. an unlabeled object. The stabilizer of an element φ ∈ S[n] is the group of relabelings that +leaves the object invariant. We will denote this stabilizer by Aut(φ). +We next want to count labeled combinatorial objects while keeping track of automorphisms. +For this, we will need more sophisticated generating functions, called cycle index series. The +terminology comes from the notion of cycle type for a permutation, which we now review. +Let n ∈ N be a positive integer. An integer partition of n is a sequence of positive integers +λ = (λ1, λ2, . . . , λℓ) such that λ1 ≥ λ2 ≥ . . . ≥ λℓ > 0 and n = λ1 + . . . + λℓ. The λi are called +parts of λ. We write λ ⊢ n or |λ| = n. The integer n is the size of the partition and ℓ, the +number of parts, is the length of the partition. We will often use the more compact notation +λ = [1m1 · · · nmn], indicating that λ has mk parts of size k (the terms with mi = 0 are usually +omitted; for example, λ = (4, 4, 1, 1, 1) is written [1342]). In this notation the length of λ is + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +8 +ℓ(λ) = m1 + · · · + mℓ, the size is |λ| = m1 + 2m2 + · · · + nmn. Each permutation π ∈ Sn +factors uniquely as a product of disjoint cycles. If the orders of these cycles are λ1, . . . , λℓ, where +λ1 ≥ . . . ≥ λℓ, then λ(π) = (λ1, . . . , λℓ) is a partition of n called the cycle type of π. +Given an integer partition λ = [1m12m2 . . . nmn] let xλ denote the monomial xm1 +1 xm2 +2 +· · · xmn +n . +A formal power series in the variables x = {x1, x2, . . .} is an infinite sum of terms aλxλ, with +aλ ∈ A, such that, for all n, we only have a finite number of terms if restrict to partitions with +|λ| ≤ n. If α is a permutation of cycle type λ we also define xα = xλ, ℓ(α) = ℓ(λ) and |α| = |λ|. +Let AS be the species of pairs (φ, α), where φ is an object of S and α an automorphism of φ: +AS[n] ={(φ, α) : φ ∈ S[n] and α ∈ Aut(φ) ≤ Sn}. +Let ω be a weight function that attaches to an object (φ, α) of AS an element ω(φ, α) ∈ A that +does not depend on the labeling. The cycle index series for S weighted by ω is the formal power +series S in x whose terms mark the cycle type of the automorphism, i.e. +S(x) = +� +n≥1 +1 +n! +� +(φ,α)∈AS[n] +ω(φ, α) xα. +In other words, [xλ]S(x) is 1/|λ|! times the weighted count of pairs (φ, α) ∈ AS such that α has +cycle type λ. The cycle index series S(x) generalizes the generating function of labeled objects +in Eq. (2), as we recover S(x) by setting ω(φ, id) = ω(φ), x1 = x and all other xk-variables to 0. +We now want to extend our cycle index series on S to one for Setk(S), and show how to +compute it from S(x). An element Φ ∈ Setk(S)[n] is a set of k elements of S with a total of n +labels. An element γ ∈ Aut(Φ) permutes the labels but preserves the set. If some elements of S +are isomorphic then γ may permute them, so γ induces a permutation γΦ ∈ Sk. We introduce +a new infinite set of variables y = {y1, y2, . . .} to mark the cycle type of γΦ, and define +SSet(x, y) = +� +n,k≥0 +1 +n! +� +(Φ,γ)∈ASetk(S)[n] +ω(Φ, γ) xγyγΦ. +Here ω(Φ, γ) = ω(φ1, γ1) . . . ω(φℓ, γℓ), where +• γΦ has cycle type λ = (λ1, λ2, . . . , λℓ) of length ℓ +• φi ∈ Φ is a representative of the ith cycle, which has size λi +• γi is the restriction of γλi to φi. +Proposition 3.2. Let S(x[k]) be the series obtained from S(x) by replacing each occurrence of +xi by xki. Then +SSet(x, y) = exp + +� +k≥1 +yk +S(x[k]) +k + + , +Ultimately, this proposition goes back to Pólya [26]. It follows from Lemma 3.1 combined +with the generating function for wreaths of S structures. We give a proof below, but refer to [3] +Chapter 4.3 for a more detailed argument, using a slightly different type of weight function ω. +To prove Proposition 3.2 we first introduce a new species ACyckS. An element of ACyckS[n] +is a pair (Φ, γ) where Φ = {φ1, . . . , φk} is a collection of objects in S with a total of n labels, and +γ ∈ Aut(Φ) is a permutation of the labels of Φ that cyclically permutes the φi, i.e. γΦ is a k-cycle +in Sk. In particular, all of the φi must be isomorphic, i.e. equivalent as unlabeled objects. +Lemma 3.3. +� +n≥1 +1 +n! +� +(Φ,γ)∈ACyck(S)[n] +ω(Φ, γ)xγ = S(x[k]) +k +. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +9 +Proof. Let (Φ, γ) be an element of ACyckS[n]. We claim that (Φ, γ) is equivalent to a tuple +(κ, π, φ, α, γ1, . . . , γk−1), where +• κ is a k-cycle in Sk, +• π is a partition of {1, . . . , n} into k blocks, each of size d, +• φ is an object of S[d], +• α is an element of Aut(φ), and +• γi is an element of Sd for each i = 1, . . . , k − 1. +By definition Φ = {φ1, . . . , φk}, where φi ∈ S[Bi] for some Bi ⊂ {1, . . . , n}, and γΦ is a k-cycle, so +we take κ = γΦ and π = {B1, . . . , Bk}. Since γ acts cyclically on Φ, all of the φi are isomorphic, +so in particular they all have the same number d of labels. The unique order-preserving bijection +Bi → {1, . . ., d} identifies each φi with an element �φi ∈ S[d]; set φ = �φk. The permutation γk +sends each φi to itself, so determines an automorphism αi of φi and therefore of �φi; set α = αk. +Note that all of the αi have the same cycle type λ = (λ1, . . . , λℓ), so the cycle type of γ is +kλ = (kλ1, . . . , kλℓ). For each i = 1, . . . , k−1, the ith power of the k-cycle γΦ maps φk to φγi +Φ(k); +let γi be the induced isomorphism γi : �φk → �φγi +Φ(k) in Sd. +Finally, note that ω(Φ, γ) = ω(φk, γk) = ω(φ, α). Since ACyck(S)[n] is empty unless k|n, we +can now write the left hand side of the equation in the statement of the lemma as, +(3) +� +n=dk≥1 +1 +n!Cn,k +� +(φ,α)∈AS[d] +ω(φ, α)xk◦α, +where k ◦ α has cycle type kλ and Cn,k is the number of partitions π of n into k equal parts +times the number of cyclic permutations κ in Sk times the number of permutations γ1, . . . , γk−1 +in (Sd)k−1, i.e. +Cn,k = 1 +k! +n! +(d!)k · (k − 1)! · (d!)k−1 = n! +d!k . +Plugging this into Eq. (3) gives the statement of the lemma. +□ +Proof of Proposition 3.2. We have +ASet(S)[n] ={(Φ, γ) : Φ is a finite set of objects of S with a total of n distinct labels, +and γ ∈ Aut(Φ) ≤ Sn}. +Since every permutation γ can be uniquely decomposed into a product of cycles, we have an +isomorphism of species +ASet(S) ∼= +� +ℓ≥1 +Setℓ + + � +k≥1 +ACyck(S) + + . +Therefore the statement follows from an application of Lemma 3.1 to the right hand side, using +the generating function of the species ACyck(S) given by Lemma 3.3 and summation over all k, +using the variables y to keep track of the cycle type of the permutation. +□ +3.3. Matchings. In order to form a forested graph with an automorphism preserving the forest, +we will start with a forest equipped with an automorphism α, then pair its leaves by a fixed-point +free involution that commutes with α. In this section we apply the counting methods from the +previous section to count the number of such involutions. +A fixed-point free involution is also called a matching; it divides the set into orbits of size 2, +so we will use the species E of sets of cardinality 2, i.e. E[2] = {1, 2} and E[n] = ∅ for n ̸= 2. +The generating function is E(x) = x2/2. +Matchings on a set of 2k elements correspond to + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +10 +elements of Setk(E)[2k]; for example the elements of Set2E[4] are {{1, 2}, {3, 4}}, {{1, 3}, {2, 4}} +and {{1, 4}, {2, 3}}. By Lemma 3.1 we have +ESet(x, y) = exp +� +y x2 +2 +� += +� +k≥0 +(2k − 1)!! x2k +(2k)!yk, +where the second equality is obtained using the expansion exp(X) = � +n≥0 Xn/n! and the +formula (2k − 1)!! = (2k)!/(k!2k). Thus we recover the (easy) fact that the number of matchings +of a set of cardinality 2k is (2k − 1)!!, and there are none if the cardinality is odd. +Now consider the species AE = AE[2] of sets of cardinality 2 with automorphisms on them. +A set of cardinality 2 has only two automorphisms, the trivial one (marked by x1x1 = x2 +1) and +the transposition (marked by x2). The cycle index series of E with trivial weight is therefore +E(x) = 1 +2! +� +(φ,α)∈AE[2] +xα = 1 +2 +� +x2 +1 + x2 +� +. +A matching of 2k elements corresponds to an element Φ ∈ Setk(E)[2k]. The automorphisms +of such an element Φ are permutations that commute with the corresponding fixed-point free +involution ι = ιΦ. We may count such permutations using Proposition 3.2, which gives +ESet(x, y) = exp + +� +k≥1 +yk +2k +� +x2 +k + x2k +� + + . +(4) +Proposition 3.4. +� +n≥0 +1 +(2n)! +� +(ι,α) +xα = exp + +� +k≥1 +1 +2k +� +x2 +k + x2k +� + + +where the sum is over all pairs (ι, α) consisting of a matching ι of {1, . . . , 2n} and a permutation +α ∈ S2n that commutes with ι. +Proof. By definition we have +ESet(x, y) = +� +n,k≥0 +1 +n! +� +(Φ,α)∈ASetk(E)[n] +xγyγφ. +Since SetkE[n] is empty unless n is even and k = n/2 we can rewrite this as +ESet(x, y) = +� +n≥0 +1 +(2n)! +� +(Φ,α)∈ASetnE[2n] +xγyγφ. +As noted previously, we may identify the pairs (ι, α) in the statement of the proposition with +elements (Φ, α) of ASetn(E)[2n]. Using Eq. (4) and setting yk = 1 for all k gives the result. +□ +Corollary 3.5. The number ηλ of matchings that commute with a given permutation α ∈ Sn of +cycle type λ = [1m12m2 . . . nmn] is given by the formula +ηλ = +n +� +k=1 +ηk,mk, +where +ηk,2s = +� +ks(2s − 1)!! +if k is odd +�s +r=0 +�2s +2r +� +kr(2r − 1)!! +if k is even + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +11 +and +ηk,2s+1 = +� +0 +if k is odd +�s +r=0 +�2s+1 +2r +� +kr(2r − 1)!! +if k is even . +Proof. Since ι commutes with α if and only if α commutes with ι, the left hand side of Proposi- +tion 3.4 can be written +� +n≥0 +1 +(2n)! +� +α∈S2n +#{ι|ι ◦ α = α ◦ ι}xα. +Now note that the number of matchings that commute with α depends only on the cycle type of +α. If we denote the set of of permutations in Sn with cycle type λ by Sλ +n, the formula becomes += +� +n≥0 +1 +(2n)! +� +λ⊢2n +| Sλ +2n |ηλxλ = +� +n≥0 +1 +n! +� +λ⊢n +| Sλ +n |ηλxλ, +where we used the fact that ηλ = 0 if |λ| is odd. For λ = [1m12m2 . . . nmn] it is easy to check +that the number of permutations in Sλ +n is +| Sλ +n | = +n! +1m1m1!2m2m2! · · · nmnmn!, +so the left hand side of Proposition 3.4 is equal to +(5) +� +n≥0 +� +λ⊢n +ηλ +1m1m1!2m2m2! · · · nmnmn!xλ. +We now turn to the right hand side. Using eX = � +m≥0 Xm/m!, (2m − 1)!! = (2m)!/(m!2m) +and (x + 1)m = �m +k=0 +�m +k +� +xk one can check that the numbers ηk,m given in the statement of the +corollary are the coefficients of xm/(kmm!) in the expansions of exp +� +x2/(2k) +� +if k is odd and +exp +� +(x2 + 2x)/(2k) +� +if k is even. Using this together with the fact that +� +k≥1 +1 +2k +� +x2 +k + x2k +� += +� +k≥1 +� +x2 +2k−1 +2(2k − 1) + x2 +2k + 2x2k +4k +� +the right hand side of Proposition 3.4 becomes +(6) +� +k≥1 + +� +m≥0 +ηk,m +kmm!xm +k + + . +Equating coefficients of (5) and (6) now gives the proposition. +□ +3.4. Rooted trees. In order to count forested graphs with automorphisms, we begin by counting +forests with automorphisms. In order to count forests with automorphisms, we begin by counting +trees with automorphisms, and in order to count trees with automorphisms, we begin by counting +rooted trees with automorphisms. +Let R be the species of leaf-labeled admissible rooted trees ρ, i.e. all internal vertices of ρ +must have valence at least 3 and R[n] is the set of all such rooted trees with n leaves. The +elements of R[3] are depicted in Figure 1. The rooted tree ρ0 with one root, one leaf and one +1-cell has no internal vertices so satisfies the definition, but it plays a special role so we call it +the special rooted tree. AR is then the species of pairs (ρ, α), where ρ ∈ R and α ∈ Aut(ρ). +Recall that an automorphism of a tree is determined by what it does to the leaves. So, for a +rooted tree ρ ∈ R[n] we can (and will) identify the group Aut(ρ) ≤ Sn with the usual simplicial +automorphisms of the tree ρ. To each pair (ρ, α) ∈ AR we assign weight ω(ρ, α) = (−1)vα(ρ), +where vα(ρ) is the number of α-orbits of internal vertices. The special rooted tree ρ0 has only the + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +12 +1 +2 +3 += +1 +3 +2 +, +2 +1 +3 +, +3 +1 +2 +, +1 +2 +3 +Figure 1. All admissible rooted trees with 3 leaves, i.e. all elements of R[3]. +The first three rooted trees have two vertices each and the fourth has one ver- +tex. Edges are colored in blue. The automorphism groups of the first, second +and third rooted tree are generated by the transpositions (2, 3), (1, 3) and (1, 2) +respectively. The automorphism group of the fourth rooted tree is the full sym- +metric group S3, i.e. it includes all permutations of the labels. +identity automorphism, and the pair (ρ0, id) has weight is (−1)0 = 1. The associated generating +function +R(x) = +� +n≥1 +1 +n! +� +(ρ,α)∈AR[n] +(−1)vα(ρ)xα +is the Frobenius characteristic of rooted trees. The first few terms are +R(x) = x1 − 1 +2(x2 +1 + x2) + 1 +63(x3 +1 + x1x2) − 1 +6(x3 +1 + 3x1x2 + 2x3) + . . . +The first term comes from the special rooted tree ρ0. The second term comes from the rooted +tree with one vertex and two leaves branching from it. We have two automorphisms that either +switch the leaf-labels or do not. The third and fourth terms in this sum come from the rooted +trees in Figure 1 and their automorphisms. +Remark 3.6. The name Frobenius characteristic comes from an interpretation of these generating +functions in the context of the representation theory of the symmetric group: We can think of +R(x) as an element of the ring of symmetric functions and each homogeneous part of R(x) as +the image of a certain representation of the symmetric group under the Frobenius characteristic +map. These representations are the vector spaces generated by the elements of R[n] with the +action of Sn which alternates with (−1)vα(ρ) as above. However, in this paper we will not make +use of this representation theoretical interpretation of these objects. +The next proposition shows that the characteristic R(x) has a remarkably simple form. +Proposition 3.7. +R(x) = +� +n≥1 +µ(n) +n +log(1 + xn), +where µ is the Möbius function. +Proof. The formula will be established by relating pairs (ρ, α) recursively to pairs (Φ, γ) consisting +of a set of rooted trees Φ and an automorphism γ ∈ Aut(Φ). +Let Φ be a collection of two or more admissible rooted trees and γ ∈ Aut(Φ). Set ω(Φ, γ) = +(−1)vγ(Φ). We can form a new admissible rooted tree ρΦ by gluing the roots of the trees in Φ +together and growing a new root from the resulting vertex. The resulting rooted tree has one +new internal vertex, which is fixed by any automorphism, and at least 2 leaves. The natural + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +13 +map Aut(Φ) → Aut(ρΦ) is a bijection, so we will use the same name γ, and we have ω(ρΦ, γ) = +−ω(Φ, γ). The only pair (ρ, α) which cannot be obtained in this way is (ρ0, id), so +R(x) = x1 − +� +k≥2 +� +n≥k +1 +n! +� +(Φ,γ)∈ASetk(R)[n] +(−1)vγ(Φ)xγ, +(7) +Proposition 3.2 tells us +RSet(x, y) = +� +k≥0 +� +n≥k +1 +n! +� +(Φ,γ)∈ASetk(R)[n] +(−1)vγ(Φ)xγyγΦ = exp + +� +k≥1 +yk +R(x[k]) +k + + , +Setting yk = 1 for all k and subtracting the terms with k = 0 or k = 1 gives the summation +term in Eq. (7) above, i.e. +R(x) = x1 − +� +exp + +� +k≥1 +R(x[k]) +k + + − 1 − R(x[1]) +� +. +(8) +Since R(x[1]) = R(x) this gives � +k≥1 R(x[k])/k = log(1 + x1) and because x[k] = (xk, x2k, . . .), +we also have � +k≥1 R(x[kn])/k = log(1+xn) for all n ≥ 1. Multiplying both sides of this equation +with µ(n)/n, summing over n ≥ 1 and using the defining recursion of the Möbius function results +in the statement. +□ +3.5. Unrooted trees. We next use this generating function for rooted trees (weighted by the +parity of vertex orbits) to find the Frobenius characteristic V(x) for the species T of unrooted +admissible trees, weighted by the parity of edge-orbits. Specifically, to a pair (t, α) ∈ AT , where +t ∈ T and α ∈ Aut(t), we assign the weight ω(t, α) = (−1)eα(t), where eα(t) is the number of +edge-orbits of α in t, and define +V(x) = +� +n≥3 +1 +n! +� +(t,α)∈AT [n] +(−1)eα(t)xα, +(9) +Note that both T [1] and T [2] are empty, so we could have started the indexing with n = 1. +Proposition 3.8. +V(x) = x1 + x2 +1 +2 − x2 +2 − (1 + x1)R(x). +Proof. Suppose t is an unrooted tree and α ∈ Aut(t). Let tα be the subtree of t spanned by the +fixed vertices. Consider first the case that tα is nonempty. +For each vertex v ∈ tα, we can think of (t, α) as a collection Φ of at least 3 rooted trees, with +v as their root, and of α as an automorphism of Φ. Note that eα(t) = vα(t) − 1 = vα(Φ). Using +Proposition 3.2 and setting yk = 1 for all k, the generating functions of such collections Φ is +exp + +� +k≥1 +R(x[k]) +k + + − 1 − R(x[1]) − R(x[2]) +2 +− R(x[1])2 +2 +, +where we have subtracted the terms that correspond to collections with fewer than 3 trees. By +Eq. (8) in the proof of Proposition 3.7 this is the same as +x1 − R(x) − R(x[2]) +2 +− R(x)2 +2 +. +(10) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +14 +If we place a root in the middle of an edge of tα we can view (t, α) as a pair {ρ1, ρ2} of rooted +trees and α as an automorphism of the pair. Since e is not adjacent to a leaf neither rooted tree +is the special rooted tree. The generating function of such rooted trees is therefore +(R(x) − x1)2 +2 +. +(11) +Here we have eα(t) = vα(t) − 1 = vα({ρ1, ρ2}). The sum of expressions (10) and (11) counts the +pair (t, α) multiple times, once for each vertex of tα with sign (−1)eα(t), and once for each edge +of tα with the opposite sign. Since tα has one more vertex than edge, this sum leaves us with +exactly one contribution from each (t, α), with sign (−1)eα(t), i.e. the sum gives the contribution +to V(x) from all such pairs. +We still have to account for pairs (t, α) with no fixed vertices, i.e. tα is empty and the only +fixed point is at the midpoint of an edge. The tree t can then be viewed as the union of two +identical rooted trees {ρ1, ρ2} rooted at this midpoint, which are exchanged by α. By Lemma 3.3 +these are counted by R(x[2])/2. Since in this case eα(t) = vα({ρ1, ρ2}), these contribute +R(x[2]) +2 +− x2 +2 +(12) +to V(x), where we have subtracted the term corresponding to the interval since that is not an +admissible unrooted tree. The sum of formulas (10), (11) and (12) now has one term for each +pair (t, α), weighted by (−1)eα(t), i.e. the sum is equal to V(x). +□ +As an immediate corollary of Proposition 3.8 and Proposition 3.7 we have +Corollary 3.9. +V(x) = x1 + x2 +1 +2 − x2 +2 − (1 + x1) +� +k≥1 +µ(k) +k +log(1 + xk). +Using log(1 + x) = � +n≥1(−1)n+1xn/n, and the definition of the Möbius function, we can +expand this power series. As µ(1) = 1, µ(2) = µ(3) = −1 and µ(4) = 0, the first coefficients are +V(x) = x1 + x2 +1 +2 − x2 +2 +− (1 + x1) +�� +x1 − x2 +1 +2 + x3 +1 +3 − x4 +1 +4 + . . . +� +− 1 +2 +� +x2 − x2 +2 +2 + . . . +� +− 1 +3 (x3 + . . .) + . . . +� += x3 +1 +6 + x1x2 +2 ++ x3 +3 − x4 +1 +12 − x2 +2 +4 + x1x3 +3 ++ . . . +where we omitted terms of total degree higher than 4. As expected there are no terms of degree +smaller than 3, as such terms would correspond to trees with fewer than 3 leaves. +We recall from Section 3.2 that setting x = (x, 0, 0, . . .) in the Frobenius characteristic V(x) for +unrooted trees recovers the generating function V (x) for unrooted trees t weighted by (−1)e(t), +where e(t) is the number of (internal) edges of t, i.e. +(13) +V (x) = x + x2 +2 − (1 + x) log(1 + x). +For a detailed exposition of this formula see [11]. +As forests are just collections of trees, Proposition 3.2 gives us the following expression for +the alternating cycle index series for the species F of forests: +VSet(x, y) = +� +n +1 +n! +� +(Φ,γ)∈AF[n] +(−1)eγ(Φ)xγyγΦ = exp + +� +k≥1 +yk +V(x[k]) +k + + . +(14) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +15 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +Figure 2. A forest with automorphism γ = (13)(24)(57)(68)(9 10) and match- +ing ι = (13)(25)(47)(68)(9 10) that commutes with γ, corresponds to a forested +graph with the automorphism that flips the graph over the horizontal axis. +3.6. Forested graphs. What we want to do with these forests is to glue their leaves together in +pairs to form admissible graphs (which we can only do if there is an even number of leaves). We +also want to keep track of the Euler characteristic of the resulting graph G; this is the number +of trees in the forest minus half the total the number of leaves. It is equivalent to keep track of +−2χ(G) = #{leaves} − 2#{trees} (which is always a positive integer) so we mark this number +with a new variable u, and define the Frobenius characteristic of admissible forests +F(u, x) = +� +s≥0 +1 +s! +� +(Φ,γ)∈AF[s] +(−1)eγ(Φ)xγus−2k(Φ), +(15) +where F is the species of forests and k(Φ) is the number of trees in the forest Φ. +Proposition 3.10. +F(u, x) = exp + +� +k≥1 +u−2k V((u · x)[k]) +k + + , +where V((u · x)[k]) means that we replace each variable xi in V(x) with ukixki. +By Eq. (9) the lowest term of V(u·x) is of order u3, and hence the lowest term of V((u·x)[k]) +is of order u3k. Hence, F(u, x) is the exponential of a power series in only positive powers of u. +Proof. Starting with the definition of VSet(x, y), the substitution xi �→ uixi sends xλ to u|λ|xλ, +and the substitution yi �→ u−2i sends yγΦ to u2k(Φ) where k(Φ) is the number of trees in Φ. +Doing both substitutions gives F(u, x). +□ +Proposition 3.11. +� +λ +ηλ[u2nxλ]F(u, x) = �en, +where we sum over all integer partitions λ and ηλ is the number of matchings that commute with +a permutation of cycle type λ. The sum is finite since [u2nxλ]F(u, x) = 0 unless 2n ≤ |λ| ≤ 6n. +Proof. Our generating function F(u, x) counts forests together with automorphisms, i.e. if we set +Fr,λ = {(Φ, γ)|Φ has |λ| leaves and k = |λ| − r +2 +components and γ has cycle type λ}, + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +16 +then +[urxλ]F(u, x) = +1 +|λ|! +� +(Φ,γ)∈Fλ,r +(−1)eγ(Φ). +Because a nonempty forest has at least one component, we have [urxλ]F(u, x) = 0 if |λ| < r, +and because every connected component of an admissible forest must have at least three leaves +we have [urxλ]F(u, x) = 0 if |λ| > 3r. In particular, for a given r there are only finitely many +integer partitions λ such that [urxλ]F(u, x) is nonzero, i.e. the terms in the sum +� +λ +ηλ[urxλ]F(x, u) +are only nonzero when r ≤ |λ| ≤ 3r. +Let Mr ⊂ Sr be the set of fixed-point free involutions on the set {1, . . . , r} and IF r,λ = +{(Φ, γ, ι)|(Φ, γ) ∈ Fr,λ and ι ∈ Mr commutes with γ}. Then ηλ[urxλ]F(x, u) is equal to 1/|λ|! +times the alternating sum of elements in IFr,λ. We can make a forested graph equipped with +an automorphism from an element (Φ, γ, ι) ∈ IFr,λ by using ι to identify the leaves of Φ in +pairs, provided r = 2n is even (see Figure 2). The result will be a forested graph (G, Φ0) with +χ(G) = −n, where Φ0 is the subforest of G consisting of just the edges and vertices of Φ, and the +leaves of Φ have become (labeled) half-edges in G\Φ0. This gives us a one-to-one correspondence +between elements of IF2n,λ and pairs ((G, Φ0), γ) consisting of a forested graph (G, Φ0) with +χ(G) = −n and an automorphism γ ∈ Aut(G, Φ0), where the half-edges of G \ Φ0 are labeled by +{1, . . . , |λ|}. The symmetric group S|λ| acts on IF2n,λ by permuting the labels. The stabilizer +of ((G, Φ0), γ) is Autγ(G, Φ0), the automorphisms of (G, Φ0) that commute with γ, and the orbit +is the unlabeled pair [G, Φ0] together with a conjugacy class [γ] of the automorphism group +Aut(G, Φ0). The weight (−1)eγ(Φ) is constant on orbits. If we now take the sum over all integer +partitions λ, the orbit-stabilizer theorem gives +� +λ +ηλ[u2nxλ]F(x, u) = +1 +|λ|! +� +λ +� +(Φ,γ,ι)∈IF2n,λ +(−1)eγ(Φ) += +� +[G,Φ0] +χ(G)=−n +� +[γ]∈Aut(G,Φ0) +1 +|Autγ(G, Φ0)|(−1)eγ(Φ0) += +� +[G,Φ0] +χ(G)=−n +1 +|Aut(G, Φ0)| +� +γ∈Aut(G,Φ0) +(−1)eγ(Φ0). +The second step uses the orbit stabilizer theorem again on the centralizer Autγ(G, Φ0). Since +the last line is the definition of �en, the proposition is proved. +□ +The following theorem summarizes the steps needed for computing e(Out(Fn)) for a given n. +Theorem 3.12. For fixed n ≥ 2, the numbers e(Out(F2)), . . . , e(Out(Fn)) can be computed by +the following steps: +(1) Calculate V(x) up to homogeneous degree 6(n−1) in x using the formula in Corollary 3.9. +(2) Calculate the coefficients [u2kxλ]F(u, x) from V(x) using the formula in Proposition 3.10 +for all pairs k, λ with k ≤ n − 1 and λ a partition of size less than 6k. +(3) Calculate the numbers �e0, . . . ,�en−1 using the formula +�ek = +� +λ +ηλ[u2kxλ]F(u, x) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +17 +from Proposition 3.11. Recall that this is a finite sum: the terms are nonzero only for +partitions λ of size 2k ≤ |λ| ≤ 6k. +(4) Recover e(Out(Fn)) from �e0, . . . ,�en−1 using the recursive formula in Corollary 2.6. +The most demanding part of the computation is the expansion of the generating function +F(u, x) for the Frobenius characteristic of admissible forests. We used Jos Vermaseren’s FORM +programming language [27] to compute this expansion up to an appropriate order and calculated +e(Out(Fn)) for n ≤ 15. The result is listed in Appendix A. We will report on an optimized +program with which the numbers e(Out(Fn)) have been computed for all n ≤ 100 in a separate +publication [9]. +3.7. Forested graphs with legs. So far we have only needed to consider forested graphs (G, Φ) +such that G has no univalent vertices. However, to determine the asymptotic behavior of the +integral Euler characteristic we will need to do a finer analysis, which involves studying pairs +(G, Φ) where G is allowed to have univalent (but not bivalent) vertices, while Φ is not allowed to +contain any edges adjacent to univalent vertices. We call such pairs forested graphs with legs. In +this section we point out that a minor modification of the counting methods from the previous +sections counts these more general forested graphs. +Recall that admissible graphs are constructed by pairing the leaves of an admissible forest Φ. +The edges of the forest then become a subforest Φ0 of the admissible graph G. To get the original +forest back you cut all the 1-cells of G that are not in Φ0; the half-edges that result are the leaves +of the original forest Φ. If the graph G has univalent vertices we are not allowing the subforest +Φ to contain adjacent 1-cells so they must be cut, which results in components with one 0-cell +and one half-edge. These are not admissible trees, so we will call these special components, and +mark them with a new variable w. A forest which is allowed to have both admissible and special +components will be called an extended forest. Matching the leaves of an extended forest results +in a graph with a univalent vertex for each special component. +Let F⋆ denote the species of extended forests, and define F ⋆(x, u, w) to be +F ⋆(u, x, w) = +� +Φ∈F ⋆ +(−1)e(Φ)us(Φ)−2k(Φ)wj(Φ) xs(Φ) +s(Φ)!, +where e(Φ) is the number of edges of Φ, s(Φ) is the number of leaves, k(Φ) is the total number +of components and j(Φ) is the number of special components, i.e. [urxswj]F ⋆(u, x, w) is the +edge-weighted count of forests with s leaves and (s − r)/2 components, of which j are special. +Recall that V (x) is the generating function for unrooted trees, weighted by the parity of their +internal edges (see Eq. (13)). +Proposition 3.13. F ⋆(u, x, w) = exp(u−1wx + u−2V (ux)). +Proof. A special component has 1 leaf, 0 edges and 1 component (which is special!), so the term +u−1wx accounts for special components. The term u−2V (ux) accounts for admissible trees as +before. Exponentiating gives the generating function for collections, as per Lemma 3.1. +□ +In [10] we showed that the rational Euler characteristic χn = χ(Out(Fn+1)) is given by +χn = +� +[G,Φ] +(−1)e(Φ) +|Aut(G, Φ)|, +where the sum is over isomorphism classes [G, Φ] of forested graphs with G connected of Euler +characteristic −n. We denoted the corresponding generating function by T (ℏ) = � +n≥1 χnℏn. +We then defined an analogous generating function T (ℏ, w) for isomorphism classes [G, Φ] of +connected forested graphs with legs, which are weighted in the same way as in the formula above. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +18 +Here the variable w marks the legs of G, i.e. [ℏnws]T (ℏ, w) is the weighted sum over isomorphism +classes [G, Φ] such that G is connected, has Euler characteristic −n, and has s legs. We can relate +T (ℏ, w) to the generating function F ⋆(u, x, w): +Lemma 3.14. +� +m≥0 +(2m − 1)!![x2m]F ⋆(u, x, w) = exp +�u−2w2 +2 ++ T (u2, w) +� +. +Proof. The coefficient of u2nws in (2m−1)!![x2m]F ⋆(u, x, w) gives the weighted count of forested +graphs with s legs and χ = −n that can be made by gluing the leaves of extended forests Φ with +2m leaves. Adding up over all m gives the count of all forested graphs with s legs and χ = −n. +The term exp(T (u2, w)) counts forests, but misses components with a single internal vertex and +two legs. Those are taken care of by adding the term (u−2w2)/2 to T (u2, w). +□ +Finally, we recall from [10] that T (ℏ, w) and T (ℏ) are related in the following way. +Proposition 3.15 ([10], Proposition 3.1). +T (ℏ, w) = T (ℏe−w) + w +2 + ℏ−1 +� +ew − 1 − w − w2 +2 +� +. +Here T (ℏe−w) accounts for connected forested graphs with negative Euler characteristic, the +term w +2 accounts for those with Euler characteristic zero, and the last term accounts for trees. +4. Asymptotics +4.1. Precise asymptotic statements. To discuss the asymptotic behavior of sequences we use +the following standard conventions. +Notation. Let {cn} be a sequence defined for all but finitely many positive integers n. The set +O(cn) consists of all such sequences {an} for which lim supn→∞ |an/cn| < ∞. The notation an = +bn + O(cn) means an − bn ∈ O(cn). Recall that the notation an ∼ bn means limn→∞ an/bn = 1. +To describe the asymptotic behavior of the numbers �en we will use the Γ function, which is +defined by Γ(x) = +� ∞ +0 +zx−1e−zdz for all x > 0. At integer arguments it agrees with the factorial +n! = Γ(n + 1). By Stirling’s formula, we have Γ(x) ∼ +√ +2πxx− 1 +2 e−x [1, Eq. (3.9)], i.e. Γ(x) grows +more than exponentially. Explicitly, we will quantify the asymptotic behavior of �en using the +sequences Bn and Ln defined by +Bn = − +1 +√ +2π +Γ +� +n − 1 +2 +� +log2 n +and +Ln = +log n +log log n. +It follows from Stirling’s formula that +Lemma 4.1. Bn ∼ −nne−n/(n log2 n). +Proof. Bn/(−nne−n/(n log2 n)) ∼ e +1 +2 (1 − 1 +2n)n−1 ∼ 1, where we used limn→∞(1 + x +n)n = ex. +□ +The remainder of the paper is devoted to proving the following theorem. +Theorem 4.2. �en has asymptotic behavior +�en = e− 1 +4 Bn + O(Bn/Ln). +Note that limn→∞ Ln = ∞, so it follows that �en ∼ e− 1 +4 Bn. By applying Stirling’s formula, +i.e. Lemma 4.1, we moreover find that �en ∼ −e− 1 +4 nne−n/(n log2 n). In [10] we defined �χn to +be the coefficient of ℏn in the series �T(ℏ) = exp(� +n≥1 χnℏn), where χn = χ(Out(Fn+1)) is the +rational Euler characteristic of Out(Fn+1), and proved + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +19 +Proposition 4.3 (Proposition 8.1 and Lemma 8.7 in [10]). +�χn = Bn + O(Bn/Ln). +Proof. We just have to substitute the asymptotic formula for the numbers vk in [10, Lemma 8.7] +(where we uncover an obvious typo: there should be no minus sign in [10, equation (8.7)]) into +the formula for �χn from [10, Proposition 8.1]. +□ +As an immediate corollary we get +Corollary 4.4. There exists a constant C, such that +| �χn | ≤ C Γ +� +n − 1 +2 +� +for all n ≥ 1. +Proof. This follows from O(�χn) = O(Bn) = O +� +Γ(n − 1 +2)/ log2 n +� +⊂ O +� +Γ(n − 1 +2) +� +. Hence, +limn→∞ | �χn /Γ(n − 1 +2)| is finite and as Γ(n − 1 +2) > 0 for n ≥ 1, �χn /Γ(n − 1 +2) stays bounded. +□ +We also showed that the numbers χn have the same asymptotic behavior as �χn [10, Proposi- +tion 8.6]. A slight modification of the proof of this given in [10] gives the following statement. +Proposition 4.5. The numbers e(Out(Fn+1)) have the same asymptotic behavior as �en, i.e. +e(Out(Fn+1)) = e− 1 +4 Bn + O(Bn/Ln). +Proof. Lemma 8.8 of [10] gives a criterion for showing the asymptotic behavior of the coef- +ficients of a series � anxn agrees with with that of the coefficients of exp(� anxn). +The +proof of Proposition 8.6 in [10] immediately following Lemma 8.8 applies almost verbatim to +show that the coefficients en defined by �∞ +n=1 en ℏn = log (�∞ +n=0 �en ℏn) have the same asymp- +totic behaviour as �en, i.e. en = e− 1 +4 Bn + O(Bn/Ln). By Corollary 2.6 of the present paper, +e(Out(Fn+1)) = en + � +d|n,d̸=1 +µ(d) +d +en/d. As n has fewer than n divisors and |µ(n)| ≤ 1, the sums +� +d|n,d̸=1 +µ(d) +d +en/d form a sequence in O(n en/2) = O(nBn/2) ⊂ O(nΓ(n/2 − 1 +2)) ⊂ O(Bn/Ln), +showing that e(Out(Fn+1)) also has the same asymptotic behavior as �en. +□ +Proof of Theorem 1.1. Follows directly from Proposition 4.5, Lemma 4.1, log(n − 1) ∼ log n and +(1 − 1/n)n−1 ∼ e−1. +□ +Remark 4.6. In fact, we prove a slightly stronger statement than Theorem 1.1 which quantifies +the error term in the asymptotic behavior. By Propositions 4.3 and 4.5, we have for large n +e(Out(Fn))/χ(Out(Fn)) = e− 1 +4 + O(log log n/ log n). +The error term O(log log n/ log n) above appears to be too pessimistic. Based on the compu- +tations of e(Out(Fn)) up to n = 100 from [9] and empirical comparison with χ(Out(Fn)), we +conjecture +e(Out(Fn))/χ(Out(Fn)) = e− 1 +4 +� +1 − 29 +32n + O +� 1 +n2 +�� +. +4.2. Relating the integral and rational Euler characteristics. Each triple [G, Φ, α] con- +sisting of a forested graph with χ(G) = −n and an automorphism α preserving Φ contributes to +�en, and the triples with α = id give the rational Euler characteristic �χn. To prove Theorem 4.2 +we will exploit the fact that we already know the asymptotics of �χn. We will eventually find +that the contributions to �en from triples [G, Φ, α] such that α fixes Φ and has order at most +2 dominate the asymptotics of �en. We will isolate these contributions in the formulas we have +for our generating functions, and prove that they dominate by bounding the relative size of the +remaining terms. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +20 +We start with the formula +�en = +� +λ +ηλ[u2nxλ]F(u, x) +from Proposition 3.11. From Proposition 3.10 we have an expression for F(u, x), so +�en = +� +λ +ηλ[u2nxλ] exp + +� +k≥1 +u−2k V((u · x)[k]) +k + + . +Recall that a permutation is called a derangement if it has no fixed points; this is equivalent to +saying that the corresponding cycle type has no parts of size 1, so we call it a deranged partition. +An integer partition λ is equivalent to a pair (m, δ) where m is the number of parts of size 1 in +λ and δ is the deranged partition obtained from λ by removing all parts of size 1. +To make an admissible graph with an automorphism from a pair (Φ, α) consisting of a forest +and an automorphism, leaves of Φ that are permuted by α in cycles of equal lengths must be +paired with each other; in particular, leaves that are fixed by α must be paired with each other. +By first pairing the fixed leaves we can reduce the expression for �en above to a sum over deranged +partitions. Specifically, if there are 2m fixed leaves there are (2m − 1)!! ways to pair them, so if +λ = (2m, δ) then ηλ = (2m − 1)!!ηδ and +�en = +� +δ +ηδ[u2nxδ] +∞ +� +m=0 +(2m − 1)!![x2m +1 ] exp + +� +k≥1 +u−2k V((u · x)[k]) +k + + += +� +δ +ηδ[u2nxδ] exp + +� +k≥2 +u−2k V((u · x)[k]) +k + + +∞ +� +m=0 +(2m − 1)!![x2m +1 ] exp +� +u−2V(u · x) +� +, +where we sum only over deranged integer partitions δ. (Note that the variable x1 does not appear +in the power series V((u·x)[k]) for k ≥ 2.) The expression in the first exponential will not change +in what follows, so we give it a name: +h1(u, x) := +� +k≥2 +u−2k V((u · x)[k]) +k += +� +k≥2 +ukx3 +k +6k ++ ukxkx2k +2k ++ ukx3k +3k +− u2kx4 +k +12k ++ . . . +and our expression reads +�en = +� +δ +ηδ[u2nxδ] exp(h1(u, x)) +∞ +� +m=0 +(2m − 1)!![x2m +1 ] exp +� +u−2V(u · x) +� +. +(16) +Next we look more closely at the term exp(u−2V(u·x)). We first separate out the contribution +to V(x) of pairs (t, α) with α = id. This is obtained by setting xi = 0 in V(x) for all i ≥ 2; +recall that this gives us the generating function V (x1) for trees without automorphisms. By +Corollary 3.9 we have +V(x) = x1 + x2 +1 +2 − x2 +2 − (1 + x1)R(x) += x1 + x2 +1 +2 − x2 +2 − (1 + x1) +� +k≥1 +µ(k) +k +log(1 + xk) += x1 + x2 +1 +2 − (1 + x1) log(1 + x1) − x2 +2 − (1 + x1) +� +k≥2 +µ(k) +k +log(1 + xk) += V (x1) − x2 +2 + (1 + x1)W(x), +(17) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +21 +where W(x) = − � +k≥2 +µ(k) +k +log(1 + xk) = −R(x) + (1 + x1) log(1 + x1). Note that W(x) counts +rooted trees with automorphisms that do not fix any leaves, in particular it does not involve the +variable x1. By Eq. (17) we have +exp +� +u−2V(u · x) +� += exp +� +u−2 � +V (ux1) − u2 x2 +2 + (1 + ux1)W(u · x) +�� += exp +�� +u−2V (ux1) + u−1x1W(u · x) +� ++ u−2 � +W(u · x) − u2 x2 +2 +�� +. +By Proposition 3.13, exp +� +u−2V (ux) + u−1wx +� += F ⋆(u, x, w), where [u2nxswj]F ⋆(u, x, w) counts +forests with j special components and s leaves that glue up to graphs with χ = −n and j legs. +Note that W(u · x) can be interpreted as a power series in u whose coefficients are polynomials +in x1, x2, . . .. This power series has no constant coefficient, so if f(w) is another power series, +then the composition f(W(u · x)) is convergent in the usual power series topology. Substituting +x1 for x and W(u · x) for w in Proposition 3.13, our formula for exp +� +u−2V(u · x +� +) becomes +exp +� +u−2V(u · x) +� += F ⋆(u, x1, W(u · x)) exp +� +u−2 � +W(u · x) − u2 x2 +2 +�� +. +Substituting the above into Eq. (16) gives the following expression: +�en = +� +δ +ηδ[u2nxδ] exp(h1(u, x))× +∞ +� +m=0 +(2m − 1)!![xm +1 ]F ⋆(u, x1, W(u · x)) exp +� +u−2 � +W(u · x) − u2 x2 +2 +�� +. +(18) +Now recall the statement of Lemma 3.14 +� +m≥0 +(2m − 1)!![x2m]F ⋆(u, x, w) = exp +�u−2w2 +2 ++ T (u2, w) +� +. +After substituting x1 for x and W(u · x) for w in this statement, Eq. (18) becomes +�en = +� +δ +ηδ[u2nxδ] exp(h1(u, x))× +exp +�u−2W(u · x)2 +2 ++ T (u2, W(u · x)) + u−2 � +W(u · x) − u2 x2 +2 +�� +(19) +By Proposition 3.15 we have a relation between T (ℏ, w) and T (ℏ). Substituting u2 for ℏ and +W(u · x) for w, this relation becomes +T (u2, W(u · x)) = T (u2e−W(u·x)) + W(u · x) +2 ++ u−2 +� +eW(u·x) − 1 − W(u · x) − W(u · x)2 +2 +� +. +Substituting this into Eq. (19) and simplifying turns our expression for �en into +� +δ +ηδ[u2nxδ] exp +� +h1(u, x) + T +� +u2e−W(u·¯x)� ++ W(u · x) +2 ++ u−2 � +eW(u·¯x) − 1 − u2 x2 +2 +�� +. +(20) +Setting +�T(ℏ) = exp(T (ℏ)) +h2(u, x) = W(u · x) +2 +h3(u, x) = u−2 � +eW(u·¯x) − 1 − u2 x2 +2 +� +, +we record Equation (20) formally as a theorem: + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +22 +v +w +(G, Φ, α) +1 +2 +3 +4 +5 +6 +G′ +G′′ +(G, Φ) +Figure 3. Reducing a forested graph (G, Φ) with automorphism α interchang- +ing v and w to a forested graph with legs. The graph G′ is obtained by cutting +edges not in Gα or Φ, and α induces the derangement (14)(23)(56). The next +graph G′′ replaces each set of trees at a vertex of Gα by a single orange leg, and +the final graph G results from contracting all separating edges in G′′ that are +not legs. +Theorem 4.7. +�en = +� +δ +ηδ[u2nxδ] �T(u2e−W(u·¯x))H(u, x), +where the sum is over all deranged integer partitions δ and +H(u, x) = exp (h1(u, x) + h2(u, x) + h3(u, x)) . +Here is a combinatorial interpretation of Theorem 4.7. Given a triple (G, Φ, α), let Gα be +the subgraph of G fixed by α, and Φα = Φ ∩ Gα. If we cut all edges of G which are not in +Φ or in Gα, we obtain a graph G′ with leaves, and α induces a derangement of these leaves +(see Figure 3). The fixed subgraph Gα is a subgraph of G′. Components of G′ that do not +intersect Gα are k-cycles of trees with k ≥ 2; these contribute the term h1 in Eq. (20). If C is a +component that does intersect Gα then (C ∩ Gα, C ∩ Φα) is a forested graph, and the rest of C +consists of k-cycles of rooted trees attached at various vertices of C ∩ Gα, where k ≥ 2. At each +of these vertices, remove all of the deranged trees that are attached there and replace them with +a single orange leg. Then contract all separating edges of the result that are not orange legs to +get an admissible forested graph (C, Φ) with legs. We can count such forested graphs using the +generating functions T (ℏe−w) (if χ(C) < 0), w +2 (if χ(C) = 0), and ℏ−1w − x2 +2 (if C is a tree but +C is not a single vertex with two half-edges attached). Replacing ℏ by u2 and w by W(u · x) has +the effect of marking the negative Euler characteristic by u2 instead of ℏ and reconstructing the +components C by adding rooted trees to the forested graphs. +To determine the asymptotic behavior of �en we will estimate the size of the coefficients of +H(u, x), and show that their contribution to �en is dominated asymptotically by the contribution +of �T(u2e−W(u·¯x)). We will also show that the sum � +δ ηδ[u2nxδ] �T(u2e−W(u·¯x)) is dominated by +contributions of deranged partitions δ with all parts of size 2. We will then be able to determine +the asymptotic behavior of �en using estimates on the size of ηλ and the fact that we know the +behavior of the coefficients �χn of �T(ℏ) from our previous work in [10]. +We note that µ(2) = µ(3) = −1 and µ(4) = 0, hence the first few terms of W(u · ¯x) are +W(u · ¯x) = − +� +k≥2 +µ(k) +k +log(1 + ukxk) = 1 +2 +� +u2x2 − u4x2 +2 +2 +� ++ u3x3 +3 ++ . . . += u2x2 +2 ++ u3x3 +3 +− u4x2 +2 +4 ++ . . . + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +23 +where higher powers of u were omitted. It follows that h2(u, x) = W(u · ¯x)/2 and +h3(u, x) = u−2 � +eW(u·¯x) − 1 − u2 x2 +2 +� += ux3 +3 − u2 x2 +2 +8 + . . . +are power series in only positive powers of u (as is h1(u, x)). +4.3. Splitting and merging Γ functions and estimating the numbers ηλ. In this section +we show that the numbers ηλ and ηk,m defined in Corollary 3.5 are bounded by Γ functions +modulated by exponentials. Recall that the Γ function satisfies +• Γ(x + 1) = xΓ(x) for all x > 0, +• Γ(1/2) = √π, and +• Γ is is log-convex, i.e. +log Γ +� +ta + (1 − t) b + 1 +2 +� +≤ t log Γ +� +a + 1 +2 +� ++ (1 − t) log Γ +� +b + 1 +2 +� +, +(21) +for all t ∈ [0, 1] and a, b ≥ 0. +In fact, these three properties determine the function Γ completely [1, Theorem 2.1]. Lemma 4.8, +Corollary 4.9 and Lemma 4.10 below follow easily using these properties. +Lemma 4.8. For all x, y, z ∈ R with 0 ≤ z ≤ y ≤ x we have +Γ +� +x + 1 +2 +� +Γ +� +y + 1 +2 +� +≤ Γ +� +x + z + 1 +2 +� +Γ +� +y − z + 1 +2 +� +. +Proof. Adding Eq. (21) to itself with a and b interchanged gives, +log Γ +� +ta + (1 − t) b + 1 +2 +� ++ log Γ +� +(1 − t) a + tb + 1 +2 +� +≤ log Γ +� +a + 1 +2 +� ++ log Γ +� +b + 1 +2 +� +. +We can choose a = x + z, b = y − z and t = 1 − z/(x − y + 2z), as t ∈ [0, 1] and a, b ≥ 0 are +fulfilled because z ≤ y ≤ x. Exponentiating gives the stated inequality. +□ +Corollary 4.9. For all x, y ∈ R with x, y ≥ 0 we have +Γ +� +x + 1 +2 +� +√π +Γ +� +y + 1 +2 +� +√π +≤ Γ +� +x + y + 1 +2 +� +√π +(22) +Proof. Specializing Lemma 4.8 with y = z and using Γ(1/2) = √π gives the bound. +□ +Lemma 4.10. The exists a constant C such that for all x, y ∈ R with x, y ≥ 0 we have +Γ +� +x + y + 1 +2 +� +≤ C1+x+yΓ +� +x + 1 +2 +� +Γ +� +y + 1 +2 +� +. +(23) +Proof. Specializing Lemma 4.8 to x = y = (x′ + y′)/2 and z = |x′ − y′|/2 gives, +Γ +�x′ + y′ + 1 +2 +�2 +≤ Γ +� +x′ + 1 +2 +� +Γ +� +y′ + 1 +2 +� +for all x′, y′ ∈ R with x′, y′ ≥ 0. +The duplication formula for the Γ function [1, Eq. (3.11)] can be written as +Γ +�z + 1 +2 +� += 2−z√πΓ(z + 1)/Γ +�z +2 + 1 +� +. +(24) +Applying this identity once to the left hand side of the inequality above results in +Γ +� +x + y + 1 +2 +� +≤ 2x+y +√π F(x + y)Γ +� +x + 1 +2 +� +Γ +� +y + 1 +2 +� +for all x, y ∈ R with x, y ≥ 0. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +24 +where F(z) = Γ +� z +2 + 1 +� +Γ +� +z + 1 +2 +� +/(Γ +� z+1 +2 +� +Γ(z+1)). To prove the upper bound in the statement +it is therefore sufficient to show that there is a constant C such that F(z) ≤ C for all z ≥ 0. +F(z) is regular for all z ≥ 0, so it is sufficient to prove the existence of such a constant for z ≥ 1. +In this case, we have F(z) ≤ 1, because z+1 +2 +≤ z and can apply Lemma 4.8 to establish +Γ +�z +2 + 1 +� +Γ +� +z + 1 +2 +� +≤ Γ +�z + 1 +2 +� +Γ(z + 1) +for all z ≥ 1. +□ +We now apply these lemmas to bound the numbers ηλ defined in Corollary 3.5. +Lemma 4.11. There exists a constant C such that +ηk,m ≤ (kC)m/2 +√π +Γ +�m + 1 +2 +� +for all k ≥ 1 and m ≥ 0. +Proof. Because ηk,0 = 1 and Γ(1/2) = √π, the statement is obvious if m = 0. In all other cases it +is sufficient to prove the bound for m large. By the standard identity (2n−1)!! = 2nΓ(n+ 1 +2)/√π +and by the definition of ηk,m the statement is true for all odd k. For even k, +ηk,m = +⌊m/2⌋ +� +r=0 +�m +2r +� +kr(2r − 1)!! = +⌊m/2⌋ +� +r=0 +�m +2r +� +kr2r Γ(r + 1 +2) +√π +≤ 2m +max +0≤r≤⌊m/2⌋(2k)r Γ(r + 1 +2) +√π +, +where we used the floor function ⌊·⌋ and +�m +2r +� +≤ 2m. The statement follows since Γ(x) is increasing +for sufficiently large x. +□ +Recall that ℓ(λ) denotes the length of λ, i.e. the number of parts of the partition. +Corollary 4.12. There is a constant C such that for all integer partitions λ, +ηλ ≤ C|λ| +√π Γ +�ℓ(λ) + 1 +2 +� +. +Proof. Let λ = [1m12m2 · · · ] and note that mk = 0 for all k > |λ|. Recall that log k ≤ k for all +k ≥ 1. Therefore, +|λ| +� +k=1 +kmk/2 = exp + + +|λ| +� +k=1 +mk/2 log k + + ≤ exp + + +|λ| +� +k=1 +kmk/2 + + = e|λ|/2. +By Lemma 4.11 there is a constant C such that +ηλ = +|λ| +� +k=1 +ηk,mk ≤ +|λ| +� +k=1 +(kC)mk/2 +√π +Γ +�mk + 1 +2 +� +. +The proof is completed by using the first inequality, the fact that � +k≥1 mk = ℓ(λ) ≤ |λ| and the +bound from Corollary 4.9. +□ +Finally, the last lemma in this section shows how the constant e− 1 +4 arises: +Lemma 4.13. For large n, +n +� +m=0 +(−1)m +2mm! η2,m = e− 1 +4 + O +� 1 +Ln +� +. +In fact this sum converges much faster than the indicated error term, but this rough error +estimate is sufficient for our purpose. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +25 +Proof. By Corollary 3.5 we have η2,m = �⌊m/2⌋ +r=0 +�m +2r +� +2r(2r − 1)!!. Using this, we find that +∞ +� +m=0 +(−1)m +2mm! η2,m = +∞ +� +m=0 +(−1)m +2mm! m! +⌊m/2⌋ +� +r=0 +2r +2rr!(m − 2r)! = +∞ +� +r=0 +1 +r! +∞ +� +m=2r +(−1)m +2m(m − 2r)! += +∞ +� +r=0 +(−1)2r +4rr! +∞ +� +m=0 +(−1)m +2mm! = e +1 +4 e− 1 +2 = e− 1 +4 . +By Lemma 4.11 we find a constant C such that the tail of this series is bounded as follows: +����� +∞ +� +m=n+1 +(−1)m +2mm! η2,m +����� ≤ +∞ +� +m=n+1 +Cm Γ( m+1 +2 ) +√πm! = Cn+1 +∞ +� +m=0 +Cm +Γ( m+n+2 +2 +) +√π(m + n + 1)! +≤ Cn+1C′1+ n+1 +2 Γ( n+2 +2 ) +√πn! +∞ +� +m=0 +CmC′ +m +2 Γ( m+1 +2 ) +√πm! ∈ O +� +CnC′ +n +2 Γ( n+1 +2 ) +n! +� +⊂ O +� 1 +Ln +� +, +where we used a constant C′ from Lemma 4.10 to split the Γ function in the numerator and +Corollary 4.9 to split the factorial function in the denominator. The convergence of the infinite +sum and the last inclusion of sets follows from Stirling’s approximation of the Γ function. +□ +4.4. A new sequence of numbers. Let +Pn = +n +� +m=0 +�χn−m +�− 1 +2(n − m) +m +� +η2,m, +where the binomial coefficient +�q +k +� +is defined by q(q−1)···(q−k+1) +k! +for integers k ≥ 0 and all q ∈ +R. In this section we will use the fact that we know the asymptotic behavior of the numbers +�χn (Proposition 4.3) together with the estimates from the previous section to determine the +asymptotic behavior of the numbers Pn. In the two sections after this one we will prove +Proposition 4.14. +�en = Pn + O +� +Γ +� +n − 7 +12 +�� +. +At the end of this section we observe that Proposition 4.14 together with the asymptotics of +Pn imply the main asymptotic result, Theorem 4.2. +In order to determine the asymptotic behavior of the Pn we need a few more estimates. +Lemma 4.15. There exists a constant C such that +1 ≤ +m−1 +� +r=0 +n − m + 2r +n − m − 1 +2 + r ≤ exp +� +C m(m + 1) +n +� +for all integers n, m with 0 ≤ m ≤ n − 1 and n ≥ 1. +Proof. The lower bound is obvious as n − m + 2r ≥ n − m − 1 +2 + r. We start by proving the +bound for all m ≥ n−1 +2 . We have +m−1 +� +r=0 +n − m + 2r +n − m − 1 +2 + r = 2m +m−1 +� +r=0 +n−m +2 ++ r +n − m − 1 +2 + r ≤ 2m = 2m n +n ≤ 2 +m(2m+1) +n +≤ 4 +m(m+1) +n +, +as n − m ≥ 1 ⇒ n−m +2 ++ r ≤ n − m − 1 +2 + r and 2m + 1 ≥ n. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +26 +We still have to prove the bound for m ≤ n−1 +2 . Recall that 1 + x ≤ ex for all x ∈ R. It follows +that +1 +1−x ≤ e +x +1−x for x < 1. Therefore, +m−1 +� +r=0 +n − m + 2r +n − m − 1 +2 + r = +m−1 +� +r=0 +1 + 2r−m +n +1 − m+ 1 +2 −r +n +≤ +m−1 +� +r=0 +exp +� +2r − m +n ++ +m+ 1 +2 −r +n +1 − m+ 1 +2 −r +n +� +≤ +m−1 +� +r=0 +exp +�2r − m +n ++ 2m + 1 +2 − r +n +� += exp +�m(m + 1) +n +� +, +where we used m ≤ n−1 +2 +⇒ 1/ +� +1 − m+ 1 +2 −r +n +� +≤ 1/ +� 1 +2 + r +n +� +≤ 2. +□ +Lemma 4.16. There exists a constant C such that +1 ≤ +log(n + 1) +log(n − m + 1) ≤ exp +� +C m(m + 1) +n +� +for all integers n, m with 0 ≤ m ≤ n − 1 and n ≥ 1. +Proof. The lower bound is obvious. We start by proving the estimate for m > n+1 +2 . Because log +grows slower than the exponential, there exists a constant C′ such that +log(n + 1) +log(n − m + 1) ≤ log(n + 1) +log 2 +≤ C′n ≤ (C′2) +n+1 +2 +≤ (C′2)m ≤ (C′4) +m(m+1) +n +. +The bounds remains to be proven for all m ≤ n+1 +2 . Again we use the inequality +1 +1−x ≤ exp +x +1−x ⇒ +log +1 +1−x ≤ +x +1−x which holds for all x ∈ [0, 1) to get +log(n + 1) +log(n − m + 1) = +1 +1 + +log(1− +m +n+1 ) +log(n+1) +≤ exp + + − +log(1− +m +n+1 ) +log(n+1) +1 + +log(1− +m +n+1 ) +log(n+1) + + = exp +�− log(1 − +m +n+1) +log(n − m + 1) +� +≤ exp +�log +1 +1− +m +n+1 +log 2 +� +≤ exp +� +1 +log 2 +m +n+1 +1 − +m +n+1 +� +≤ exp +� +2 +log 2 +m +n + 1 +� +, +where we used 1/ +� +1 − +m +n+1 +� +≤ 1 +2 in the last step. +□ +Recall the sequences Bn = − +1 +√ +2π +Γ(n− 1 +2 ) +log2 n +and Ln = +log n +log log n from the beginning of Section 4. +Because these sequences are not defined for n = 1, it is convenient to use following ones instead +B′ +n = − +1 +√ +2π +Γ(n − 1 +2) +log2(n + 1) +L′ +n = +log(n + 1) +log log(n + e). +It is clear that O(B′ +n) = O(Bn) and O(L′ +n) = O(Ln), but we also have control over the error: +Lemma 4.17. B′ +n = Bn + O +� Bn +n +� +and 1/L′ +n = 1/Ln + O +� +1 +nLn +� +. +Proof. This follows from the fact that the following elementary limits exist: +lim +n→∞ +1/ log2(n + 1) − 1/ log2 n +1/(n log2 n) +and +lim +n→∞ +log log(n + e)/ log(n + 1) − log log n/ log n +log log n/(n log n) +. +This can be shown, for instance, by using log(n + x) − log n ≤ x/n. +□ +Lemma 4.18. Let +Qn,m = �χn−m /B′ +n−m − 1. +There exists a constant C such that |Qn,m| ≤ Cm+1/L′ +n for all n, m with m ≥ 0 and n ≥ m + 1. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +27 +Proof. By Proposition 4.3 and Lemma 4.17 we have �χn −B′ +n ∈ O(B′ +n/L′ +n). Because B′ +n, �χn and +L′ +n are finite for all n ≥ 1, there exists a constant C′ such that |�χn/B′ +n − 1| ≤ C′/L′ +n for all +n ≥ 1. It follows that for all integers n, m with m ≥ 0 and n ≥ m + 1. +|Qn,m| = +���χn−m /B′ +n−m − 1 +�� ≤ C′/L′ +n−m ≤ C′ exp +� +C′′ m(m + 1) +n +� log log(n − m + e) +log(n + 1) +, +where we used a constant C′′ from Lemma 4.16. Using m + 1 ≤ n and the monotonicity of log +gives the bound. +□ +Lemma 4.19. Let +Rn,m = (−1)m2mm! +�− 1 +2(n − m) +m +� +B′ +n−m/B′ +n − 1. +There exists a constant C such that |Rn,m| ≤ Cm+1 +n +for all n, m with m ≥ 0 and n ≥ m + 1. +Proof. Use Γ(n − 1 +2) = Γ(n − m − 1 +2) �m−1 +r=0 (n − m − 1 +2 + r) and +� q +m +� += +1 +m! +�m−1 +r=0 (q − r) to get +Rn,m + 1 = (−1)m2m +log2(n + 1) +log2(n − m + 1) +m−1 +� +r=0 +− 1 +2(n − m) − r +n − m − 1 +2 + r += +log2(n + 1) +log2(n − m + 1) +m−1 +� +r=0 +n − m + 2r +n − m − 1 +2 + r. +Therefore, by Lemma 4.15 and 4.16 there exists a constant C′ such that +1 ≤ 1 + Rn,m ≤ exp +� +C′ m(m + 1) +n +� +for all integers n, m with m ≥ 0 and n ≥ m + 1. Because 1 − x ≤ e−x ⇒ ex ≤ 1 + xex for +all x ∈ R, we have |Rn,m| ≤ C′ m(m+1) +n +exp +� +C′ m(m+1) +n +� +≤ C′ m(m+1) +n +exp (mC′) for all m ≥ 0 +and n ≥ m + 1. The statement follows from the fact that we can find a constant C′′ such that +C′ m(m+1) +n +exp +� +C′ m(m+1) +n +� +≤ C′′m+1 +n +for all m ≥ 0 and n ≥ m + 1. +□ +We are now ready to estimate the numbers Pn. +Proposition 4.20. +Pn = e− 1 +4 B′ +n + O (B′ +n/L′ +n) . +Proof. With Qn,m and Rn,m from Lemmas 4.18 and 4.19 we have +Pn = +n +� +m=0 +�χn−m +�− 1 +2(n − m) +m +� +η2,m = B′ +n +n +� +m=0 +(−1)m +2mm! (1 + Qn,m)(1 + Rn,m)η2,m +By Lemma 4.11 there exists a constant C such that η2,m ≤ CmΓ( m+1 +2 ) for all m ≥ 0. From +Lemma 4.18 and 4.19 we get a constant C′ such that +n +� +m=0 +���� +(−1)m +2mm! (Qn,m + Rn,m + Qn,mRn,m)η2,m +���� +≤ C′ +� 1 +L′n ++ 1 +n + +1 +nL′n +� +∞ +� +m=0 +1 +2mm!(CC′)mΓ +�m + 1 +2 +� +∈ O (1/L′ +n) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +28 +where the sum over m is convergent due to the factorial m! = Γ(m + 1). Hence, +Pn = B′ +n +n +� +m=0 +(−1)m +2mm! η2,m + O (B′ +n/L′ +n) . +An application of Lemma 4.13 and using O(1/Ln) = O(1/L′ +n) concludes the proof. +□ +Proof of Theorem 4.2. We want to show that �en = e− 1 +4 Bn + O(Bn/Ln). By Proposition 4.14 +(which we will prove in the next section), �en = Pn + O (Γ (n − 7/12)) , and by Proposition 4.20 +Pn = e− 1 +4 B′ +n+O (B′ +n/L′ +n) . We now use the elementary fact that O (Γ (n − 7/12)) ⊂ O (B′ +n/L′ +n) = +O (Bn/Ln) together with the asymptotic equality B′ +n = Bn + O (Bn/n) from Lemma 4.17 to +finish the proof. +□ +4.5. Estimates for Proposition 4.14. We still need to prove Proposition 4.14. To do so we +need to establish that the contribution of H(u, x) in Theorem 4.7 is negligible to the asymptotic +behaviour of the numbers �en and the power series �T(u2e−W(u·¯x)) only contributes partially. We +first show that the coefficients of �T(u2e−W(u·¯x)) can be written explicitly using the numbers �χn. +Lemma 4.21. Let r ≥ 0 and δ be a deranged partition. Then +[urxδ] �T +� +u2e−W(u·¯x)� += �χ r−|δ| +2 +r +� +k=2 +�(r − |δ|) µ(k) +2k +mk(δ) +� +, +where we agree that �χ r−|δ| +2 += 0 if r−|δ| is odd or negative. In particular, [u0xδ] �T +� +u2e−W(u·¯x)� += +1 if δ = ∅ and 0 otherwise, and [u1xδ] �T +� +u2e−W(u·¯x)� += 0 for all δ. +Proof. Use �T(ℏ) = � �χn ℏn and W(x) = − � +k≥2 +µ(k) +k +log(1 + xk) to get +�T +� +u2e−W(u·¯x)� += +� +n≥0 +�χn u2n � +k≥2 +� +1 + ukxk +�n µ(k) +k += +� +n≥0 +�χn u2n � +k≥2 +� +mk≥0 +�n µ(k) +k +mk +� +ukmkxmk +k += +� +n≥0 +� +δ +�χn u2n+|δ|xδ � +k≥2 +�n µ(k) +k +mk(δ) +� +. +□ +Using this explicit formula we can now obtain a bound on the coefficients of �T(u2e−W(u·¯x)): +Corollary 4.22. There exists a constant C, such that for all r ≥ 2 and deranged partitions +δ = [2m23m3 · · · ] with |δ| ≤ r, +���[urxδ] �T +� +u2e−W(u·¯x)���� ≤ C +Γ +� +r−|δ|+2ℓ(δ)−1 +2 +� +m2!m3! +. +Proof. We start with the special cases where δ is a deranged partition of r or r − 1. In both +cases r ≥ 2 implies δ ̸= ∅, i.e. ℓ(δ) ≥ 1 and the argument of the Γ function on the right hand +side of the statement is positive. By Lemma 4.21 and the agreement that �χ 1 +2 = 0 and +� 0 +m +� += 0 +for all m ≥ 1, we find that the left hand side vanishes in these cases and the statement follows. +For |δ| ≤ r − 2 we can apply Corollary 4.4 to the statement of Lemma 4.21 to get a constant +C such that for all deranged partitions δ = [2m23m3 · · · ], +���[urxδ] �T +� +u2e−W(u·¯x)���� ≤ CΓ +�r − |δ| − 1 +2 +� ����� +r +� +k=2 +�(r − |δ|) µ(k) +2k +mk +������ . +(25) + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +29 +From the standard expression for the binomial coefficients +� q +m +� += q(q−1)···(q−m+1) +m! +, we get +����� +r +� +k=2 +�(r − |δ|) µ(k) +2k +mk +������ = +r +� +k=2 +��� +�mk−1 +s=0 +� +(r − |δ|) µ(k) +2k − s +���� +mk! +≤ +�r +k=2 +�mk−1 +s=0 +� +(r − |δ|) 1 +2k + s +� +m2!m3! +, +where we used |µ(n)| ≤ 1. By using xΓ(x) = Γ(x + 1) repeatedly we get +Γ +�r − |δ| − 1 +2 +� += +Γ +� +r−|δ|+2ℓ(δ)−1 +2 +� +�ℓ(δ)−1 +s=0 +� +r−|δ|−1 +2 ++ s +�. +Combining these observations with Eq. (25) gives +���[urxδ] �T +� +u2e−W(u·¯x)���� ≤ C +Γ +� +r−|δ|+2ℓ(δ)−1 +2 +� +m2!m3! +�r +k=2 +�mk−1 +s=0 +� +(r − |δ|) 1 +2k + s +� +�ℓ(δ)−1 +s=0 +� +r−|δ|−1 +2 ++ s +� +. +It is easy to verify that �r +k=2 +�mk−1 +s=0 +� +(r − |δ|) 1 +2k + s +� +≤ �ℓ(δ)−1 +s=0 +� +r−|δ|−1 +2 ++ s +� +by using ℓ(δ) = +�|δ| +k=2 mk and r − |δ| ≥ 2. +□ +It is substantially harder to prove a good estimate for the coefficients of H(u, x). It turns out +to be convenient to include the numbers ηλ in our estimate. +For integer partitions λ, λ′, we define λ ∪ λ′ to be the partition of |λ| + |λ′| that contains the +union of all parts of λ and λ′, i.e. mk(λ ∪ λ′) = mk(λ) + mk(λ′) for all k. +Proposition 4.23. There is a constant C such that for all r ≥ 0 and deranged partitions δ, +� +δ′ +ηδ∪δ′ +���[urxδ′]H(u, x) +��� ≤ C1+r+|δ|Γ +�ℓ(δ) + 5 +6r + 1 +2 +� +, +where we sum over all deranged integer partitions δ′. +The proof of this proposition will occupy the remainder of this subsection. +Recall that the series H(u, x) is defined by H = exp (h1 + h2 + h3) where +h1(u, x) = +� +k≥2 +u−2k V((u · x)[k]) +k +, +h2(u, x) = W(u · x) +2 +, +h3(u, x) = u−2 � +eW(u·¯x) − 1 − u2 x2 +2 +� +. +Lemma 4.24. [urxδ] log H(u, x) = 0 unless either +(1) r ≥ 2 and |δ| = r +(2) r ≥ 1 and |δ| = r + 2 or +(3) |δ| = r +2k for some k ≥ 2 dividing r and δ = kµ = (kµ1, . . . , kµℓ) for some partition µ. +Proof. The nonzero terms of of h2 are all of the form (1), the nonzero terms of h3 are of the +form (2) and the nonzero terms of h1 are of the form (3). +□ +Let J be the set of pairs (r, δ) satisfying the conditions of Lemma 4.24. +Corollary 4.25. For all (r, δ) ∈ J, we have |δ| ≤ 3r. +Proof. The statement is obvious for the first two cases of Lemma 4.24, and follows in the third +case because k divides r, so is at most r. +□ + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +30 +Remark 4.26. This corollary also follows immediately from the combinatorics: An admissible +graph G with χ(G) = −n has at most 3n edges, so if pairing all leaves of some forest produces an +an admissible graph with −2χ(G) = r then the forest cannot have more than 3r(= 6n) leaves. +Lemma 4.27. All coefficients of log H have absolute value less than 1. +Proof. Since the nonzero terms of h1, h2 and h3 have no monomials in common, we can consider +their coefficients separately. It is clear from their definitions that the coefficients of W(x) and +V(x) are less than 1, so the coefficients of h2 and h1 are as well. For h3, we have +eW(x) = +� +k≥2 +(1 + xk)− µ(k) +k += +� +k≥2 + +� +m≥0 +�− µ(k) +k +m +� +xm +k + + += +� +k≥2 + +� +m≥0 +− µ(k) +k +1 +· − µ(k) +k +− 1 +2 +· · · − µ(k) +k +− m + 1 +m +xm +k + + . +The magnitude of the fractions | − µ(k) +k +−ℓ+1 +ℓ +| is always smaller than 1, because |µ(k)| ≤ 1. Hence +all coefficients of eW(x) (and therefore of h3) are also less than 1. +□ +The next lemma shows that we can bound the coefficients of H(u, x) without determining the +explicit values of those coefficients. +Lemma 4.28. Let J be the set of pairs (r, δ) satisfying the conditions of Lemma 4.24. For all +C ≥ 1 +� +ℓ(δ′)=ℓ′ +C|δ′| ���[ur′xδ′]H(u, x) +��� ≤ C3r′[ur′xℓ′] exp + + � +(r,δ)∈J +urxℓ(δ) + + , +where we sum over all deranged partitions δ′ of length ℓ′ on the left hand side. +Proof. By Lemma 4.27 and the positivity of the expansion exp(x) = �∞ +n=0 +xn +n! , +C|δ′| ���[ur′xδ′]H(u, x) +��� ≤ C|δ′|[ur′xδ′] exp + + � +(r,δ)∈J +urxδ + + . +By Corollary 4.25 the coefficient extraction operator has support only for |δ| ≤ 3r. Since C ≥ 1, +C|δ| ≤ C3r. Summing over all deranged partitions δ of the same length on the left is equivalent +to substituting xi = x for all i on the right. +□ +Our next estimate will make use of the of the following rough bound on integer partitions. +Lemma 4.29. The number of integer partitions of size at most n is smaller than 2n. +Proof. Writing a string of length n of 1’s with either a + or a comma in between gives a compo- +sition of n, i.e. an integer partition of n with an ordering of the parts. There are 2n−1 different +such compositions of n. Ordering the elements of a composition gives a many-to-one function to +integer partitions, which is clearly surjective for all n ≥ 1. Thus the number of integer partitions +of n is bounded by 2n−1 for n ≥ 1. The number of integer partitions of size at most n is therefore +smaller than 1+20 +21 +. . .+2n−1, where the initial 1 accounts for the empty integer partition. +Since 20 + . . . + 2n−1 = 2n − 1, the statement follows. +□ +Now let K be the subset of pairs (r, δ) in J for which 5 +6r < ℓ(δ). + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +31 +Lemma 4.30. There is a constant C such that for all r′, ℓ′ ≥ 0, +[ur′xℓ′] exp + + +� +(r,δ)∈J\K +urxℓ(δ) + + ≤ Cr′. +Moreover, [ur′xℓ′] exp +�� +(r,δ)∈J\K urxℓ(δ)� += 0 when 5 +6r′ < ℓ′. +Proof. The second statement follows immediately from the definition of K. +By Corollary 4.25, the set J \ K is contained in the set of all pairs (r, λ) where r ≥ 1 +and λ is a partition such that |λ| ≤ 3r. +By Lemma 4.29 this implies that � +(r,δ)∈J\K ur is +bounded coefficientwise by � +r≥1 23rur. Since log r ≤ r for all r ≥ 1, the series � +r≥1 23rur = +� +r≥1 ur8relog r/r, is bounded coefficientwise by � +r≥1 ur(e8)r/r. Summing over ℓ′ on the left +hand side of the inequality in the statement of the lemma is equivalent to setting x = 1. Hence, +[ur′xℓ′] exp + + +� +(r,δ)∈J\K +urxℓ(δ) + + ≤ [ur′] exp + + +� +(r,δ)∈J\K +ur + + ≤ [ur′] exp + +� +r≥1 +(8e)r +r +ur + + = (8e)r′, +where we used log +1 +1−x = � +r≥1 +xr +r in the last step. +□ +Corollary 4.31. There is a constant C such that for all r′ ≥ 0 and z ∈ R with z ≥ 0, +� +ℓ′≥0 +Γ +�z + ℓ′ + 1 +2 +� +[ur′xℓ′] exp + + +� +(r,δ)∈J\K +urxℓ(δ) + + ≤ Cr′+1Γ +�z + 5 +6r′ + 1 +2 +� +. +Proof. By Lemma 4.30 there is a constant C′ such that +� +ℓ′≥0 +Γ +�z + ℓ′ + 1 +2 +� +[ur′xℓ′] exp + + +� +(r,δ)∈J\K +urxℓ(δ) + + ≤ C′r′ +� +0≤ℓ′≤⌊ 5 +6 r′⌋ +Γ +�z + ℓ′ + 1 +2 +� +. +By Corollary 4.9, Γ +� +z+ℓ′+1 +2 +� +Γ +� 5 +6 r′−ℓ′+1 +2 +� +≤ √πΓ +� z+ 5 +6 r′+1 +2 +� +for all 0 ≤ ℓ′ ≤ 5 +6r′. Hence, +� +0≤ℓ′≤⌊ 5 +6 r′⌋ +Γ +�z + ℓ′ + 1 +2 +� +≤ √πΓ +�z + 5 +6r′ + 1 +2 +� +� +0≤ℓ′≤⌊ 5 +6 r′⌋ +1 +Γ +� 5 +6 r′−ℓ′+1 +2 +�. +The sum over ℓ′ is bounded by a constant independent of r′ as +� +0≤ℓ′≤⌊ 5 +6 r′⌋ +1 +Γ +� 5 +6 r′−ℓ′+1 +2 +� = +� +0≤ℓ′≤⌊ 5 +6 r′⌋ +1 +Γ +� 5 +6 r′−⌊ 5 +6 r′⌋+ℓ′+1 +2 +� ≤ +C′′ +5 +6 r′−⌊ 5 +6 r′⌋ +Γ +� 5 +6 r′−⌊ 5 +6 r′⌋+1 +2 +� +� +0≤ℓ′≤∞ +C′′ℓ′ +Γ +� ℓ′+1 +2 +�, +where we used Lemma 4.10 to split the Γ function. +□ +Lemma 4.32. The set K is a subset of +K = {(1, [31]), (2, [23]), (2, [2141]), (2, [22]), (3, [33]), (4, [24])}. +Proof. By Lemma 4.24, for all (r, δ) ∈ J we have |δ| = r + 2k for some k ≥ 0, and if k ̸= 0 then +k divides r and δ = kµ for some partition µ. Recall that K is the subset of pairs (r, δ) in J for +which 5 +6r < ℓ(δ). Since each part of a deranged partition has size at least 2, ℓ(δ) ≤ |δ|/2, so +5 +6r < ℓ(δ) implies 2r < 6k. We look for pairs (r, δ) that fulfill all these conditions. +If k = 0, there are no solutions. +If k = 1, the only solutions are r ∈ {1, 2} and the possible deranged partitions δ are δ ∈ {[31]} +for r = 1 and δ ∈ {[22], [41]} for r = 2. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +32 +If k ≥ 2, the additional constraint that k divides r implies that (r, k) ∈ {(2, 2), (3, 3), (4, 2)}. +For both (r, k) = (2, 2) and (r, k) = (3, 3), we have |µ| = 3, which means µ ∈ {[13], [1121], [31]}. +Hence, δ ∈ {[23], [2141], [61]} for (r, k) = (2, 2) and δ ∈ {[33], [3161], [91]} for (r, k) = (3, 3). +For (r, k) = (4, 2) we have δ = 2µ with µ ∈ {[14], [1221], [22], [1131], [41]} and therefore δ ∈ +{[24], [2241], [42], [2161], [81]}. +The requirement ℓ(δ) > 5 +6r now eliminates all solutions that are not in the list K. +□ +Corollary 4.33. For all (r, δ) ∈ K, we have 5 +6r < ℓ(δ) < 5 +6r + 2. +Proof. This can be verified by checking all 6 elements in Corollary 4.32. +□ +Lemma 4.34. Fix (r, δ) ∈ K. There is a constant C such that for r′ ≥ 0 and z ∈ R with z ≥ 0, +� +ℓ′≥0 +[ur′xℓ′] exp(urxℓ(δ))Γ +�z + ℓ′ + 1 +2 +� +≤ Cz+r′+1Γ +�z + 5 +6r′ + 1 +2 +� +. +Proof. We can expand the left hand side to get +� +ℓ′≥0 +[ur′xℓ′] exp(urxℓ(δ))Γ +�z + ℓ′ + 1 +2 +� += +� +ℓ′≥0 +[ur′xℓ′] +� +k≥0 +urkxℓ(δ)k +k! +Γ +�z + ℓ′ + 1 +2 +� +. +The sum over k only contributes if rk = r′. Hence, if r divides r′ we get +� +ℓ′≥0 +[ur′xℓ′] exp(urxℓ(δ))Γ +�z + ℓ′ + 1 +2 +� += +1 +(r′/r)!Γ +� +z + ℓ(δ) r′ +r + 1 +2 +� +and zero otherwise. By Lemma 4.10 there exists a constant C′ such that for α with 0 ≤ α ≤ ℓ(δ), +1 +(r′/r)!Γ +� +z + ℓ(δ) r′ +r + 1 +2 +� +≤ +1 +(r′/r)!C′ +z +2 +ℓ(δ) r′ +2r +1Γ +� +α r′ +2r + 1 +2 +� +Γ +� +z + (ℓ(δ) − α) r′ +r + 1 +2 +� +. +Recall that n! = Γ(n + 1). Hence, Γ +� +α r′ +2r + 1 +2 +� +/(r′/r)! is bounded for all r′ ≥ 0 if α ≤ 2. We +may choose α = ℓ(δ) − 5 +6r, which fulfills 0 ≤ α ≤ 2 as 5 +6r ≤ ℓ(δ) ≤ 2 + 5 +6r by Corollary 4.33. +□ +Lemma 4.35. Let +Ar′ = +� +ℓ′≥0 +Γ +�ℓ′ + 1 +2 +� +[ur′xℓ′] exp + + � +(r,δ)∈J +urxℓ(δ) + + . +There is a constant C such that Ar′ ≤ Cr′+1Γ +� 5 +6 r′+1 +2 +� +for all r′ ≥ 0. +Proof. We can split the exponential in the definition of Ar′, +exp + + � +(r,δ)∈J +urxℓ(δ) + + = exp + + � +(r,δ)∈K +urxℓ(δ) + + exp + + +� +(r,δ)∈J\K +urxℓ(δ) + + +By Lemma 4.32, exp +�� +(r,δ)∈K urxℓ(δ)� +is bounded by exp +�� +(r,δ)∈K urxℓ(δ)� +coefficientwise. +Let (r1, δ1), . . . , (r6, δ6) denote the elements of K in Lemma 4.32. With this notation, +exp + + � +(r,δ)∈K +urxℓ(δ) + + = +6 +� +i=1 +eurixℓ(δi). + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +33 +We may use this to write Ar′ as +Ar′ = +� +Γ +��7 +i=1 ℓ′ +i + 1 +2 +� � 6 +� +i=1 +[ur′ +ixℓ′ +i]eurixℓ(δi) +� +[ur′ +7xℓ′ +7] exp + + +� +(r,δ)∈J\K +urxℓ(δ) + + , +where we have to sum over all tuples of integers ℓ′ +1, . . . , ℓ′ +7, r′ +1, . . . , r′ +7 ≥ 0 with r′ +1 + . . . + r′ +7 = r′. +Applying Corollary 4.31 once and Lemma 4.34 six times results in the bound +Ar′ ≤ +� +C′ +�7 +i=1 r′ +i+1Γ +� +5 +6 +�7 +i=1 r′ +i + 1 +2 +� +, +where C′ is an appropriate constant, whose existence follows from Lemma 4.34 and Corollary 4.31, +and we sum over all r′ +1, . . . , r′ +7 ≥ 0 with r′ +1 + . . . + r′ +7 = r′. The terms in the sum are constant +and their number is +�r′+7−1 +r′ +� +, which is smaller than 2r′+6 by the binomial theorem. +□ +Proof of Proposition 4.23. By Corollary 4.12 and Lemma 4.10 there are constants C′ and C′′ +such that for all δ and δ′, +ηδ∪δ′ ≤ C′|δ|+|δ′| +√π +Γ +�ℓ(δ) + ℓ(δ′) + 1 +2 +� +≤ C′|δ|+|δ′|C′′ℓ(δ)+ℓ(δ′) +√π +Γ +�ℓ(δ) + 1 +2 +� +Γ +�ℓ(δ′) + 1 +2 +� +. +We may assume that �C = C′C′′ > 1 and use the fact that ℓ(δ) ≤ |δ| and ℓ(δ′) ≤ |δ′| to obtain +� +δ′ +ηδ∪δ′ +���[ur′xδ′]H(u, x) +��� ≤ +�C|δ| +√π Γ +�ℓ(δ) + 1 +2 +� � +ℓ′≥0 +Γ +�ℓ′ + 1 +2 +� +� +ℓ(δ′)=ℓ′ +�C|δ′| ���[ur′xδ′]H(u, x) +��� +for all r′ ≥ 0 and deranged partitions δ. By Lemma 4.28 this last expression is bounded by +≤ +�C|δ| +√π Γ +�ℓ(δ) + 1 +2 +� � +ℓ′≥0 +Γ +�ℓ′ + 1 +2 +� +�C3r′[ur′xℓ′] exp + + � +(r,δ)∈J +urxℓ(δ) + + , +for all r′ and δ. The statement follows by estimating the sum over ℓ′ using Lemma 4.35. +□ +4.6. Proof of Proposition 4.14. +Lemma 4.36. There exists a constant C such that for all n, r ≥ 0 with r ≤ 2n−2 and deranged +partitions δ = [2m2 · · · ] with |δ| ≤ 2n − r, +���[u2n−rxδ] �T +� +u2e−W(u·¯x)���� +� +δ′ +ηδ∪δ′ +���[urxδ′]H(u, x) +��� +≤ +C1+r+|δ| +Γ +� m2 +2 + 1 +�Γ +� +n − +1 +6(r + |δ| − 2m2) + 1 +2 +� +. +Proof. By Corollary 4.22 and Proposition 4.23 we can find constants C1 and C2 such that for all +2n − r ≥ 2 and deranged partitions δ = [2m23m3 · · · ] with |δ| ≤ 2n − r, +���[u2n−rxδ] �T +� +u2e−W(u·¯x)���� +� +δ′ +ηδ∪δ′ +���[urxδ′]H(u, x) +��� +≤ C1C1+r+|δ| +2 +Γ +� +n − r+|δ|−2ℓ(δ)+1 +2 +� +m2!m3! +Γ +�ℓ(δ) + 5 +6r + 1 +2 +� +. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +34 +Recall that ℓ(δ) = �|δ| +k=2 mk. By Lemma 4.10 there is a constant C3 such that +Γ +�ℓ(δ) + 5 +6r + 1 +2 +� +≤ C +1+ +ℓ(δ)+ 5 +6 r +2 +3 +C1+m2+m3 +3 +Γ +�m2 + 1 +2 +� +Γ +�m3 + 1 +2 +� +Γ +�ℓ(δ) − m2 − m3 + 5 +6r + 1 +2 +� +. +By the duplication formula of the Γ function (Eq. (24)), Γ +� m+1 +2 +� +/m! = √π/(2mΓ(m/2 + 1)). +Combining all this and using Corollary 4.9 to merge the Γ functions, we find that +���[u2n−rxδ] �T +� +u2e−W(u·¯x)���� +� +δ′ +ηδ∪δ′ +���[urxδ′]H(u, x) +��� +≤ √π +3 C1C1+r+|δ| +2 +C +2+m2+m3+ +ℓ(δ)+ 5 +6 r +2 +3 +2m2+m3 +Γ +� +n − +1 +6 r+|δ|+m2+m3−3ℓ(δ)+1 +2 +� +Γ (m2/2 + 1) Γ (m3/2 + 1) +. +Moreover, we can use Corollary 4.9 to get the bound, +Γ +� +n − +1 +6r + |δ| + m2 + m3 − 3ℓ(δ) + 1 +2 +� +≤ √π +Γ +� +n − +1 +6 (r+|δ|−2m2)+1 +2 +� +Γ +� 5 +6 |δ|+ 4 +3 m2+m3−3ℓ(δ)+1 +2 +�, +where the denominator is bounded form below as 5 +6|δ| + 4 +3m2 + m3 − 3ℓ(δ) ≥ 0 for all deranged +partitions δ, which follows from |δ| = �|δ| +k=2 kmk, and Γ(x) does not vanish for x > 0. +□ +Corollary 4.37. For given n ≥ 1 and s ≤ 2n, let +Bn,s = +� +ηδ∪δ′ +� +[u2n−rxδ] �T +� +u2e−W(u·¯x)�� � +[urxδ′]H(u, x) +� +, +where the sum is over all integers r and all pairs of deranged partitions (δ, δ′) with 0 ≤ r ≤ 2n−2, +δ = [2m23m3 · · · ] and the restriction that r + �|δ| +k=3 kmk = r + |δ| − 2m2 = s. We have +2n +� +s=1 +Bn,s ∈ O +� +Γ +� +n − 7 +12 +�� +. +Proof. By Lemma 4.36 there exists a constant C such that +|Bn,s| ≤ +� +C1+r+|δ| +Γ +� m2 +2 + 1 +�Γ +� +n − +1 +6s + 1 +2 +� +, +where the sum runs over all integers r and deranged partitions δ with 0 ≤ r ≤ 2n − 2 and +r + �|δ| +k=3 kmk = s. This suggests to treat the parts of size 2 of the partition δ separately: We +also have the bound +|Bn,s| ≤ +� +r≥0,δ(3) +r+|δ(3)|=s +� +m2≥0 +C1+r+|δ(3)|+2m2 +Γ +� m2 +2 + 1 +� +Γ +� +n − +1 +6s + 1 +2 +� +, +where we sum over pairs (r, δ(3)) where r is an integer ≥ 0 and δ(3) is an integer partition where +each part has size at least 3 with r + |δ(3)| = s. There are at most as many such pairs (r, δ(3)) as +there are integer partitions of s and, by Lemma 4.29, there are fewer than 2s integer partitions +of s. Therefore, +|Bn,s| ≤ 2s � +m2≥0 +C1+s+2m2 +Γ +� m2 +2 + 1 +�Γ +� +n − +1 +6s + 1 +2 +� += 2sC′C1+sΓ +� +n − +1 +6s + 1 +2 +� +, + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +35 +where C′ = � +m2≥0 +C2m2 +Γ( +m2 +2 +1), which is obviously convergent. By Corollary 4.9, we have for all +2n ≥ s ≥ 1, Γ +� +n − +1 +6 s+1 +2 +� +≤ √πΓ +� +n − 7 +12 +� +/Γ +� 1 +6 (s−1)+1 +2 +� +. Hence +2n +� +s=1 +|Bn,s| ≤ √π +2n +� +s=1 +2sC′C1+s +Γ +� +n − 7 +12 +� +Γ +� 1 +6 (s−1)+1 +2 +� ≤ √πCC′Γ +� +n − 7 +12 +� ∞ +� +s=1 +2sCs +Γ +� 1 +6 (s−1)+1 +2 +�, +where the sum over s is convergent. +□ +Proof of Proposition 4.14. By Theorem 4.7 we have +�en = +2n +� +r=0 +� +δ,δ′ +ηδ∪δ′ +� +[u2n−rxδ] �T +� +u2e−W(u·¯x)�� � +[urxδ′]H(u, x) +� +where we sum over all pairs of deranged partitions δ and δ′. +By Lemma 4.21, the expression [u2n−rxδ] �T +� +u2e−W(u·¯x)� +vanishes if r = 2n − 1 or if r = 2n +and δ ̸= ∅, while [u1x∅] �T +� +u2e−W(u·¯x)� += 1. Therefore +2n +� +r=2n−1 +� +δ,δ′ +ηδ∪δ′ +� +[u2n−rxδ] �T +� +u2e−W(u·¯x)�� � +[urxδ′]H(u, x) +� += +� +δ′ +ηδ′[u2nxδ′]H(u, x). +By Proposition 4.23, the right hand side is bounded by C1+2nΓ +� 5 +6n + 1 +2 +� +⊂ O +� +Γ +� +n − 7 +12 +�� +, so +�en = +2n−2 +� +r=0 +� +δ,δ′ +ηδ∪δ′ +� +[u2n−rxδ] �T +� +u2e−W(u · ¯x) +�� � +[urxδ′]H(u, x) +� ++ O +� +Γ +� +n − 7 +12 +�� +. +(26) +Using the notation and statement of Corollary 4.37, this becomes +�en = +2n +� +s=0 +Bn,s + O +� +Γ +� +n − 7 +12 +�� += Bn,0 + O +� +Γ +� +n − 7 +12 +�� +where Bn,0 is given by the expression in Eq. (26) restricted to summands where r + |δ| − 2m2 = +r+� +k≥3 kmk(δ) = 0. The sum over r therefore trivializes and we only have to account for r = 0 +and the sum over deranged partitions reduces to a sum over partitions that only have parts of +size 2. We may write this as, +� +m2≥0 +� +δ′ +η[2m2]∪δ′ +� +[u2nxm2 +2 x0 +3x0 +4 · · · ] �T +� +u2e−W(u·¯x)�� � +[u0xδ′]H(u, x) +� += +� +m2≥0 +η[2m2 ][u2nxm2 +2 x0 +3x0 +4 · · · ] �T +� +u2e−W(u·¯x)� +, +where we used the fact that [u0x∅]H(u, x) = 1 and [u0xδ′]H(u, x) = 0 if δ′ ̸= ∅. By Lemma 4.21, +[u2nxm2 +2 x0 +3x0 +4 · · · ] �T +� +u2e−W(u·¯x)� += �χ 2n−2m2 +2 +�(2n − 2m2) µ(2) +4 +m2 +� +. +Using µ(2) = −1 and η[2m2 ] = η2,m2 gives the statement. +□ + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +36 +5. The odd forested graph complex +As remarked in Section 2.3, the Euler characteristic e(Out(Fn)) is equal to the Euler char- +acteristic of Kontsevich’s Lie graph complex, which is equal to the Euler characteristic of the +forested graph complex. In [22] Kontsevich defined an odd version1 of the Lie graph complex. +The two graph complexes differ only by the definition of the orientation of a graph. +Recall that the (even) forested graph complex is generated by all even forested graphs (see +Section 2.3), which are forested graphs with no automorphisms α that induce an odd permutation +αΦ : EΦ → EΦ on the forest edges EΦ. In other words, all automorphisms α of an even forested +graph satisfy sign(αΦ) = 1. +Every automorphism α of a graph G also induces an automorphism αH : H1(G, Z) → H1(G, Z). +An odd forested graph is a forested graph all of whose automorphisms α satisfy +sign(αΦ) det(αH) = 1. +The complex spanned by such odd forested graphs computes H∗(Out(Fn+1); �Q), where �Q is the +representation obtained by composing the canonical group homomorphism Out(Fn) → GLn(Z) +with the determinant map. The techniques developed in this paper can be used almost verbatim +to compute the associated Euler characteristic eodd(Out(Fn)) = � +k(−1)kHk(Out(Fn), �Q), which +is equal to the Euler characteristic of Kontsevich’s odd Lie graph complex. +As in Proposition 2.2 we define +� +[G,Φ] +(−1)e(Φ) = eodd(Out(Fn+1)), +where we sum over all isomorphism classes of connected odd forested graphs [G, Φ] of Euler +characteristic χ(G) = −n. As we did in Section 2.3 we also define +�eodd +n += +� +[G,Φ] +(−1)e(Φ), +(27) +where we sum over all isomorphism classes of possibly disconnected odd forested graphs of Euler +characteristic χ(G) = −n. Theorem 2.5 works for odd forested graphs as well as for even ones, +giving +� +n≥0 +�eodd +n +ℏn = +∞ +� +n=1 +� +1 +1 − ℏn +�eodd(Out(Fn+1)) +and we can compute eodd(Out(Fn+1)) from the coefficients �eodd +n +. +Recall that for a forested graph (G, Φ) and an automorphism α ∈ Aut(G, Φ), the number +eα(Φ) denotes the number of cycles of the permutation on the forest edges, αΦ : EΦ → EΦ. +Lemma 5.1. For a forested graph (G, Φ), the sum � +α∈Aut(G,Φ) det(αH)(−1)eα(Φ) is equal to +(−1)e(Φ)|Aut(G, Φ)| if (G, Φ) is odd or 0 otherwise. +Proof. The argument for Lemma 2.4 applies. +□ +Each automorphism α of a forested graph (G, Φ) also provides us with a permutation αEG\Φ : +EG \ EΦ → EG \ EΦ that permutes the non-forest edges, a set of permutations (αe)e∈EG\EΦ of +order 1 or 2 that might change the orientation of each non-forest edge, and a permutation αH0(Φ) +that permutes the connected components of the forest. +1In [28], Willwacher used the opposite notions of “even” and “odd” orientation of a graph, so that Kontsevich’s +“odd” graph complexes are Willwacher’s “even” graph complexes. + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +37 +Lemma 5.2. For a connected forested graph (G, Φ) and an automorphism α : (G, Φ) → (G, Φ), +det(αH) = sign(αH0(Φ))sign(αEG\EΦ) +� +e∈EG\EΦ +sign(αe). +Proof. Let ZE and ZV be the Z-vector spaces generated by the edge and vertex set of (G, Φ). +We have the exact sequence +0 → H1(G, Z) → ZE → ZV → H0(G, Z) → 0. +The spaces ZE and ZV come with a natural bilinear form, hence we can dualize the usual +boundary operator ∂1 : ZE → ZV to ∂∗ +1 : ZV → ZE. We get the isomorphism ZV ⊕H1(G, Z) → +ZE⊕H0(G, Z) given by (v, c) �→ (c+∂∗ +1v, ∂0v). An automorphism of G also gives automorphisms +on all the vector spaces in this discussion. As determinant are multiplicative under direct sums, +det(αZV ⊕H1(G,Z)) = det(αZV ) det(αH1(G,Z)) = det(αZE) det(αH0(G,Z)) = det(αZE⊕H0(G,Z)). +Since G is connected H0(G, Z) is one-dimensional and α cannot change its orientation. Therefore, +det(αH0(G,Z)) = 1 and det(αZV ) det(αH1(G,Z)) = det(αZE). The vector space ZE can be decom- +posed ZE = ZEΦ ⊕ Z(EG \ EΦ) and α acts block-wise on both summands as it does not mix for- +est and non-forested edges. It follows that det(αZV ) det(αH1(G,Z)) = det(αZEΦ) det(αZ(EG\EΦ)). +From Φ, we get the short exact sequence, 0 → ZEΦ → ZV → H0(Φ, Z) → 0, where we used the +fact that a forest has no first homology. Consequently, det(αZV ) = det(αZEΦ) det(αH0(Φ,Z)) and +det(αZEΦ) det(αH0(Φ,Z)) det(αH1(G,Z)) = det(αZEΦ) det(αZ(EG\EΦ)). +Ordering and directing the edges EG \ EΦ gives a basis of Z(EG \ EΦ). The orientation can +be changed by switching two edges or reversing the direction of one edge. Hence, det(αZEΦ) = +sign(αEG\EΦ) � +e∈EG\EΦ sign(αe). Fixing an ordering of the connected components of Φ gives a +basis of H0(Φ, Z). Therefore, det(αH0(Φ,Z)) is equal to the sign of the permutation that α induces +on the components of Φ. The fact that det X = ±1 for all X ∈ GLn(Z) gives the lemma. +□ +Combining Lemmas 5.1 and 5.2 with Eq. (27) results in +Theorem 5.3. +�eodd +n += +� +[G,Φ] +1 +|Aut(G, Φ)| +� +α∈Aut(G,Φ) +sign(αH0(Φ))sign(αEG\EΦ) + + +� +e∈EG\EΦ +sign(αe) + + (−1)eα(Φ). +Here, we sum over all forested graphs [G, Φ] of Euler characteristic χ(G) = −n. +This statement is the odd version of Theorem 2.1. As in Section 3 we can produce a formula +for �eodd by counting forests and matchings separately before combining both expressions to give +a counting formula for forested graphs. In the odd case, we have to change a couple of signs in +the derivation to accommodate the additional sign factors in the statement above. +We define an odd version of our forest generating function F in Eq. (15) using the same +notation and by weighting each odd permutation of the connected components of the forest with +a sign: +Fodd(u, x) = +� +s≥0 +1 +s! +� +(Φ,γ)∈AF[s] +sign(γH0(Φ))(−1)eγ(Φ)xγus−2k(Φ) +(28) +where sign(γH0(Φ)) is the sign of the permutation induced by γ on the connected components of +Φ. This factor will account for the sign(αH0(Φ)) term in Theorem 5.3. Using the same argument +as for Proposition 3.10, but accounting for a sign flip for each even cycle on the set of components +or equivalently, by setting yk = (−1)k+1 in Eq. (14), results in + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +38 +Proposition 5.4. +Fodd(u, x) = exp + +� +k≥1 +(−1)k+1u−2k V((u · x)[k]) +k + + . +To account for the factor sign(αEG\EΦ) +�� +e∈EG\EΦ sign(αe) +� +in Theorem 5.3, we have to use +a signed version of the matchings that we introduced in Section 3.3. A permutation α ∈ S2n +and a fixed-point free involution on {1, . . . , 2n} such that α ◦ ι = ι ◦ α give rise to a permutation +αι of the orbits of ι and a set of n permutations αe1, . . . , αen that permute the elements of each +individual orbit of ι. We define +ηodd +α += +� +ι◦α=α◦ι +sign(αι) +n +� +k=1 +sign(αek), +where we sum over all such pairs ι and α. +Using these definitions with the argument for Proposition 3.11 and Theorem 5.3, we get +Proposition 5.5. +� +λ +ηodd +λ +[u2nxλ]Fodd(u, x) = �eodd +n +. +Following the argument in Section 3.3 for the derivation of a formula for ηλ, the cycle index +series for the signed version of a matching of two points is given by +Eodd(x) = 1 +2! +� +(φ,α)∈AE[2] +sign(α)xα = 1 +2 +� +x2 +1 − x2 +� +. +As in Section 3.3, we can use Proposition 3.2 to get a generating function for the numbers ηodd +α +: +Lemma 5.6. +� +n≥0 +1 +(2n)! +� +α∈S2n +ηodd +α += exp + +� +k≥1 +(−1)k+1 +2k +� +x2 +k − x2k +� + + , +Proof. Set yk = (−1)k+1 after applying Proposition 3.2 as each even cycle of the permutation +induced on the components is counted with a minus sign this way and the sign of a permutation +is equal to (−1)# of even cycles. +□ +Repeating the computation for Corollary 3.5 while accounting for the changed signs we find, +Corollary 5.7. If λ = [1m12m2 . . . nmn], then +ηodd +λ += +n +� +k=1 +ηodd +k,mk, +where +ηodd +k,2s = +� +ks(2s − 1)!! +if k is odd +�s +r=0(−1)r�2s +2r +� +kr(2r − 1)!! +if k is even +and +ηodd +k,2s+1 = +� +0 +if k is odd +− �s +r=0(−1)r�2s+1 +2r +� +kr(2r − 1)!! +if k is even . + +THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS +39 +Via exactly the same procedure described in Theorem 3.12, but substituting the odd versions +of the respective series, we get an effective algorithm for computing the numbers eodd(Out(Fn)). +The first few values are listed in Appendix A. +The analytic argument in Section 4 also works in the odd case, as the relevant coefficients of +F and Fodd agree, the values of ηα and ηodd +α +are equal for the trivial permutation and |ηodd +α +| ≤ ηα +for all α. The modified signs only have a nontrivial consequence in Lemma 4.13. Instead of the +statement of Lemma 4.13 we find that in the odd case +n +� +m=0 +(−1)m +2mm! ηodd +2,m = e +1 +4 + O +� 1 +Ln +� +, +after repeating the computation using the numbers from Corollary 5.7. The remaining proof +is completely equivalent up to the substitution of the relevant number e− 1 +4 → e +1 +4 . Following +through the argument again results in the odd version of Theorem 1.1: +Theorem 5.8. The Euler characteristic eodd(Out(Fn)) has the leading asymptotic behaviour +eodd(Out(Fn)) ∼ −e +1 +4 +�n +e +�n +1 +(n log n)2 as n → ∞. +Proof of Theorem 1.3. Use Theorem 5.8, [10, Thm. A] and Stirling’s formula (Lemma 4.1). +□ +References +[1] Emil Artin. The Gamma Function. Holt, Rinehart and Winston, 1964. +[2] Laurent Bartholdi. The rational homology of the outer automorphism group of F7. New York J. Math., +22:191–197, 2016. +[3] François Bergeron, Gilbert Labelle, and Pierre Leroux. Combinatorial species and tree-like structures. 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Table of χ(Out(Fn)), e(Out(Fn)) and eodd(Out(Fn)) for n ≤ 15 +n +χ(Out(Fn)) +e(Out(Fn)) +eodd(Out(Fn)) +2 +− 1 +24 ≈ −0.042 +1 +0 +3 +− 1 +48 ≈ −0.021 +1 +0 +4 +− 161 +5760 ≈ −0.028 +2 +−1 +5 +− 367 +5760 ≈ −0.064 +1 +0 +6 +− 120257 +580608 ≈ −0.21 +2 +−1 +7 +− 39793 +45360 ≈ −0.88 +1 +−1 +8 +− 6389072441 +1393459200 ≈ −4.6 +1 +−8 +9 +− 993607187 +34836480 ≈ −29 +−21 +−38 +10 +− 5048071877071 +24524881920 +≈ −206 +−124 +−275 +11 +− 9718190078959 +5748019200 +≈ −1691 +−1202 +−2224 +12 +− 375393773534736899347 +24103053950976000 +≈ −15575 +−10738 +−20358 +13 +− 2495397080915203519 +15692092416000 +≈ −159023 +−112901 +−207321 +14 +− 1031156416543036906701911 +578473294823424000 +≈ −1782548 +−1271148 +−2320136 +15 +− 6147011108414481406421 +282457663488000 +≈ −21762593 +−15668391 +−28287408 +The numbers χ(Out(Fn)) were computed using [10, Proposition 8.5], the numbers e(Out(Fn)) +were computed as described in Theorem 3.12 and the numbers eodd(Out(Fn)) similarly as dis- +cussed in Section 5. See [9] for the technical computer programming details of these computations. +Michael Borinsky, ETH Zürich, Institute for Theoretical Studies, Clausiusstr. 47, 8092 Zürich, +Switzerland +Karen Vogtmann, University of Warwick, Mathematics Institute, Zeeman Building, Coventry CV4 +7AL, United Kingdom + diff --git a/ddAzT4oBgHgl3EQfLvvB/content/tmp_files/load_file.txt b/ddAzT4oBgHgl3EQfLvvB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bbeeccfa4839c435257b63fca7c7df27c3afed99 --- /dev/null +++ b/ddAzT4oBgHgl3EQfLvvB/content/tmp_files/load_file.txt @@ -0,0 +1,1776 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf,len=1775 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='01121v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='AT] 3 Jan 2023 THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS MICHAEL BORINSKY AND KAREN VOGTMANN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich’s Lie graph complex have the same rational cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We show that the associated Euler characteristic grows like −e−1/4 (n/e)n/(n log n)2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Introduction The moduli space MGn of finite metric graphs with fundamental group Fn was introduced in [17] as a tool for studying the group Out(Fn) of outer automorphisms of a free group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By the main result in that paper MGn is the quotient of a contractible space, called Outer space, on which Out(Fn) acts with finite stabilizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Thus the homology of Out(Fn) with trivial rational coefficients is equal to the homology of MGn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Kontsevich showed in [22, 23] that the homology of MGn can also be identified with the cohomology of his Lie graph complex, which can in turn be identified with the primitive part of the cohomology of the Lie algebra of symplectic derivations of a free Lie algebra (see [15] for a detailed exposition of Kontsevich’s results).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In [5] Berglund and Madsen found this Lie algebra in a very different context, and proved that its cohomology is a sub-algebra of the cohomology of the block diffeomorphism group of even-dimensional products of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In more recent years, algebraic geometers have studied MGn as a tropical analog of the classical moduli space Mn of smooth complex curves of genus n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The simplicial completion of Outer space descends to a natural compactification of MGn, which the tropical geometers have dubbed the moduli space of tropical curves, by analogy with the Deligne-Mumford compactification of Mn (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In yet another context, MGn may be considered a natural parameter space for the n-loop contribution to certain Feynman amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This direction has been explored, for example, by Bloch, Berghoff and Kreimer [7, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this paper we prove a formula for the Euler characteristic of MGn, and then determine its asymptotic growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The asymptotic result depends on our results in [10], where we determined the asymptotic growth rate of the rational or virtual Euler characteristic χ(Out(Fn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This is a rational number closely related to the alternation sum of the Betti numbers, but which has better group-theoretic properties, making it easier to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The rational Euler characteristic of Out(Fn) coincides with the number Kontsevich referred to as the orbifold Euler characteristic of his Lie graph complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The actual alternating sum of the Betti numbers is denoted e(Out(Fn)) in this paper to distinguish it from χ(Out(Fn)), and is called the integral Euler characteristic to conform with other terminology in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If we are primarily interested in the cohomology of the space MGn or (equivalently) the group Out(Fn), then the number e(Out(Fn)) = e(MGn) is clearly more relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Brown [12] showed that the rational and integral Euler characteristics of a group Γ are closely related, namely e(Γ) can be calculated from the rational Euler characteristics of centralizers of finite-order elements by the formula e(Γ) = � [α] χ(C(α)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 1 THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 2 Here the sum is over representatives α for the conjugacy classes of finite-order elements (including the identity), and C(α) is the centralizer of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The number e(Out(Fn)) was calculated for n ≤ 11 by Morita, Sakasai and Suzuki [25], using methods from symplectic representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In the present paper we use Brown’s formula, results on centralizers from [24], an adaptation of Joyal’s theory of species [20] and further development of the asymptotic methods of [10] to first give an effective formula for e(Out(Fn)) and then to determine its asymptotic growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The effective formula is developed in Sections 2-3 and summarized in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Based on it, we wrote a computer program to compute e(Out(Fn)) for n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The results are listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Further optimizations of this program by Jos Vermaseren enabled the computation of the numbers e(Out(Fn)) for all n ≤ 100 [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Section 4 is devoted to proving the following asymptotic result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The integral Euler characteristic e(Out(Fn)) has the asymptotic behaviour e(Out(Fn)) ∼ −e− 1 4 �n e �n 1 (n log n)2 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here the notation an ∼ bn means that limn→∞ an/bn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In particular this verifies the fact, suggested by our results on the rational Euler characteristic in [10], that there is a huge amount of cohomology in odd dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since the cohomology of Out(Fn) is a direct summand of the cohomology of Aut(Fn), we reach the same conclusion for Aut(Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The cohomology Hk(Out(Fn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Q) is known to vanish for both k < 4n/5 and k > 2n−3 (see [14] and the references there), thus all of this cohomology must be concentrated in dimensions 4n/5 ≤ k ≤ 2n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The only odd-dimensional class known to date occurs in H11(Out(F8)) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It is interesting to note that this is the largest possible dimension, the virtual cohomological dimension of Out(F8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This is in contrast to the fact that the groups GLn(Z) and mapping class groups of punctured surfaces, both of which are often considered analogs of Out(Fn), have no rational cohomology in their virtual cohomological dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Comparing Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 with our results in [10] on χ(Out(Fn)) gives Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The ratio limn→∞ e(Out(Fn))/χ(Out(Fn)) = e− 1 4 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Use Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1, [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A] and Stirling’s formula (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', [1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ This solves Problem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5 of the paper [25] by Morita, Sakasai and Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' More precise asymptotic statements and comments about the rate of convergence are given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In his original paper [22] Kontsevich introduced commutative and associative graph complexes in addition to the Lie graph complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The methods of the present paper can be modified to compute the Euler characteristics for both of these other graph complexes as well as to determine their asymptotic behavior (see [8]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' they can also be used to do the same for other moduli spaces of graphs, such as colored graphs or graphs with leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As Kontsevich noted, the associative graph complex computes the homology of mapping class groups of punctured surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Both the rational and integral Euler characteristics of these groups for a once-punctured surface were originally computed by Harer and Zagier [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' They also computed the asymptotics and deduced the existence of lots of cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Getzler and Kapranov partially extended this to surfaces with more than one puncture (without determining the asymptotics) as an application of their general theory of modular operads [18], and we remark that our method of finding the generating function is similar to theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In [22] Kontsevich also defined odd versions of his graph complexes, and noted that in the Lie case the primitive part of the homology computes the cohomology of Out(Fn) with twisted coefficients �Q, where the twisting is given by composing the natural map from Out(Fn) to GLn(Z) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 3 with the determinant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This odd version of Lie graph homology occurs, for example, in the study of diffeomorphism groups of odd-dimensional products of spheres [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In a final section, Section 5, we explain how to modify our results to compute the Euler characteristic of this odd Lie graph complex, which we denote eodd(Out(Fn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The results also extend to the analysis of the asymptotics, and we find Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The ratio limn→∞ eodd(Out(Fn))/χ(Out(Fn)) = e 1 4 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Acknowledgments We are grateful to Jos Vermaseren for generous FORM programming help and to Thomas Willwacher for illuminating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' MB was supported by Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A first formula for e(Out(Fn)) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The rational Euler characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As noted in the introduction, Brown’s theorem says we can compute e(Out(Fn)) by adding up the rational Euler characteristics of centralizers of finite-order elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' One way to compute the rational Euler characteristic of a group Γ is to find a contractible cell complex Y on which Γ acts properly and cocompactly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' the rational Euler characteristic χ(Γ) is then given by the formula χ(Γ) = � [σ] (−1)dim(σ) |Stab(σ)| , where we sum over all orbits [σ] of cells, σ is a representative from the orbit and Stab(σ) is its stabilizer under the group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Fortunately, we have such complexes Y for centralizers of finite-order elements of Out(Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The entire group Out(Fn) centralizes the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall from [17] that Out(Fn) acts properly and cocompactly on the spine Kn of Outer space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Kn is a contractible cube complex with one k-dimensional cube for each equivalence class [G, Φ, g] of triples (G, Φ, g), where (G, Φ) is a connected forested graph, and g: Fn → π1(G) is an isomorphism, called a marking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here by a graph we mean a CW-complex of dimension 0 or 1, a forest is a graph without cycles, and a subforest of G is a subcomplex that is a forest and contains all of the vertices of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We say a graph is of rank n if its fundamental group is of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A graph is admissible if it has no isolated, univalent or bivalent vertices, and a forested graph is a pair (G, Φ) consisting of an admissible graph G and a subforest Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Two triples (G, Φ, g) and (G′, Φ′, g′) are equivalent if there is a graph isomorphism h: G → G′ sending Φ to Φ′ and inducing an isomorphism h∗ : π(G) → π(G′), such that g−1h∗g = id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The spine K = Kn is contractible, and the action of Out(Fn) on K simply changes the marking g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Thus there is one orbit for each isomorphism class [G, Φ] of forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The stabilizer of a cube [G, Φ, g] is isomorphic to Aut(G, Φ), the automorphisms of G that preserve Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Thus, χ(Out(Fn)) = χ(C(id)) = � [G,Φ] (−1)e(Φ) |Aut(G, Φ)|, where e(Φ) is the number of edges in Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The equivariant spine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In [24] an equivariant version of K was introduced, which can be used to study the centralizer of any finite order element of Out(Fn) (in fact the centralizer of any finite-order subgroup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We briefly summarize the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A graph G is said to realize a finite-order automorphism α if there is some marking g: Fn → π1(G) and automorphism fα of G such that g−1(fα)∗g = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Every finite-order element of Out(Fn) can be realized on some admissible graph G, [16, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This translates to the statement that the action of α on K has at least one fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The centralizer C(α) acts on the entire fixed-point set Kα of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows from [24] that this fixed point set is contractible, cocompact and has the structure of a cube complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Specifically, Kα has one cube for each equivalence class of triples (G, Φ, g)α where (G, g) realizes α and Φ is a (possibly empty) forest in G that is invariant under the action of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here (G, Φ, g)α is equivalent to (G′, Φ′, g′)α if there is an α-invariant automorphism h: G → G′ sending Φ to Φ′ such that g−1h∗g = id, and again we write [G, Φ, g]α for the equivalence class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The dimension of the cube [G, Φ, g]α is the number eα(Φ) of edge-orbits in Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The stabilizer Stab[G, Φ, g]α is isomorphic to Autα(G, Φ), the automorphisms of G that com- mute with the action of α and send Φ to itself, so χ(C(α)) = � [G,Φ]α (−1)eα(Φ) |Autα(G, Φ)|, where [G, Φ]α runs over isomorphism classes of pairs that realize α with a connected graph of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Brown’s theorem [12] then gives e(Out(Fn)) = � [α] � [G,Φ]α (−1)eα(Φ) |Autα(G, Φ)|, where [α] runs over conjugacy classes of finite-order elements α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The group Aut(G, Φ) acts on itself by conjugation, so the orbit-stabilizer theorem gives |Aut(G, Φ)| = |(orbit of α)| · |StabAut(G,Φ)(α)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since StabAut(G,Φ)(α) = Autα(G, Φ) this gives: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' e(Out(Fn)) = � [G,Φ] 1 |Aut(G, Φ)| � α∈Aut(G,Φ) (−1)eα(Φ), where we sum over all isomorphism classes [G, Φ] of connected forested graphs of rank n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Disconnected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that the Euler characteristic of a connected graph of rank n is χ(G) = 1 − n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The graph’s Euler characteristic is often better behaved than its rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For instance, the formula in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 is easier to work with if we drop the requirement that G be connected and shift the index by one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' define �en = � [G,Φ] 1 |Aut(G, Φ)| � α∈Aut(G,Φ) (−1)eα(Φ) (1) where we sum over all isomorphism classes [G, Φ] of (not necessarily connected) forested graphs with χ(G) = −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this section we show how to recover e(Out(Fn+1)) once the numbers �en are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We begin by deriving new formulas for e(Out(Fn+1)) and �en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Define a forested graph (G, Φ) to be even if G has no automorphisms that induce an odd permutation of the edges of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If we sum over all isomorphism classes [G, Φ] — (1) —of even forested graphs with χ(G) = −n, then � [G,Φ] (−1)e(Φ) = �en .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 5 (2) —of connected even forested graphs with χ(G) = −n, then � [G,Φ] (−1)e(Φ) = e(Out(Fn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Conant and Vogtmann showed in [15] that Kontsevich’s Lie graph complex, as defined in [22] is quasi-isomorphic to the complex spanned by even forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Thus this propositions shows that e(Out(Fn)) is equal to the Euler characteristic of the Lie graph complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 is an immediate consequence of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum � α∈Aut(G,Φ)(−1)eα(Φ) vanishes if (G, Φ) has an automorphism that induces an odd permutation on Φ and is equal to (−1)e(Φ)|Aut(G, Φ)| otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An element α ∈ Aut(G, Φ) induces a permutation αΦ ∈ Se(Φ) on the set of edges of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By definition, the number eα(Φ) is equal to the number of cycles of in the cycle decomposition of αΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sign of a permutation is the parity of the number of its even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since the parity of the number of odd cycles of a permutation on an n-element set is equal to the parity of n, we have (−1)eα(Φ) = sign(αΦ)(−1)e(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sign function gives a homomorphism from Aut(G, Φ) to the cyclic group of order 2, which is surjective if and only if Aut(G, Φ) contains an odd permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If it is surjective, then half of the elements of Aut(G, Φ) have each sign, so � α∈Aut(G,Φ) sign(αΦ) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' otherwise � α∈Aut(G,Φ) sign(αΦ) = (−1)e(Φ)|Aut(G, Φ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Use Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (1) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Each pair consisting of a forested graph and an automorphism contributes to the sum in the definition of �en, so evaluating this sum means doing a weighted count of forested graphs and their automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will do this counting by means of generating functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' formal power series whose coefficients encode the counts we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can also use formal power series to describe the relation between �en and e(Out(Fn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By standard topological quantum field theory convention, we use the symbol ℏ as the formal variable that marks the negative Euler characteristic of the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � n≥0 �en ℏn = ∞ � n=1 � 1 1 − ℏn �e(Out(Fn+1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For any (possibly disconnected) admissible graph G and subforest Φ ⊂ G, set θ(G, Φ) = (−1)e(Φ)ℏ−χ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since an admissible graph is either trivial or has strictly negative Euler charac- teristic, θ(G, Φ) ∈ Q[[ℏ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The function θ factors over connected components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' θ((G1, Φ1) ⊔ (G2, Φ2)) = θ(G1, Φ1)θ(G2, Φ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2(1), � n≥0 �en ℏn = � [G,Φ] even θ(G, Φ), where we sum over all isomorphism classes of (possibly disconnected) even forested graphs [G, Φ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Each isomorphism class [G, Φ] can be described by giving a set of isomorphism classes of connected graphs together with the multiplicity with which each connected class appears in the disconnected class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In the sum above, a component [g, ϕ] of [G, Φ] such that ϕ has an odd number of edges can appear at most once, as otherwise [G, Φ] would have an odd automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 6 Components with an even number of edges in the forest can appear with any multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum of the function θ over all even admissible forested graphs hence satisfies the following identity: � n≥0 �en ℏn = � [G,Φ] even θ(G, Φ) = \uf8eb \uf8ec \uf8ec \uf8ed � [g,ϕ] e(ϕ) odd (1 + θ(g, ϕ)) \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed � [g,ϕ] e(ϕ) even � m≥0 θ(g, ϕ)m \uf8f6 \uf8f7 \uf8f7 \uf8f8 where the [g, ϕ] are isomorphism classes of connected forested graphs and e(ϕ) denotes the number of edges of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using the fact that θ(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ϕ) is −ℏ−χ(g) if [g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ϕ] is odd and ℏ−χ(g) if [g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ϕ] is even and evaluating the sum over m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' we obtain � n≥0 �en ℏn = \uf8eb \uf8ec \uf8ec \uf8ed � [g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='ϕ] e(ϕ) odd � 1 − ℏ−χ(g)� \uf8f6 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ed � [g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='ϕ] e(ϕ) even 1 1 − ℏ−χ(g) \uf8f6 \uf8f7 \uf8f7 \uf8f8 = � n≥1 (1 − ℏn)βn odd (1 − ℏn)βn even ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' where βn odd and βn even are the numbers of connected forested admissible graphs without odd edge- automorphisms with Euler characteristic −n with an odd or even number of edges in the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A connected graph with Euler characteristic −n has rank n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2(2), βn even − βn odd = e(Out(Fn+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ We can now explain how to calculate the numbers e(Out(Fn)) recursively from the numbers �en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that the classical Möbius function µ is defined recursively for positive integers by µ(1) = 1 and � d|n µ(d) = 0 for all n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let � n≥1 en ℏn = log f(ℏ) be the logarithm of the series f(ℏ) = � n≥0 �en ℏn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The numbers en for n ≥ 1 are given recursively by en = �en − 1 n n−1 � k=1 k ek �en−k and e(Out(Fn+1)) for n ≥ 1 by e(Out(Fn+1)) = � d|n µ(d) d en/d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The recursive expression for en in terms of �en follows by taking the derivative of log f(x) with respect to ℏ and using (log f(x))′ = f ′(x)/f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Taking the logarithm of the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5, using log(1/(1 − x)) = �∞ n=1 xn/n and the definition of the Möbius function gives the second formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An effective formula for e(Out(Fn)) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Admissible trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In the last section we reduced the problem of computing e(Out(Fn)) to the problem of doing a weighted count of forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In order to count forested graphs we will first count trees, then forests, then ways of matching the leaves of the forests to form forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Throughout the rest of the paper we fix the following conventions and terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By a tree we mean a connected graph with no cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A forest is a disjoint union of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We give univalent vertices of trees and forests a special role and call them leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A tree or forest is admissible if it has no isolated or bivalent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We require admissible trees to have at least one vertex, so a single edge which connects two leaves is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The edges of a tree or forest are the 1-cells that are not attached to leaves (sometimes called internal edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A rooted tree is a tree where one distinguished univalent vertex takes the role of the root and the other THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 7 univalent vertices are leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Our trees and rooted trees and forests will be labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' That means each leaf shall be decorated with a unique element from some given finite set U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The set of all admissible labeled (or labeled rooted) trees forms a combinatorial species in the sense of Joyal [20] (see also [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since our counting problems fit neatly into the theory of species, we review the relevant parts of this theory in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Species and generating functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A combinatorial species is a functor from the group- oid of finite sets to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' More explicitly, a species S associates to every finite set U of labels, a finite set S[U] of combinatorial objects such that every bijection U → V gives rise to a bijection of sets S[U] → S[V ] in a way compatible with composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In the case U = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', n} we write S[n] for S[U].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An element of S[U] for some U is an S-object, or more informally, an object of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A set partition of U with k blocks is a collection π = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , Bk} of k mutually disjoint subsets of U such that � B∈π B = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' From a given species S we can construct a new species Setk(S) by associating to a set U all collections of objects {φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , φk} such that φ1 ∈ S[B1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , φk ∈ S[Bk] for some partition of U into blocks B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The functor Setk(S) is the species of sets of size k which contain objects of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To make the formulas more compact we agree that Set0(S)[0] contains one element, the empty set, and that Setk(S)[0] is empty for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let A be an algebra and ω : S[U] → A a function that associates some element of A to each object of S[U] in a way that is independent of the labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can extend the weight function to Setk(S) by setting the weight of the empty set to 1 and ω({φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , φk}) = �k ℓ=1 ω(φℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' When working with a formal power series F(x) = � aixi we will frequently use the coefficient extraction operator [xn] to extract the coefficient of xn, so [xn]F(x) = an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We now define two formal power series, in A[[x]] and A[[x, y]] respectively, by S(x) = � n≥1 xn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � φ∈S[n] ω(φ) and SSet(x, y) = � n,k≥0 xn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' yk � Φ∈Setk(S)[n] ω(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (2) These are exponential generating functions for weighted counts: [xn]S(x) is 1/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' times the number of objects φ ∈ S[n], counted with weight ω(φ), and [xnyk]S(x, y) is 1/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' times the number of elements Φ ∈ Setk(S)[n], counted with weight ω(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will make frequent use of the following exponential formula, a standard lemma that relates the power series S and SSet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' SSet(x, y) = exp (y S(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Use the expansion exp(X) = � n≥0 Xn/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' and the fact that the number of set partitions of a set of cardinality n into k blocks with sizes m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , mk is given by n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='/(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' mk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ For every species S there is a natural action of Sn on S[n] that permutes the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The orbit of an element φ ∈ S[n] under this action is an isomorphism class of combinatorial objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' an unlabeled object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The stabilizer of an element φ ∈ S[n] is the group of relabelings that leaves the object invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will denote this stabilizer by Aut(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We next want to count labeled combinatorial objects while keeping track of automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For this, we will need more sophisticated generating functions, called cycle index series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The terminology comes from the notion of cycle type for a permutation, which we now review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let n ∈ N be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An integer partition of n is a sequence of positive integers λ = (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , λℓ) such that λ1 ≥ λ2 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ≥ λℓ > 0 and n = λ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' + λℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The λi are called parts of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We write λ ⊢ n or |λ| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The integer n is the size of the partition and ℓ, the number of parts, is the length of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will often use the more compact notation λ = [1m1 · · · nmn], indicating that λ has mk parts of size k (the terms with mi = 0 are usually omitted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' for example, λ = (4, 4, 1, 1, 1) is written [1342]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this notation the length of λ is THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 8 ℓ(λ) = m1 + · · · + mℓ, the size is |λ| = m1 + 2m2 + · · · + nmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Each permutation π ∈ Sn factors uniquely as a product of disjoint cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If the orders of these cycles are λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , λℓ, where λ1 ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ≥ λℓ, then λ(π) = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , λℓ) is a partition of n called the cycle type of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Given an integer partition λ = [1m12m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' nmn] let xλ denote the monomial xm1 1 xm2 2 · · xmn n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A formal power series in the variables x = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='} is an infinite sum of terms aλxλ, with aλ ∈ A, such that, for all n, we only have a finite number of terms if restrict to partitions with |λ| ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If α is a permutation of cycle type λ we also define xα = xλ, ℓ(α) = ℓ(λ) and |α| = |λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let AS be the species of pairs (φ, α), where φ is an object of S and α an automorphism of φ: AS[n] ={(φ, α) : φ ∈ S[n] and α ∈ Aut(φ) ≤ Sn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let ω be a weight function that attaches to an object (φ, α) of AS an element ω(φ, α) ∈ A that does not depend on the labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The cycle index series for S weighted by ω is the formal power series S in x whose terms mark the cycle type of the automorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' S(x) = � n≥1 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (φ,α)∈AS[n] ω(φ, α) xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In other words, [xλ]S(x) is 1/|λ|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' times the weighted count of pairs (φ, α) ∈ AS such that α has cycle type λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The cycle index series S(x) generalizes the generating function of labeled objects in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (2), as we recover S(x) by setting ω(φ, id) = ω(φ), x1 = x and all other xk-variables to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We now want to extend our cycle index series on S to one for Setk(S), and show how to compute it from S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An element Φ ∈ Setk(S)[n] is a set of k elements of S with a total of n labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An element γ ∈ Aut(Φ) permutes the labels but preserves the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If some elements of S are isomorphic then γ may permute them, so γ induces a permutation γΦ ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We introduce a new infinite set of variables y = {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='} to mark the cycle type of γΦ, and define SSet(x, y) = � n,k≥0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈ASetk(S)[n] ω(Φ, γ) xγyγΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here ω(Φ, γ) = ω(φ1, γ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ω(φℓ, γℓ), where γΦ has cycle type λ = (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , λℓ) of length ℓ φi ∈ Φ is a representative of the ith cycle, which has size λi γi is the restriction of γλi to φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let S(x[k]) be the series obtained from S(x) by replacing each occurrence of xi by xki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Then SSet(x, y) = exp \uf8eb \uf8ed� k≥1 yk S(x[k]) k \uf8f6 \uf8f8 , Ultimately, this proposition goes back to Pólya [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 combined with the generating function for wreaths of S structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We give a proof below, but refer to [3] Chapter 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 for a more detailed argument, using a slightly different type of weight function ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 we first introduce a new species ACyckS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An element of ACyckS[n] is a pair (Φ, γ) where Φ = {φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , φk} is a collection of objects in S with a total of n labels, and γ ∈ Aut(Φ) is a permutation of the labels of Φ that cyclically permutes the φi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' γΦ is a k-cycle in Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In particular, all of the φi must be isomorphic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' equivalent as unlabeled objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � n≥1 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈ACyck(S)[n] ω(Φ, γ)xγ = S(x[k]) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let (Φ, γ) be an element of ACyckS[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We claim that (Φ, γ) is equivalent to a tuple (κ, π, φ, α, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , γk−1), where κ is a k-cycle in Sk, π is a partition of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , n} into k blocks, each of size d, φ is an object of S[d], α is an element of Aut(φ), and γi is an element of Sd for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By definition Φ = {φ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , φk}, where φi ∈ S[Bi] for some Bi ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , n}, and γΦ is a k-cycle, so we take κ = γΦ and π = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , Bk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since γ acts cyclically on Φ, all of the φi are isomorphic, so in particular they all have the same number d of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The unique order-preserving bijection Bi → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', d} identifies each φi with an element �φi ∈ S[d];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' set φ = �φk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The permutation γk sends each φi to itself, so determines an automorphism αi of φi and therefore of �φi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' set α = αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Note that all of the αi have the same cycle type λ = (λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , λℓ), so the cycle type of γ is kλ = (kλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , kλℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , k−1, the ith power of the k-cycle γΦ maps φk to φγi Φ(k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' let γi be the induced isomorphism γi : �φk → �φγi Φ(k) in Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Finally, note that ω(Φ, γ) = ω(φk, γk) = ω(φ, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since ACyck(S)[n] is empty unless k|n, we can now write the left hand side of the equation in the statement of the lemma as, (3) � n=dk≥1 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='Cn,k � (φ,α)∈AS[d] ω(φ, α)xk◦α, where k ◦ α has cycle type kλ and Cn,k is the number of partitions π of n into k equal parts times the number of cyclic permutations κ in Sk times the number of permutations γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , γk−1 in (Sd)k−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Cn,k = 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' )k · (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' · (d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' )k−1 = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Plugging this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (3) gives the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We have ASet(S)[n] ={(Φ, γ) : Φ is a finite set of objects of S with a total of n distinct labels, and γ ∈ Aut(Φ) ≤ Sn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since every permutation γ can be uniquely decomposed into a product of cycles, we have an isomorphism of species ASet(S) ∼= � ℓ≥1 Setℓ \uf8eb \uf8ed � k≥1 ACyck(S) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore the statement follows from an application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 to the right hand side, using the generating function of the species ACyck(S) given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 and summation over all k, using the variables y to keep track of the cycle type of the permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In order to form a forested graph with an automorphism preserving the forest, we will start with a forest equipped with an automorphism α, then pair its leaves by a fixed-point free involution that commutes with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this section we apply the counting methods from the previous section to count the number of such involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A fixed-point free involution is also called a matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' it divides the set into orbits of size 2, so we will use the species E of sets of cardinality 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' E[2] = {1, 2} and E[n] = ∅ for n ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The generating function is E(x) = x2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Matchings on a set of 2k elements correspond to THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 10 elements of Setk(E)[2k];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' for example the elements of Set2E[4] are {{1, 2}, {3, 4}}, {{1, 3}, {2, 4}} and {{1, 4}, {2, 3}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 we have ESet(x, y) = exp � y x2 2 � = � k≥0 (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' x2k (2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='yk, where the second equality is obtained using the expansion exp(X) = � n≥0 Xn/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' and the formula (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = (2k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='/(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Thus we recover the (easy) fact that the number of matchings of a set of cardinality 2k is (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', and there are none if the cardinality is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Now consider the species AE = AE[2] of sets of cardinality 2 with automorphisms on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A set of cardinality 2 has only two automorphisms, the trivial one (marked by x1x1 = x2 1) and the transposition (marked by x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The cycle index series of E with trivial weight is therefore E(x) = 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (φ,α)∈AE[2] xα = 1 2 � x2 1 + x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A matching of 2k elements corresponds to an element Φ ∈ Setk(E)[2k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The automorphisms of such an element Φ are permutations that commute with the corresponding fixed-point free involution ι = ιΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We may count such permutations using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2, which gives ESet(x, y) = exp \uf8eb \uf8ed� k≥1 yk 2k � x2 k + x2k � \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (4) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � n≥0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (ι,α) xα = exp \uf8eb \uf8ed� k≥1 1 2k � x2 k + x2k � \uf8f6 \uf8f8 where the sum is over all pairs (ι, α) consisting of a matching ι of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , 2n} and a permutation α ∈ S2n that commutes with ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By definition we have ESet(x, y) = � n,k≥0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,α)∈ASetk(E)[n] xγyγφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since SetkE[n] is empty unless n is even and k = n/2 we can rewrite this as ESet(x, y) = � n≥0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,α)∈ASetnE[2n] xγyγφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As noted previously, we may identify the pairs (ι, α) in the statement of the proposition with elements (Φ, α) of ASetn(E)[2n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (4) and setting yk = 1 for all k gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The number ηλ of matchings that commute with a given permutation α ∈ Sn of cycle type λ = [1m12m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' nmn] is given by the formula ηλ = n � k=1 ηk,mk, where ηk,2s = � ks(2s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if k is odd �s r=0 �2s 2r � kr(2r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if k is even THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 11 and ηk,2s+1 = � 0 if k is odd �s r=0 �2s+1 2r � kr(2r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if k is even .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since ι commutes with α if and only if α commutes with ι, the left hand side of Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4 can be written � n≥0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � α∈S2n #{ι|ι ◦ α = α ◦ ι}xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Now note that the number of matchings that commute with α depends only on the cycle type of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If we denote the set of of permutations in Sn with cycle type λ by Sλ n, the formula becomes = � n≥0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � λ⊢2n | Sλ 2n |ηλxλ = � n≥0 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � λ⊢n | Sλ n |ηλxλ, where we used the fact that ηλ = 0 if |λ| is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For λ = [1m12m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' nmn] it is easy to check that the number of permutations in Sλ n is | Sλ n | = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 1m1m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2m2m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' · · · nmnmn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', so the left hand side of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4 is equal to (5) � n≥0 � λ⊢n ηλ 1m1m1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2m2m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' · · · nmnmn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='xλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We now turn to the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using eX = � m≥0 Xm/m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = (2m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='/(m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2m) and (x + 1)m = �m k=0 �m k � xk one can check that the numbers ηk,m given in the statement of the corollary are the coefficients of xm/(kmm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=') in the expansions of exp � x2/(2k) � if k is odd and exp � (x2 + 2x)/(2k) � if k is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using this together with the fact that � k≥1 1 2k � x2 k + x2k � = � k≥1 � x2 2k−1 2(2k − 1) + x2 2k + 2x2k 4k � the right hand side of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4 becomes (6) � k≥1 \uf8eb \uf8ed� m≥0 ηk,m kmm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='xm k \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Equating coefficients of (5) and (6) now gives the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Rooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In order to count forested graphs with automorphisms, we begin by counting forests with automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In order to count forests with automorphisms, we begin by counting trees with automorphisms, and in order to count trees with automorphisms, we begin by counting rooted trees with automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let R be the species of leaf-labeled admissible rooted trees ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' all internal vertices of ρ must have valence at least 3 and R[n] is the set of all such rooted trees with n leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The elements of R[3] are depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The rooted tree ρ0 with one root, one leaf and one 1-cell has no internal vertices so satisfies the definition, but it plays a special role so we call it the special rooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' AR is then the species of pairs (ρ, α), where ρ ∈ R and α ∈ Aut(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that an automorphism of a tree is determined by what it does to the leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' So, for a rooted tree ρ ∈ R[n] we can (and will) identify the group Aut(ρ) ≤ Sn with the usual simplicial automorphisms of the tree ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To each pair (ρ, α) ∈ AR we assign weight ω(ρ, α) = (−1)vα(ρ), where vα(ρ) is the number of α-orbits of internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The special rooted tree ρ0 has only the THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 12 1 2 3 = 1 3 2 , 2 1 3 , 3 1 2 , 1 2 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' All admissible rooted trees with 3 leaves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' all elements of R[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The first three rooted trees have two vertices each and the fourth has one ver- tex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Edges are colored in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The automorphism groups of the first, second and third rooted tree are generated by the transpositions (2, 3), (1, 3) and (1, 2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The automorphism group of the fourth rooted tree is the full sym- metric group S3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' it includes all permutations of the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' identity automorphism, and the pair (ρ0, id) has weight is (−1)0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The associated generating function R(x) = � n≥1 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (ρ,α)∈AR[n] (−1)vα(ρ)xα is the Frobenius characteristic of rooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The first few terms are R(x) = x1 − 1 2(x2 1 + x2) + 1 63(x3 1 + x1x2) − 1 6(x3 1 + 3x1x2 + 2x3) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The first term comes from the special rooted tree ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The second term comes from the rooted tree with one vertex and two leaves branching from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We have two automorphisms that either switch the leaf-labels or do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The third and fourth terms in this sum come from the rooted trees in Figure 1 and their automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The name Frobenius characteristic comes from an interpretation of these generating functions in the context of the representation theory of the symmetric group: We can think of R(x) as an element of the ring of symmetric functions and each homogeneous part of R(x) as the image of a certain representation of the symmetric group under the Frobenius characteristic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' These representations are the vector spaces generated by the elements of R[n] with the action of Sn which alternates with (−1)vα(ρ) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' However, in this paper we will not make use of this representation theoretical interpretation of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The next proposition shows that the characteristic R(x) has a remarkably simple form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' R(x) = � n≥1 µ(n) n log(1 + xn), where µ is the Möbius function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The formula will be established by relating pairs (ρ, α) recursively to pairs (Φ, γ) consisting of a set of rooted trees Φ and an automorphism γ ∈ Aut(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let Φ be a collection of two or more admissible rooted trees and γ ∈ Aut(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Set ω(Φ, γ) = (−1)vγ(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can form a new admissible rooted tree ρΦ by gluing the roots of the trees in Φ together and growing a new root from the resulting vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The resulting rooted tree has one new internal vertex, which is fixed by any automorphism, and at least 2 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The natural THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 13 map Aut(Φ) → Aut(ρΦ) is a bijection, so we will use the same name γ, and we have ω(ρΦ, γ) = −ω(Φ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The only pair (ρ, α) which cannot be obtained in this way is (ρ0, id), so R(x) = x1 − � k≥2 � n≥k 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈ASetk(R)[n] (−1)vγ(Φ)xγ, (7) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 tells us RSet(x, y) = � k≥0 � n≥k 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈ASetk(R)[n] (−1)vγ(Φ)xγyγΦ = exp \uf8eb \uf8ed� k≥1 yk R(x[k]) k \uf8f6 \uf8f8 , Setting yk = 1 for all k and subtracting the terms with k = 0 or k = 1 gives the summation term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (7) above, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' R(x) = x1 − � exp \uf8eb \uf8ed� k≥1 R(x[k]) k \uf8f6 \uf8f8 − 1 − R(x[1]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (8) Since R(x[1]) = R(x) this gives � k≥1 R(x[k])/k = log(1 + x1) and because x[k] = (xk, x2k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ), we also have � k≥1 R(x[kn])/k = log(1+xn) for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Multiplying both sides of this equation with µ(n)/n, summing over n ≥ 1 and using the defining recursion of the Möbius function results in the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Unrooted trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We next use this generating function for rooted trees (weighted by the parity of vertex orbits) to find the Frobenius characteristic V(x) for the species T of unrooted admissible trees, weighted by the parity of edge-orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Specifically, to a pair (t, α) ∈ AT , where t ∈ T and α ∈ Aut(t), we assign the weight ω(t, α) = (−1)eα(t), where eα(t) is the number of edge-orbits of α in t, and define V(x) = � n≥3 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (t,α)∈AT [n] (−1)eα(t)xα, (9) Note that both T [1] and T [2] are empty, so we could have started the indexing with n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' V(x) = x1 + x2 1 2 − x2 2 − (1 + x1)R(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Suppose t is an unrooted tree and α ∈ Aut(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let tα be the subtree of t spanned by the fixed vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Consider first the case that tα is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For each vertex v ∈ tα, we can think of (t, α) as a collection Φ of at least 3 rooted trees, with v as their root, and of α as an automorphism of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Note that eα(t) = vα(t) − 1 = vα(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 and setting yk = 1 for all k, the generating functions of such collections Φ is exp \uf8eb \uf8ed� k≥1 R(x[k]) k \uf8f6 \uf8f8 − 1 − R(x[1]) − R(x[2]) 2 − R(x[1])2 2 , where we have subtracted the terms that correspond to collections with fewer than 3 trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (8) in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7 this is the same as x1 − R(x) − R(x[2]) 2 − R(x)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (10) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 14 If we place a root in the middle of an edge of tα we can view (t, α) as a pair {ρ1, ρ2} of rooted trees and α as an automorphism of the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since e is not adjacent to a leaf neither rooted tree is the special rooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The generating function of such rooted trees is therefore (R(x) − x1)2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (11) Here we have eα(t) = vα(t) − 1 = vα({ρ1, ρ2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum of expressions (10) and (11) counts the pair (t, α) multiple times, once for each vertex of tα with sign (−1)eα(t), and once for each edge of tα with the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since tα has one more vertex than edge, this sum leaves us with exactly one contribution from each (t, α), with sign (−1)eα(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' the sum gives the contribution to V(x) from all such pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We still have to account for pairs (t, α) with no fixed vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' tα is empty and the only fixed point is at the midpoint of an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The tree t can then be viewed as the union of two identical rooted trees {ρ1, ρ2} rooted at this midpoint, which are exchanged by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 these are counted by R(x[2])/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since in this case eα(t) = vα({ρ1, ρ2}), these contribute R(x[2]) 2 − x2 2 (12) to V(x), where we have subtracted the term corresponding to the interval since that is not an admissible unrooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum of formulas (10), (11) and (12) now has one term for each pair (t, α), weighted by (−1)eα(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' the sum is equal to V(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ As an immediate corollary of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7 we have Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' V(x) = x1 + x2 1 2 − x2 2 − (1 + x1) � k≥1 µ(k) k log(1 + xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using log(1 + x) = � n≥1(−1)n+1xn/n, and the definition of the Möbius function, we can expand this power series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As µ(1) = 1, µ(2) = µ(3) = −1 and µ(4) = 0, the first coefficients are V(x) = x1 + x2 1 2 − x2 2 − (1 + x1) �� x1 − x2 1 2 + x3 1 3 − x4 1 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � − 1 2 � x2 − x2 2 2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � − 1 3 (x3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=') + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � = x3 1 6 + x1x2 2 + x3 3 − x4 1 12 − x2 2 4 + x1x3 3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' where we omitted terms of total degree higher than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As expected there are no terms of degree smaller than 3, as such terms would correspond to trees with fewer than 3 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We recall from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 that setting x = (x, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=') in the Frobenius characteristic V(x) for unrooted trees recovers the generating function V (x) for unrooted trees t weighted by (−1)e(t), where e(t) is the number of (internal) edges of t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (13) V (x) = x + x2 2 − (1 + x) log(1 + x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For a detailed exposition of this formula see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As forests are just collections of trees, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 gives us the following expression for the alternating cycle index series for the species F of forests: VSet(x, y) = � n 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈AF[n] (−1)eγ(Φ)xγyγΦ = exp \uf8eb \uf8ed� k≥1 yk V(x[k]) k \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (14) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 15 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A forest with automorphism γ = (13)(24)(57)(68)(9 10) and match- ing ι = (13)(25)(47)(68)(9 10) that commutes with γ, corresponds to a forested graph with the automorphism that flips the graph over the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' What we want to do with these forests is to glue their leaves together in pairs to form admissible graphs (which we can only do if there is an even number of leaves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We also want to keep track of the Euler characteristic of the resulting graph G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' this is the number of trees in the forest minus half the total the number of leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It is equivalent to keep track of −2χ(G) = #{leaves} − 2#{trees} (which is always a positive integer) so we mark this number with a new variable u, and define the Frobenius characteristic of admissible forests F(u, x) = � s≥0 1 s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈AF[s] (−1)eγ(Φ)xγus−2k(Φ), (15) where F is the species of forests and k(Φ) is the number of trees in the forest Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' F(u, x) = exp \uf8eb \uf8ed� k≥1 u−2k V((u · x)[k]) k \uf8f6 \uf8f8 , where V((u · x)[k]) means that we replace each variable xi in V(x) with ukixki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (9) the lowest term of V(u·x) is of order u3, and hence the lowest term of V((u·x)[k]) is of order u3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, F(u, x) is the exponential of a power series in only positive powers of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Starting with the definition of VSet(x, y), the substitution xi �→ uixi sends xλ to u|λ|xλ, and the substitution yi �→ u−2i sends yγΦ to u2k(Φ) where k(Φ) is the number of trees in Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Doing both substitutions gives F(u, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � λ ηλ[u2nxλ]F(u, x) = �en, where we sum over all integer partitions λ and ηλ is the number of matchings that commute with a permutation of cycle type λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum is finite since [u2nxλ]F(u, x) = 0 unless 2n ≤ |λ| ≤ 6n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Our generating function F(u, x) counts forests together with automorphisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if we set Fr,λ = {(Φ, γ)|Φ has |λ| leaves and k = |λ| − r 2 components and γ has cycle type λ}, THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 16 then [urxλ]F(u, x) = 1 |λ|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈Fλ,r (−1)eγ(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Because a nonempty forest has at least one component, we have [urxλ]F(u, x) = 0 if |λ| < r, and because every connected component of an admissible forest must have at least three leaves we have [urxλ]F(u, x) = 0 if |λ| > 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In particular, for a given r there are only finitely many integer partitions λ such that [urxλ]F(u, x) is nonzero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' the terms in the sum � λ ηλ[urxλ]F(x, u) are only nonzero when r ≤ |λ| ≤ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let Mr ⊂ Sr be the set of fixed-point free involutions on the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , r} and IF r,λ = {(Φ, γ, ι)|(Φ, γ) ∈ Fr,λ and ι ∈ Mr commutes with γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Then ηλ[urxλ]F(x, u) is equal to 1/|λ|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' times the alternating sum of elements in IFr,λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can make a forested graph equipped with an automorphism from an element (Φ, γ, ι) ∈ IFr,λ by using ι to identify the leaves of Φ in pairs, provided r = 2n is even (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The result will be a forested graph (G, Φ0) with χ(G) = −n, where Φ0 is the subforest of G consisting of just the edges and vertices of Φ, and the leaves of Φ have become (labeled) half-edges in G\\Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This gives us a one-to-one correspondence between elements of IF2n,λ and pairs ((G, Φ0), γ) consisting of a forested graph (G, Φ0) with χ(G) = −n and an automorphism γ ∈ Aut(G, Φ0), where the half-edges of G \\ Φ0 are labeled by {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , |λ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The symmetric group S|λ| acts on IF2n,λ by permuting the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The stabilizer of ((G, Φ0), γ) is Autγ(G, Φ0), the automorphisms of (G, Φ0) that commute with γ, and the orbit is the unlabeled pair [G, Φ0] together with a conjugacy class [γ] of the automorphism group Aut(G, Φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The weight (−1)eγ(Φ) is constant on orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If we now take the sum over all integer partitions λ, the orbit-stabilizer theorem gives � λ ηλ[u2nxλ]F(x, u) = 1 |λ|!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � λ � (Φ,γ,ι)∈IF2n,λ (−1)eγ(Φ) = � [G,Φ0] χ(G)=−n � [γ]∈Aut(G,Φ0) 1 |Autγ(G, Φ0)|(−1)eγ(Φ0) = � [G,Φ0] χ(G)=−n 1 |Aut(G, Φ0)| � γ∈Aut(G,Φ0) (−1)eγ(Φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The second step uses the orbit stabilizer theorem again on the centralizer Autγ(G, Φ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since the last line is the definition of �en, the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ The following theorem summarizes the steps needed for computing e(Out(Fn)) for a given n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For fixed n ≥ 2, the numbers e(Out(F2)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , e(Out(Fn)) can be computed by the following steps: (1) Calculate V(x) up to homogeneous degree 6(n−1) in x using the formula in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (2) Calculate the coefficients [u2kxλ]F(u, x) from V(x) using the formula in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 for all pairs k, λ with k ≤ n − 1 and λ a partition of size less than 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (3) Calculate the numbers �e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ,�en−1 using the formula �ek = � λ ηλ[u2kxλ]F(u, x) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 17 from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that this is a finite sum: the terms are nonzero only for partitions λ of size 2k ≤ |λ| ≤ 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (4) Recover e(Out(Fn)) from �e0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ,�en−1 using the recursive formula in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The most demanding part of the computation is the expansion of the generating function F(u, x) for the Frobenius characteristic of admissible forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We used Jos Vermaseren’s FORM programming language [27] to compute this expansion up to an appropriate order and calculated e(Out(Fn)) for n ≤ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The result is listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will report on an optimized program with which the numbers e(Out(Fn)) have been computed for all n ≤ 100 in a separate publication [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Forested graphs with legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' So far we have only needed to consider forested graphs (G, Φ) such that G has no univalent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' However, to determine the asymptotic behavior of the integral Euler characteristic we will need to do a finer analysis, which involves studying pairs (G, Φ) where G is allowed to have univalent (but not bivalent) vertices, while Φ is not allowed to contain any edges adjacent to univalent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We call such pairs forested graphs with legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this section we point out that a minor modification of the counting methods from the previous sections counts these more general forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that admissible graphs are constructed by pairing the leaves of an admissible forest Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The edges of the forest then become a subforest Φ0 of the admissible graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To get the original forest back you cut all the 1-cells of G that are not in Φ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' the half-edges that result are the leaves of the original forest Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If the graph G has univalent vertices we are not allowing the subforest Φ to contain adjacent 1-cells so they must be cut, which results in components with one 0-cell and one half-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' These are not admissible trees, so we will call these special components, and mark them with a new variable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A forest which is allowed to have both admissible and special components will be called an extended forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Matching the leaves of an extended forest results in a graph with a univalent vertex for each special component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let F⋆ denote the species of extended forests, and define F ⋆(x, u, w) to be F ⋆(u, x, w) = � Φ∈F ⋆ (−1)e(Φ)us(Φ)−2k(Φ)wj(Φ) xs(Φ) s(Φ)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', where e(Φ) is the number of edges of Φ, s(Φ) is the number of leaves, k(Φ) is the total number of components and j(Φ) is the number of special components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [urxswj]F ⋆(u, x, w) is the edge-weighted count of forests with s leaves and (s − r)/2 components, of which j are special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that V (x) is the generating function for unrooted trees, weighted by the parity of their internal edges (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' F ⋆(u, x, w) = exp(u−1wx + u−2V (ux)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A special component has 1 leaf, 0 edges and 1 component (which is special!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ), so the term u−1wx accounts for special components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The term u−2V (ux) accounts for admissible trees as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Exponentiating gives the generating function for collections, as per Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ In [10] we showed that the rational Euler characteristic χn = χ(Out(Fn+1)) is given by χn = � [G,Φ] (−1)e(Φ) |Aut(G, Φ)|, where the sum is over isomorphism classes [G, Φ] of forested graphs with G connected of Euler characteristic −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We denoted the corresponding generating function by T (ℏ) = � n≥1 χnℏn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We then defined an analogous generating function T (ℏ, w) for isomorphism classes [G, Φ] of connected forested graphs with legs, which are weighted in the same way as in the formula above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 18 Here the variable w marks the legs of G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [ℏnws]T (ℏ, w) is the weighted sum over isomorphism classes [G, Φ] such that G is connected, has Euler characteristic −n, and has s legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can relate T (ℏ, w) to the generating function F ⋆(u, x, w): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � m≥0 (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [x2m]F ⋆(u, x, w) = exp �u−2w2 2 + T (u2, w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The coefficient of u2nws in (2m−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [x2m]F ⋆(u, x, w) gives the weighted count of forested graphs with s legs and χ = −n that can be made by gluing the leaves of extended forests Φ with 2m leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Adding up over all m gives the count of all forested graphs with s legs and χ = −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The term exp(T (u2, w)) counts forests, but misses components with a single internal vertex and two legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Those are taken care of by adding the term (u−2w2)/2 to T (u2, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Finally, we recall from [10] that T (ℏ, w) and T (ℏ) are related in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='15 ([10], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' T (ℏ, w) = T (ℏe−w) + w 2 + ℏ−1 � ew − 1 − w − w2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here T (ℏe−w) accounts for connected forested graphs with negative Euler characteristic, the term w 2 accounts for those with Euler characteristic zero, and the last term accounts for trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Asymptotics 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Precise asymptotic statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To discuss the asymptotic behavior of sequences we use the following standard conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let {cn} be a sequence defined for all but finitely many positive integers n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The set O(cn) consists of all such sequences {an} for which lim supn→∞ |an/cn| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The notation an = bn + O(cn) means an − bn ∈ O(cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that the notation an ∼ bn means limn→∞ an/bn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To describe the asymptotic behavior of the numbers �en we will use the Γ function, which is defined by Γ(x) = � ∞ 0 zx−1e−zdz for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' At integer arguments it agrees with the factorial n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = Γ(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Stirling’s formula, we have Γ(x) ∼ √ 2πxx− 1 2 e−x [1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Γ(x) grows more than exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Explicitly, we will quantify the asymptotic behavior of �en using the sequences Bn and Ln defined by Bn = − 1 √ 2π Γ � n − 1 2 � log2 n and Ln = log n log log n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows from Stirling’s formula that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Bn ∼ −nne−n/(n log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Bn/(−nne−n/(n log2 n)) ∼ e 1 2 (1 − 1 2n)n−1 ∼ 1, where we used limn→∞(1 + x n)n = ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ The remainder of the paper is devoted to proving the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �en has asymptotic behavior �en = e− 1 4 Bn + O(Bn/Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Note that limn→∞ Ln = ∞, so it follows that �en ∼ e− 1 4 Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By applying Stirling’s formula, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1, we moreover find that �en ∼ −e− 1 4 nne−n/(n log2 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In [10] we defined �χn to be the coefficient of ℏn in the series �T(ℏ) = exp(� n≥1 χnℏn), where χn = χ(Out(Fn+1)) is the rational Euler characteristic of Out(Fn+1), and proved THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 19 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 (Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7 in [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �χn = Bn + O(Bn/Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We just have to substitute the asymptotic formula for the numbers vk in [10, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7] (where we uncover an obvious typo: there should be no minus sign in [10, equation (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7)]) into the formula for �χn from [10, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ As an immediate corollary we get Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C, such that | �χn | ≤ C Γ � n − 1 2 � for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This follows from O(�χn) = O(Bn) = O � Γ(n − 1 2)/ log2 n � ⊂ O � Γ(n − 1 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, limn→∞ | �χn /Γ(n − 1 2)| is finite and as Γ(n − 1 2) > 0 for n ≥ 1, �χn /Γ(n − 1 2) stays bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ We also showed that the numbers χn have the same asymptotic behavior as �χn [10, Proposi- tion 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A slight modification of the proof of this given in [10] gives the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The numbers e(Out(Fn+1)) have the same asymptotic behavior as �en, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' e(Out(Fn+1)) = e− 1 4 Bn + O(Bn/Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8 of [10] gives a criterion for showing the asymptotic behavior of the coef- ficients of a series � anxn agrees with with that of the coefficients of exp(� anxn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6 in [10] immediately following Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8 applies almost verbatim to show that the coefficients en defined by �∞ n=1 en ℏn = log (�∞ n=0 �en ℏn) have the same asymp- totic behaviour as �en, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' en = e− 1 4 Bn + O(Bn/Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6 of the present paper, e(Out(Fn+1)) = en + � d|n,d̸=1 µ(d) d en/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As n has fewer than n divisors and |µ(n)| ≤ 1, the sums � d|n,d̸=1 µ(d) d en/d form a sequence in O(n en/2) = O(nBn/2) ⊂ O(nΓ(n/2 − 1 2)) ⊂ O(Bn/Ln), showing that e(Out(Fn+1)) also has the same asymptotic behavior as �en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Follows directly from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1, log(n − 1) ∼ log n and (1 − 1/n)n−1 ∼ e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In fact, we prove a slightly stronger statement than Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 which quantifies the error term in the asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5, we have for large n e(Out(Fn))/χ(Out(Fn)) = e− 1 4 + O(log log n/ log n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The error term O(log log n/ log n) above appears to be too pessimistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Based on the compu- tations of e(Out(Fn)) up to n = 100 from [9] and empirical comparison with χ(Out(Fn)), we conjecture e(Out(Fn))/χ(Out(Fn)) = e− 1 4 � 1 − 29 32n + O � 1 n2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Relating the integral and rational Euler characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Each triple [G, Φ, α] con- sisting of a forested graph with χ(G) = −n and an automorphism α preserving Φ contributes to �en, and the triples with α = id give the rational Euler characteristic �χn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 we will exploit the fact that we already know the asymptotics of �χn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will eventually find that the contributions to �en from triples [G, Φ, α] such that α fixes Φ and has order at most 2 dominate the asymptotics of �en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will isolate these contributions in the formulas we have for our generating functions, and prove that they dominate by bounding the relative size of the remaining terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 20 We start with the formula �en = � λ ηλ[u2nxλ]F(u, x) from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 we have an expression for F(u, x), so �en = � λ ηλ[u2nxλ] exp \uf8eb \uf8ed� k≥1 u−2k V((u · x)[k]) k \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that a permutation is called a derangement if it has no fixed points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' this is equivalent to saying that the corresponding cycle type has no parts of size 1, so we call it a deranged partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An integer partition λ is equivalent to a pair (m, δ) where m is the number of parts of size 1 in λ and δ is the deranged partition obtained from λ by removing all parts of size 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To make an admissible graph with an automorphism from a pair (Φ, α) consisting of a forest and an automorphism, leaves of Φ that are permuted by α in cycles of equal lengths must be paired with each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' in particular, leaves that are fixed by α must be paired with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By first pairing the fixed leaves we can reduce the expression for �en above to a sum over deranged partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Specifically, if there are 2m fixed leaves there are (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ways to pair them, so if λ = (2m, δ) then ηλ = (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='ηδ and �en = � δ ηδ[u2nxδ] ∞ � m=0 (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [x2m 1 ] exp \uf8eb \uf8ed� k≥1 u−2k V((u · x)[k]) k \uf8f6 \uf8f8 = � δ ηδ[u2nxδ] exp \uf8eb \uf8ed� k≥2 u−2k V((u · x)[k]) k \uf8f6 \uf8f8 ∞ � m=0 (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [x2m 1 ] exp � u−2V(u · x) � , where we sum only over deranged integer partitions δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (Note that the variable x1 does not appear in the power series V((u·x)[k]) for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=') The expression in the first exponential will not change in what follows, so we give it a name: h1(u, x) := � k≥2 u−2k V((u · x)[k]) k = � k≥2 ukx3 k 6k + ukxkx2k 2k + ukx3k 3k − u2kx4 k 12k + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' and our expression reads �en = � δ ηδ[u2nxδ] exp(h1(u, x)) ∞ � m=0 (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [x2m 1 ] exp � u−2V(u · x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (16) Next we look more closely at the term exp(u−2V(u·x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We first separate out the contribution to V(x) of pairs (t, α) with α = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This is obtained by setting xi = 0 in V(x) for all i ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' recall that this gives us the generating function V (x1) for trees without automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9 we have V(x) = x1 + x2 1 2 − x2 2 − (1 + x1)R(x) = x1 + x2 1 2 − x2 2 − (1 + x1) � k≥1 µ(k) k log(1 + xk) = x1 + x2 1 2 − (1 + x1) log(1 + x1) − x2 2 − (1 + x1) � k≥2 µ(k) k log(1 + xk) = V (x1) − x2 2 + (1 + x1)W(x), (17) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 21 where W(x) = − � k≥2 µ(k) k log(1 + xk) = −R(x) + (1 + x1) log(1 + x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Note that W(x) counts rooted trees with automorphisms that do not fix any leaves, in particular it does not involve the variable x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (17) we have exp � u−2V(u · x) � = exp � u−2 � V (ux1) − u2 x2 2 + (1 + ux1)W(u · x) �� = exp �� u−2V (ux1) + u−1x1W(u · x) � + u−2 � W(u · x) − u2 x2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13, exp � u−2V (ux) + u−1wx � = F ⋆(u, x, w), where [u2nxswj]F ⋆(u, x, w) counts forests with j special components and s leaves that glue up to graphs with χ = −n and j legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Note that W(u · x) can be interpreted as a power series in u whose coefficients are polynomials in x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='. This power series has no constant coefficient, so if f(w) is another power series, then the composition f(W(u · x)) is convergent in the usual power series topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Substituting x1 for x and W(u · x) for w in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13, our formula for exp � u−2V(u · x � ) becomes exp � u−2V(u · x) � = F ⋆(u, x1, W(u · x)) exp � u−2 � W(u · x) − u2 x2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Substituting the above into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (16) gives the following expression: �en = � δ ηδ[u2nxδ] exp(h1(u, x))× ∞ � m=0 (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [xm 1 ]F ⋆(u, x1, W(u · x)) exp � u−2 � W(u · x) − u2 x2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (18) Now recall the statement of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14 � m≥0 (2m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [x2m]F ⋆(u, x, w) = exp �u−2w2 2 + T (u2, w) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' After substituting x1 for x and W(u · x) for w in this statement, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (18) becomes �en = � δ ηδ[u2nxδ] exp(h1(u, x))× exp �u−2W(u · x)2 2 + T (u2, W(u · x)) + u−2 � W(u · x) − u2 x2 2 �� (19) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='15 we have a relation between T (ℏ, w) and T (ℏ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Substituting u2 for ℏ and W(u · x) for w, this relation becomes T (u2, W(u · x)) = T (u2e−W(u·x)) + W(u · x) 2 + u−2 � eW(u·x) − 1 − W(u · x) − W(u · x)2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (19) and simplifying turns our expression for �en into � δ ηδ[u2nxδ] exp � h1(u, x) + T � u2e−W(u·¯x)� + W(u · x) 2 + u−2 � eW(u·¯x) − 1 − u2 x2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (20) Setting �T(ℏ) = exp(T (ℏ)) h2(u, x) = W(u · x) 2 h3(u, x) = u−2 � eW(u·¯x) − 1 − u2 x2 2 � , we record Equation (20) formally as a theorem: THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 22 v w (G, Φ, α) 1 2 3 4 5 6 G′ G′′ (G, Φ) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Reducing a forested graph (G, Φ) with automorphism α interchang- ing v and w to a forested graph with legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The graph G′ is obtained by cutting edges not in Gα or Φ, and α induces the derangement (14)(23)(56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The next graph G′′ replaces each set of trees at a vertex of Gα by a single orange leg, and the final graph G results from contracting all separating edges in G′′ that are not legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �en = � δ ηδ[u2nxδ] �T(u2e−W(u·¯x))H(u, x), where the sum is over all deranged integer partitions δ and H(u, x) = exp (h1(u, x) + h2(u, x) + h3(u, x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here is a combinatorial interpretation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Given a triple (G, Φ, α), let Gα be the subgraph of G fixed by α, and Φα = Φ ∩ Gα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If we cut all edges of G which are not in Φ or in Gα, we obtain a graph G′ with leaves, and α induces a derangement of these leaves (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The fixed subgraph Gα is a subgraph of G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Components of G′ that do not intersect Gα are k-cycles of trees with k ≥ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' these contribute the term h1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If C is a component that does intersect Gα then (C ∩ Gα, C ∩ Φα) is a forested graph, and the rest of C consists of k-cycles of rooted trees attached at various vertices of C ∩ Gα, where k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' At each of these vertices, remove all of the deranged trees that are attached there and replace them with a single orange leg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Then contract all separating edges of the result that are not orange legs to get an admissible forested graph (C, Φ) with legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can count such forested graphs using the generating functions T (ℏe−w) (if χ(C) < 0), w 2 (if χ(C) = 0), and ℏ−1w − x2 2 (if C is a tree but C is not a single vertex with two half-edges attached).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Replacing ℏ by u2 and w by W(u · x) has the effect of marking the negative Euler characteristic by u2 instead of ℏ and reconstructing the components C by adding rooted trees to the forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To determine the asymptotic behavior of �en we will estimate the size of the coefficients of H(u, x), and show that their contribution to �en is dominated asymptotically by the contribution of �T(u2e−W(u·¯x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will also show that the sum � δ ηδ[u2nxδ] �T(u2e−W(u·¯x)) is dominated by contributions of deranged partitions δ with all parts of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We will then be able to determine the asymptotic behavior of �en using estimates on the size of ηλ and the fact that we know the behavior of the coefficients �χn of �T(ℏ) from our previous work in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We note that µ(2) = µ(3) = −1 and µ(4) = 0, hence the first few terms of W(u · ¯x) are W(u · ¯x) = − � k≥2 µ(k) k log(1 + ukxk) = 1 2 � u2x2 − u4x2 2 2 � + u3x3 3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = u2x2 2 + u3x3 3 − u4x2 2 4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 23 where higher powers of u were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows that h2(u, x) = W(u · ¯x)/2 and h3(u, x) = u−2 � eW(u·¯x) − 1 − u2 x2 2 � = ux3 3 − u2 x2 2 8 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' are power series in only positive powers of u (as is h1(u, x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Splitting and merging Γ functions and estimating the numbers ηλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this section we show that the numbers ηλ and ηk,m defined in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5 are bounded by Γ functions modulated by exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that the Γ function satisfies Γ(x + 1) = xΓ(x) for all x > 0, Γ(1/2) = √π, and Γ is is log-convex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' log Γ � ta + (1 − t) b + 1 2 � ≤ t log Γ � a + 1 2 � + (1 − t) log Γ � b + 1 2 � , (21) for all t ∈ [0, 1] and a, b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In fact, these three properties determine the function Γ completely [1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 below follow easily using these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For all x, y, z ∈ R with 0 ≤ z ≤ y ≤ x we have Γ � x + 1 2 � Γ � y + 1 2 � ≤ Γ � x + z + 1 2 � Γ � y − z + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Adding Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (21) to itself with a and b interchanged gives, log Γ � ta + (1 − t) b + 1 2 � + log Γ � (1 − t) a + tb + 1 2 � ≤ log Γ � a + 1 2 � + log Γ � b + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can choose a = x + z, b = y − z and t = 1 − z/(x − y + 2z), as t ∈ [0, 1] and a, b ≥ 0 are fulfilled because z ≤ y ≤ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Exponentiating gives the stated inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For all x, y ∈ R with x, y ≥ 0 we have Γ � x + 1 2 � √π Γ � y + 1 2 � √π ≤ Γ � x + y + 1 2 � √π (22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Specializing Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8 with y = z and using Γ(1/2) = √π gives the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The exists a constant C such that for all x, y ∈ R with x, y ≥ 0 we have Γ � x + y + 1 2 � ≤ C1+x+yΓ � x + 1 2 � Γ � y + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (23) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Specializing Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8 to x = y = (x′ + y′)/2 and z = |x′ − y′|/2 gives, Γ �x′ + y′ + 1 2 �2 ≤ Γ � x′ + 1 2 � Γ � y′ + 1 2 � for all x′, y′ ∈ R with x′, y′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The duplication formula for the Γ function [1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11)] can be written as Γ �z + 1 2 � = 2−z√πΓ(z + 1)/Γ �z 2 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (24) Applying this identity once to the left hand side of the inequality above results in Γ � x + y + 1 2 � ≤ 2x+y √π F(x + y)Γ � x + 1 2 � Γ � y + 1 2 � for all x, y ∈ R with x, y ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 24 where F(z) = Γ � z 2 + 1 � Γ � z + 1 2 � /(Γ � z+1 2 � Γ(z+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To prove the upper bound in the statement it is therefore sufficient to show that there is a constant C such that F(z) ≤ C for all z ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' F(z) is regular for all z ≥ 0, so it is sufficient to prove the existence of such a constant for z ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this case, we have F(z) ≤ 1, because z+1 2 ≤ z and can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8 to establish Γ �z 2 + 1 � Γ � z + 1 2 � ≤ Γ �z + 1 2 � Γ(z + 1) for all z ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ We now apply these lemmas to bound the numbers ηλ defined in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C such that ηk,m ≤ (kC)m/2 √π Γ �m + 1 2 � for all k ≥ 1 and m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Because ηk,0 = 1 and Γ(1/2) = √π, the statement is obvious if m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In all other cases it is sufficient to prove the bound for m large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By the standard identity (2n−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = 2nΓ(n+ 1 2)/√π and by the definition of ηk,m the statement is true for all odd k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For even k, ηk,m = ⌊m/2⌋ � r=0 �m 2r � kr(2r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = ⌊m/2⌋ � r=0 �m 2r � kr2r Γ(r + 1 2) √π ≤ 2m max 0≤r≤⌊m/2⌋(2k)r Γ(r + 1 2) √π , where we used the floor function ⌊·⌋ and �m 2r � ≤ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The statement follows since Γ(x) is increasing for sufficiently large x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Recall that ℓ(λ) denotes the length of λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' the number of parts of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There is a constant C such that for all integer partitions λ, ηλ ≤ C|λ| √π Γ �ℓ(λ) + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let λ = [1m12m2 · · · ] and note that mk = 0 for all k > |λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that log k ≤ k for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore, |λ| � k=1 kmk/2 = exp \uf8eb \uf8ed |λ| � k=1 mk/2 log k \uf8f6 \uf8f8 ≤ exp \uf8eb \uf8ed |λ| � k=1 kmk/2 \uf8f6 \uf8f8 = e|λ|/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11 there is a constant C such that ηλ = |λ| � k=1 ηk,mk ≤ |λ| � k=1 (kC)mk/2 √π Γ �mk + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The proof is completed by using the first inequality, the fact that � k≥1 mk = ℓ(λ) ≤ |λ| and the bound from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Finally, the last lemma in this section shows how the constant e− 1 4 arises: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For large n, n � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' η2,m = e− 1 4 + O � 1 Ln � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In fact this sum converges much faster than the indicated error term, but this rough error estimate is sufficient for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5 we have η2,m = �⌊m/2⌋ r=0 �m 2r � 2r(2r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using this, we find that ∞ � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' η2,m = ∞ � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ⌊m/2⌋ � r=0 2r 2rr!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (m − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = ∞ � r=0 1 r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ∞ � m=2r (−1)m 2m(m − 2r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = ∞ � r=0 (−1)2r 4rr!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ∞ � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = e 1 4 e− 1 2 = e− 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11 we find a constant C such that the tail of this series is bounded as follows: ����� ∞ � m=n+1 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' η2,m ����� ≤ ∞ � m=n+1 Cm Γ( m+1 2 ) √πm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = Cn+1 ∞ � m=0 Cm Γ( m+n+2 2 ) √π(m + n + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ≤ Cn+1C′1+ n+1 2 Γ( n+2 2 ) √πn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ∞ � m=0 CmC′ m 2 Γ( m+1 2 ) √πm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ∈ O � CnC′ n 2 Γ( n+1 2 ) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � ⊂ O � 1 Ln � , where we used a constant C′ from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 to split the Γ function in the numerator and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9 to split the factorial function in the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The convergence of the infinite sum and the last inclusion of sets follows from Stirling’s approximation of the Γ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A new sequence of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let Pn = n � m=0 �χn−m �− 1 2(n − m) m � η2,m, where the binomial coefficient �q k � is defined by q(q−1)···(q−k+1) k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' for integers k ≥ 0 and all q ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In this section we will use the fact that we know the asymptotic behavior of the numbers �χn (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3) together with the estimates from the previous section to determine the asymptotic behavior of the numbers Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In the two sections after this one we will prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �en = Pn + O � Γ � n − 7 12 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' At the end of this section we observe that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14 together with the asymptotics of Pn imply the main asymptotic result, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In order to determine the asymptotic behavior of the Pn we need a few more estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C such that 1 ≤ m−1 � r=0 n − m + 2r n − m − 1 2 + r ≤ exp � C m(m + 1) n � for all integers n, m with 0 ≤ m ≤ n − 1 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The lower bound is obvious as n − m + 2r ≥ n − m − 1 2 + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We start by proving the bound for all m ≥ n−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We have m−1 � r=0 n − m + 2r n − m − 1 2 + r = 2m m−1 � r=0 n−m 2 + r n − m − 1 2 + r ≤ 2m = 2m n n ≤ 2 m(2m+1) n ≤ 4 m(m+1) n , as n − m ≥ 1 ⇒ n−m 2 + r ≤ n − m − 1 2 + r and 2m + 1 ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 26 We still have to prove the bound for m ≤ n−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that 1 + x ≤ ex for all x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows that 1 1−x ≤ e x 1−x for x < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore, m−1 � r=0 n − m + 2r n − m − 1 2 + r = m−1 � r=0 1 + 2r−m n 1 − m+ 1 2 −r n ≤ m−1 � r=0 exp � 2r − m n + m+ 1 2 −r n 1 − m+ 1 2 −r n � ≤ m−1 � r=0 exp �2r − m n + 2m + 1 2 − r n � = exp �m(m + 1) n � , where we used m ≤ n−1 2 ⇒ 1/ � 1 − m+ 1 2 −r n � ≤ 1/ � 1 2 + r n � ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C such that 1 ≤ log(n + 1) log(n − m + 1) ≤ exp � C m(m + 1) n � for all integers n, m with 0 ≤ m ≤ n − 1 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The lower bound is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We start by proving the estimate for m > n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Because log grows slower than the exponential, there exists a constant C′ such that log(n + 1) log(n − m + 1) ≤ log(n + 1) log 2 ≤ C′n ≤ (C′2) n+1 2 ≤ (C′2)m ≤ (C′4) m(m+1) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The bounds remains to be proven for all m ≤ n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Again we use the inequality 1 1−x ≤ exp x 1−x ⇒ log 1 1−x ≤ x 1−x which holds for all x ∈ [0, 1) to get log(n + 1) log(n − m + 1) = 1 1 + log(1− m n+1 ) log(n+1) ≤ exp \uf8eb \uf8ed − log(1− m n+1 ) log(n+1) 1 + log(1− m n+1 ) log(n+1) \uf8f6 \uf8f8 = exp �− log(1 − m n+1) log(n − m + 1) � ≤ exp �log 1 1− m n+1 log 2 � ≤ exp � 1 log 2 m n+1 1 − m n+1 � ≤ exp � 2 log 2 m n + 1 � , where we used 1/ � 1 − m n+1 � ≤ 1 2 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Recall the sequences Bn = − 1 √ 2π Γ(n− 1 2 ) log2 n and Ln = log n log log n from the beginning of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Because these sequences are not defined for n = 1, it is convenient to use following ones instead B′ n = − 1 √ 2π Γ(n − 1 2) log2(n + 1) L′ n = log(n + 1) log log(n + e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It is clear that O(B′ n) = O(Bn) and O(L′ n) = O(Ln), but we also have control over the error: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' B′ n = Bn + O � Bn n � and 1/L′ n = 1/Ln + O � 1 nLn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This follows from the fact that the following elementary limits exist: lim n→∞ 1/ log2(n + 1) − 1/ log2 n 1/(n log2 n) and lim n→∞ log log(n + e)/ log(n + 1) − log log n/ log n log log n/(n log n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This can be shown, for instance, by using log(n + x) − log n ≤ x/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let Qn,m = �χn−m /B′ n−m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C such that |Qn,m| ≤ Cm+1/L′ n for all n, m with m ≥ 0 and n ≥ m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='17 we have �χn −B′ n ∈ O(B′ n/L′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Because B′ n, �χn and L′ n are finite for all n ≥ 1, there exists a constant C′ such that |�χn/B′ n − 1| ≤ C′/L′ n for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows that for all integers n, m with m ≥ 0 and n ≥ m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' |Qn,m| = ���χn−m /B′ n−m − 1 �� ≤ C′/L′ n−m ≤ C′ exp � C′′ m(m + 1) n � log log(n − m + e) log(n + 1) , where we used a constant C′′ from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using m + 1 ≤ n and the monotonicity of log gives the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let Rn,m = (−1)m2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �− 1 2(n − m) m � B′ n−m/B′ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C such that |Rn,m| ≤ Cm+1 n for all n, m with m ≥ 0 and n ≥ m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Use Γ(n − 1 2) = Γ(n − m − 1 2) �m−1 r=0 (n − m − 1 2 + r) and � q m � = 1 m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �m−1 r=0 (q − r) to get Rn,m + 1 = (−1)m2m log2(n + 1) log2(n − m + 1) m−1 � r=0 − 1 2(n − m) − r n − m − 1 2 + r = log2(n + 1) log2(n − m + 1) m−1 � r=0 n − m + 2r n − m − 1 2 + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='16 there exists a constant C′ such that 1 ≤ 1 + Rn,m ≤ exp � C′ m(m + 1) n � for all integers n, m with m ≥ 0 and n ≥ m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Because 1 − x ≤ e−x ⇒ ex ≤ 1 + xex for all x ∈ R, we have |Rn,m| ≤ C′ m(m+1) n exp � C′ m(m+1) n � ≤ C′ m(m+1) n exp (mC′) for all m ≥ 0 and n ≥ m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The statement follows from the fact that we can find a constant C′′ such that C′ m(m+1) n exp � C′ m(m+1) n � ≤ C′′m+1 n for all m ≥ 0 and n ≥ m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ We are now ready to estimate the numbers Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Pn = e− 1 4 B′ n + O (B′ n/L′ n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' With Qn,m and Rn,m from Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='18 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='19 we have Pn = n � m=0 �χn−m �− 1 2(n − m) m � η2,m = B′ n n � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (1 + Qn,m)(1 + Rn,m)η2,m By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11 there exists a constant C such that η2,m ≤ CmΓ( m+1 2 ) for all m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='18 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='19 we get a constant C′ such that n � m=0 ���� (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (Qn,m + Rn,m + Qn,mRn,m)η2,m ���� ≤ C′ � 1 L′n + 1 n + 1 nL′n � ∞ � m=0 1 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (CC′)mΓ �m + 1 2 � ∈ O (1/L′ n) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 28 where the sum over m is convergent due to the factorial m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = Γ(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, Pn = B′ n n � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' η2,m + O (B′ n/L′ n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An application of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13 and using O(1/Ln) = O(1/L′ n) concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We want to show that �en = e− 1 4 Bn + O(Bn/Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14 (which we will prove in the next section), �en = Pn + O (Γ (n − 7/12)) , and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='20 Pn = e− 1 4 B′ n+O (B′ n/L′ n) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We now use the elementary fact that O (Γ (n − 7/12)) ⊂ O (B′ n/L′ n) = O (Bn/Ln) together with the asymptotic equality B′ n = Bn + O (Bn/n) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='17 to finish the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Estimates for Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We still need to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To do so we need to establish that the contribution of H(u, x) in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7 is negligible to the asymptotic behaviour of the numbers �en and the power series �T(u2e−W(u·¯x)) only contributes partially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We first show that the coefficients of �T(u2e−W(u·¯x)) can be written explicitly using the numbers �χn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let r ≥ 0 and δ be a deranged partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Then [urxδ] �T � u2e−W(u·¯x)� = �χ r−|δ| 2 r � k=2 �(r − |δ|) µ(k) 2k mk(δ) � , where we agree that �χ r−|δ| 2 = 0 if r−|δ| is odd or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In particular, [u0xδ] �T � u2e−W(u·¯x)� = 1 if δ = ∅ and 0 otherwise, and [u1xδ] �T � u2e−W(u·¯x)� = 0 for all δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Use �T(ℏ) = � �χn ℏn and W(x) = − � k≥2 µ(k) k log(1 + xk) to get �T � u2e−W(u·¯x)� = � n≥0 �χn u2n � k≥2 � 1 + ukxk �n µ(k) k = � n≥0 �χn u2n � k≥2 � mk≥0 �n µ(k) k mk � ukmkxmk k = � n≥0 � δ �χn u2n+|δ|xδ � k≥2 �n µ(k) k mk(δ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Using this explicit formula we can now obtain a bound on the coefficients of �T(u2e−W(u·¯x)): Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C, such that for all r ≥ 2 and deranged partitions δ = [2m23m3 · · · ] with |δ| ≤ r, ���[urxδ] �T � u2e−W(u·¯x)���� ≤ C Γ � r−|δ|+2ℓ(δ)−1 2 � m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='m3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We start with the special cases where δ is a deranged partition of r or r − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In both cases r ≥ 2 implies δ ̸= ∅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ℓ(δ) ≥ 1 and the argument of the Γ function on the right hand side of the statement is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='21 and the agreement that �χ 1 2 = 0 and � 0 m � = 0 for all m ≥ 1, we find that the left hand side vanishes in these cases and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For |δ| ≤ r − 2 we can apply Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4 to the statement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='21 to get a constant C such that for all deranged partitions δ = [2m23m3 · · · ], ���[urxδ] �T � u2e−W(u·¯x)���� ≤ CΓ �r − |δ| − 1 2 � ����� r � k=2 �(r − |δ|) µ(k) 2k mk ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (25) THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 29 From the standard expression for the binomial coefficients � q m � = q(q−1)···(q−m+1) m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , we get ����� r � k=2 �(r − |δ|) µ(k) 2k mk ������ = r � k=2 ��� �mk−1 s=0 � (r − |δ|) µ(k) 2k − s ���� mk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ≤ �r k=2 �mk−1 s=0 � (r − |δ|) 1 2k + s � m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='m3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , where we used |µ(n)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By using xΓ(x) = Γ(x + 1) repeatedly we get Γ �r − |δ| − 1 2 � = Γ � r−|δ|+2ℓ(δ)−1 2 � �ℓ(δ)−1 s=0 � r−|δ|−1 2 + s �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Combining these observations with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (25) gives ���[urxδ] �T � u2e−W(u·¯x)���� ≤ C Γ � r−|δ|+2ℓ(δ)−1 2 � m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='m3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �r k=2 �mk−1 s=0 � (r − |δ|) 1 2k + s � �ℓ(δ)−1 s=0 � r−|δ|−1 2 + s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It is easy to verify that �r k=2 �mk−1 s=0 � (r − |δ|) 1 2k + s � ≤ �ℓ(δ)−1 s=0 � r−|δ|−1 2 + s � by using ℓ(δ) = �|δ| k=2 mk and r − |δ| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ It is substantially harder to prove a good estimate for the coefficients of H(u, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It turns out to be convenient to include the numbers ηλ in our estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For integer partitions λ, λ′, we define λ ∪ λ′ to be the partition of |λ| + |λ′| that contains the union of all parts of λ and λ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' mk(λ ∪ λ′) = mk(λ) + mk(λ′) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There is a constant C such that for all r ≥ 0 and deranged partitions δ, � δ′ ηδ∪δ′ ���[urxδ′]H(u, x) ��� ≤ C1+r+|δ|Γ �ℓ(δ) + 5 6r + 1 2 � , where we sum over all deranged integer partitions δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The proof of this proposition will occupy the remainder of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that the series H(u, x) is defined by H = exp (h1 + h2 + h3) where h1(u, x) = � k≥2 u−2k V((u · x)[k]) k , h2(u, x) = W(u · x) 2 , h3(u, x) = u−2 � eW(u·¯x) − 1 − u2 x2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [urxδ] log H(u, x) = 0 unless either (1) r ≥ 2 and |δ| = r (2) r ≥ 1 and |δ| = r + 2 or (3) |δ| = r +2k for some k ≥ 2 dividing r and δ = kµ = (kµ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , kµℓ) for some partition µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The nonzero terms of of h2 are all of the form (1), the nonzero terms of h3 are of the form (2) and the nonzero terms of h1 are of the form (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Let J be the set of pairs (r, δ) satisfying the conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For all (r, δ) ∈ J, we have |δ| ≤ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The statement is obvious for the first two cases of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24, and follows in the third case because k divides r, so is at most r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 30 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This corollary also follows immediately from the combinatorics: An admissible graph G with χ(G) = −n has at most 3n edges, so if pairing all leaves of some forest produces an an admissible graph with −2χ(G) = r then the forest cannot have more than 3r(= 6n) leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' All coefficients of log H have absolute value less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since the nonzero terms of h1, h2 and h3 have no monomials in common, we can consider their coefficients separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It is clear from their definitions that the coefficients of W(x) and V(x) are less than 1, so the coefficients of h2 and h1 are as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For h3, we have eW(x) = � k≥2 (1 + xk)− µ(k) k = � k≥2 \uf8eb \uf8ed� m≥0 �− µ(k) k m � xm k \uf8f6 \uf8f8 = � k≥2 \uf8eb \uf8ed� m≥0 − µ(k) k 1 − µ(k) k − 1 2 · · − µ(k) k − m + 1 m xm k \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The magnitude of the fractions | − µ(k) k −ℓ+1 ℓ | is always smaller than 1, because |µ(k)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence all coefficients of eW(x) (and therefore of h3) are also less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ The next lemma shows that we can bound the coefficients of H(u, x) without determining the explicit values of those coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let J be the set of pairs (r, δ) satisfying the conditions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For all C ≥ 1 � ℓ(δ′)=ℓ′ C|δ′| ���[ur′xδ′]H(u, x) ��� ≤ C3r′[ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J urxℓ(δ) \uf8f6 \uf8f8 , where we sum over all deranged partitions δ′ of length ℓ′ on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='27 and the positivity of the expansion exp(x) = �∞ n=0 xn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , C|δ′| ���[ur′xδ′]H(u, x) ��� ≤ C|δ′|[ur′xδ′] exp \uf8eb \uf8ed � (r,δ)∈J urxδ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='25 the coefficient extraction operator has support only for |δ| ≤ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since C ≥ 1, C|δ| ≤ C3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Summing over all deranged partitions δ of the same length on the left is equivalent to substituting xi = x for all i on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Our next estimate will make use of the of the following rough bound on integer partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The number of integer partitions of size at most n is smaller than 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Writing a string of length n of 1’s with either a + or a comma in between gives a compo- sition of n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' an integer partition of n with an ordering of the parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There are 2n−1 different such compositions of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Ordering the elements of a composition gives a many-to-one function to integer partitions, which is clearly surjective for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Thus the number of integer partitions of n is bounded by 2n−1 for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The number of integer partitions of size at most n is therefore smaller than 1+20 +21 +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='+2n−1, where the initial 1 accounts for the empty integer partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since 20 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' + 2n−1 = 2n − 1, the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Now let K be the subset of pairs (r, δ) in J for which 5 6r < ℓ(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 31 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There is a constant C such that for all r′, ℓ′ ≥ 0, [ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J\\K urxℓ(δ) \uf8f6 \uf8f8 ≤ Cr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Moreover, [ur′xℓ′] exp �� (r,δ)∈J\\K urxℓ(δ)� = 0 when 5 6r′ < ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The second statement follows immediately from the definition of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='25, the set J \\ K is contained in the set of all pairs (r, λ) where r ≥ 1 and λ is a partition such that |λ| ≤ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='29 this implies that � (r,δ)∈J\\K ur is bounded coefficientwise by � r≥1 23rur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since log r ≤ r for all r ≥ 1, the series � r≥1 23rur = � r≥1 ur8relog r/r, is bounded coefficientwise by � r≥1 ur(e8)r/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Summing over ℓ′ on the left hand side of the inequality in the statement of the lemma is equivalent to setting x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, [ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J\\K urxℓ(δ) \uf8f6 \uf8f8 ≤ [ur′] exp \uf8eb \uf8ed � (r,δ)∈J\\K ur \uf8f6 \uf8f8 ≤ [ur′] exp \uf8eb \uf8ed� r≥1 (8e)r r ur \uf8f6 \uf8f8 = (8e)r′, where we used log 1 1−x = � r≥1 xr r in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There is a constant C such that for all r′ ≥ 0 and z ∈ R with z ≥ 0, � ℓ′≥0 Γ �z + ℓ′ + 1 2 � [ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J\\K urxℓ(δ) \uf8f6 \uf8f8 ≤ Cr′+1Γ �z + 5 6r′ + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='30 there is a constant C′ such that � ℓ′≥0 Γ �z + ℓ′ + 1 2 � [ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J\\K urxℓ(δ) \uf8f6 \uf8f8 ≤ C′r′ � 0≤ℓ′≤⌊ 5 6 r′⌋ Γ �z + ℓ′ + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9, Γ � z+ℓ′+1 2 � Γ � 5 6 r′−ℓ′+1 2 � ≤ √πΓ � z+ 5 6 r′+1 2 � for all 0 ≤ ℓ′ ≤ 5 6r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, � 0≤ℓ′≤⌊ 5 6 r′⌋ Γ �z + ℓ′ + 1 2 � ≤ √πΓ �z + 5 6r′ + 1 2 � � 0≤ℓ′≤⌊ 5 6 r′⌋ 1 Γ � 5 6 r′−ℓ′+1 2 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum over ℓ′ is bounded by a constant independent of r′ as � 0≤ℓ′≤⌊ 5 6 r′⌋ 1 Γ � 5 6 r′−ℓ′+1 2 � = � 0≤ℓ′≤⌊ 5 6 r′⌋ 1 Γ � 5 6 r′−⌊ 5 6 r′⌋+ℓ′+1 2 � ≤ C′′ 5 6 r′−⌊ 5 6 r′⌋ Γ � 5 6 r′−⌊ 5 6 r′⌋+1 2 � � 0≤ℓ′≤∞ C′′ℓ′ Γ � ℓ′+1 2 �, where we used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 to split the Γ function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The set K is a subset of K = {(1, [31]), (2, [23]), (2, [2141]), (2, [22]), (3, [33]), (4, [24])}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24, for all (r, δ) ∈ J we have |δ| = r + 2k for some k ≥ 0, and if k ̸= 0 then k divides r and δ = kµ for some partition µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that K is the subset of pairs (r, δ) in J for which 5 6r < ℓ(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since each part of a deranged partition has size at least 2, ℓ(δ) ≤ |δ|/2, so 5 6r < ℓ(δ) implies 2r < 6k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We look for pairs (r, δ) that fulfill all these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If k = 0, there are no solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If k = 1, the only solutions are r ∈ {1, 2} and the possible deranged partitions δ are δ ∈ {[31]} for r = 1 and δ ∈ {[22], [41]} for r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 32 If k ≥ 2, the additional constraint that k divides r implies that (r, k) ∈ {(2, 2), (3, 3), (4, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For both (r, k) = (2, 2) and (r, k) = (3, 3), we have |µ| = 3, which means µ ∈ {[13], [1121], [31]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, δ ∈ {[23], [2141], [61]} for (r, k) = (2, 2) and δ ∈ {[33], [3161], [91]} for (r, k) = (3, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For (r, k) = (4, 2) we have δ = 2µ with µ ∈ {[14], [1221], [22], [1131], [41]} and therefore δ ∈ {[24], [2241], [42], [2161], [81]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The requirement ℓ(δ) > 5 6r now eliminates all solutions that are not in the list K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For all (r, δ) ∈ K, we have 5 6r < ℓ(δ) < 5 6r + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This can be verified by checking all 6 elements in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Fix (r, δ) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There is a constant C such that for r′ ≥ 0 and z ∈ R with z ≥ 0, � ℓ′≥0 [ur′xℓ′] exp(urxℓ(δ))Γ �z + ℓ′ + 1 2 � ≤ Cz+r′+1Γ �z + 5 6r′ + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can expand the left hand side to get � ℓ′≥0 [ur′xℓ′] exp(urxℓ(δ))Γ �z + ℓ′ + 1 2 � = � ℓ′≥0 [ur′xℓ′] � k≥0 urkxℓ(δ)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Γ �z + ℓ′ + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum over k only contributes if rk = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, if r divides r′ we get � ℓ′≥0 [ur′xℓ′] exp(urxℓ(δ))Γ �z + ℓ′ + 1 2 � = 1 (r′/r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='Γ � z + ℓ(δ) r′ r + 1 2 � and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 there exists a constant C′ such that for α with 0 ≤ α ≤ ℓ(δ), 1 (r′/r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='Γ � z + ℓ(δ) r′ r + 1 2 � ≤ 1 (r′/r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='C′ z 2 +ℓ(δ) r′ 2r +1Γ � α r′ 2r + 1 2 � Γ � z + (ℓ(δ) − α) r′ r + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = Γ(n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, Γ � α r′ 2r + 1 2 � /(r′/r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' is bounded for all r′ ≥ 0 if α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We may choose α = ℓ(δ) − 5 6r, which fulfills 0 ≤ α ≤ 2 as 5 6r ≤ ℓ(δ) ≤ 2 + 5 6r by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let Ar′ = � ℓ′≥0 Γ �ℓ′ + 1 2 � [ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J urxℓ(δ) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There is a constant C such that Ar′ ≤ Cr′+1Γ � 5 6 r′+1 2 � for all r′ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We can split the exponential in the definition of Ar′, exp \uf8eb \uf8ed � (r,δ)∈J urxℓ(δ) \uf8f6 \uf8f8 = exp \uf8eb \uf8ed � (r,δ)∈K urxℓ(δ) \uf8f6 \uf8f8 exp \uf8eb \uf8ed � (r,δ)∈J\\K urxℓ(δ) \uf8f6 \uf8f8 By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='32, exp �� (r,δ)∈K urxℓ(δ)� is bounded by exp �� (r,δ)∈K urxℓ(δ)� coefficientwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let (r1, δ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , (r6, δ6) denote the elements of K in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' With this notation, exp \uf8eb \uf8ed � (r,δ)∈K urxℓ(δ) \uf8f6 \uf8f8 = 6 � i=1 eurixℓ(δi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 33 We may use this to write Ar′ as Ar′ = � Γ ��7 i=1 ℓ′ i + 1 2 � � 6 � i=1 [ur′ ixℓ′ i]eurixℓ(δi) � [ur′ 7xℓ′ 7] exp \uf8eb \uf8ed � (r,δ)∈J\\K urxℓ(δ) \uf8f6 \uf8f8 , where we have to sum over all tuples of integers ℓ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , ℓ′ 7, r′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , r′ 7 ≥ 0 with r′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' + r′ 7 = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Applying Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='31 once and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='34 six times results in the bound Ar′ ≤ � C′ �7 i=1 r′ i+1Γ � 5 6 �7 i=1 r′ i + 1 2 � , where C′ is an appropriate constant, whose existence follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='34 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='31, and we sum over all r′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , r′ 7 ≥ 0 with r′ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' + r′ 7 = r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The terms in the sum are constant and their number is �r′+7−1 r′ � , which is smaller than 2r′+6 by the binomial theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 there are constants C′ and C′′ such that for all δ and δ′, ηδ∪δ′ ≤ C′|δ|+|δ′| √π Γ �ℓ(δ) + ℓ(δ′) + 1 2 � ≤ C′|δ|+|δ′|C′′ℓ(δ)+ℓ(δ′) √π Γ �ℓ(δ) + 1 2 � Γ �ℓ(δ′) + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We may assume that �C = C′C′′ > 1 and use the fact that ℓ(δ) ≤ |δ| and ℓ(δ′) ≤ |δ′| to obtain � δ′ ηδ∪δ′ ���[ur′xδ′]H(u, x) ��� ≤ �C|δ| √π Γ �ℓ(δ) + 1 2 � � ℓ′≥0 Γ �ℓ′ + 1 2 � � ℓ(δ′)=ℓ′ �C|δ′| ���[ur′xδ′]H(u, x) ��� for all r′ ≥ 0 and deranged partitions δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='28 this last expression is bounded by ≤ �C|δ| √π Γ �ℓ(δ) + 1 2 � � ℓ′≥0 Γ �ℓ′ + 1 2 � �C3r′[ur′xℓ′] exp \uf8eb \uf8ed � (r,δ)∈J urxℓ(δ) \uf8f6 \uf8f8 , for all r′ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The statement follows by estimating the sum over ℓ′ using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There exists a constant C such that for all n, r ≥ 0 with r ≤ 2n−2 and deranged partitions δ = [2m2 · · · ] with |δ| ≤ 2n − r, ���[u2n−rxδ] �T � u2e−W(u·¯x)���� � δ′ ηδ∪δ′ ���[urxδ′]H(u, x) ��� ≤ C1+r+|δ| Γ � m2 2 + 1 �Γ � n − 1 6(r + |δ| − 2m2) + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='22 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='23 we can find constants C1 and C2 such that for all 2n − r ≥ 2 and deranged partitions δ = [2m23m3 · · · ] with |δ| ≤ 2n − r, ���[u2n−rxδ] �T � u2e−W(u·¯x)���� � δ′ ηδ∪δ′ ���[urxδ′]H(u, x) ��� ≤ C1C1+r+|δ| 2 Γ � n − r+|δ|−2ℓ(δ)+1 2 � m2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='m3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Γ �ℓ(δ) + 5 6r + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 34 Recall that ℓ(δ) = �|δ| k=2 mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 there is a constant C3 such that Γ �ℓ(δ) + 5 6r + 1 2 � ≤ C 1+ ℓ(δ)+ 5 6 r 2 3 C1+m2+m3 3 Γ �m2 + 1 2 � Γ �m3 + 1 2 � Γ �ℓ(δ) − m2 − m3 + 5 6r + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By the duplication formula of the Γ function (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (24)), Γ � m+1 2 � /m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' = √π/(2mΓ(m/2 + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Combining all this and using Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9 to merge the Γ functions, we find that ���[u2n−rxδ] �T � u2e−W(u·¯x)���� � δ′ ηδ∪δ′ ���[urxδ′]H(u, x) ��� ≤ √π 3 C1C1+r+|δ| 2 C 2+m2+m3+ ℓ(δ)+ 5 6 r 2 3 2m2+m3 Γ � n − 1 6 r+|δ|+m2+m3−3ℓ(δ)+1 2 � Γ (m2/2 + 1) Γ (m3/2 + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Moreover, we can use Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9 to get the bound, Γ � n − 1 6r + |δ| + m2 + m3 − 3ℓ(δ) + 1 2 � ≤ √π Γ � n − 1 6 (r+|δ|−2m2)+1 2 � Γ � 5 6 |δ|+ 4 3 m2+m3−3ℓ(δ)+1 2 �, where the denominator is bounded form below as 5 6|δ| + 4 3m2 + m3 − 3ℓ(δ) ≥ 0 for all deranged partitions δ, which follows from |δ| = �|δ| k=2 kmk, and Γ(x) does not vanish for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For given n ≥ 1 and s ≤ 2n, let Bn,s = � ηδ∪δ′ � [u2n−rxδ] �T � u2e−W(u·¯x)�� � [urxδ′]H(u, x) � , where the sum is over all integers r and all pairs of deranged partitions (δ, δ′) with 0 ≤ r ≤ 2n−2, δ = [2m23m3 · · · ] and the restriction that r + �|δ| k=3 kmk = r + |δ| − 2m2 = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We have 2n � s=1 Bn,s ∈ O � Γ � n − 7 12 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='36 there exists a constant C such that |Bn,s| ≤ � C1+r+|δ| Γ � m2 2 + 1 �Γ � n − 1 6s + 1 2 � , where the sum runs over all integers r and deranged partitions δ with 0 ≤ r ≤ 2n − 2 and r + �|δ| k=3 kmk = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This suggests to treat the parts of size 2 of the partition δ separately: We also have the bound |Bn,s| ≤ � r≥0,δ(3) r+|δ(3)|=s � m2≥0 C1+r+|δ(3)|+2m2 Γ � m2 2 + 1 � Γ � n − 1 6s + 1 2 � , where we sum over pairs (r, δ(3)) where r is an integer ≥ 0 and δ(3) is an integer partition where each part has size at least 3 with r + |δ(3)| = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' There are at most as many such pairs (r, δ(3)) as there are integer partitions of s and, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='29, there are fewer than 2s integer partitions of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore, |Bn,s| ≤ 2s � m2≥0 C1+s+2m2 Γ � m2 2 + 1 �Γ � n − 1 6s + 1 2 � = 2sC′C1+sΓ � n − 1 6s + 1 2 � , THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 35 where C′ = � m2≥0 C2m2 Γ( m2 2 +1), which is obviously convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9, we have for all 2n ≥ s ≥ 1, Γ � n − 1 6 s+1 2 � ≤ √πΓ � n − 7 12 � /Γ � 1 6 (s−1)+1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence 2n � s=1 |Bn,s| ≤ √π 2n � s=1 2sC′C1+s Γ � n − 7 12 � Γ � 1 6 (s−1)+1 2 � ≤ √πCC′Γ � n − 7 12 � ∞ � s=1 2sCs Γ � 1 6 (s−1)+1 2 �, where the sum over s is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7 we have �en = 2n � r=0 � δ,δ′ ηδ∪δ′ � [u2n−rxδ] �T � u2e−W(u·¯x)�� � [urxδ′]H(u, x) � where we sum over all pairs of deranged partitions δ and δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='21, the expression [u2n−rxδ] �T � u2e−W(u·¯x)� vanishes if r = 2n − 1 or if r = 2n and δ ̸= ∅, while [u1x∅] �T � u2e−W(u·¯x)� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore 2n � r=2n−1 � δ,δ′ ηδ∪δ′ � [u2n−rxδ] �T � u2e−W(u·¯x)�� � [urxδ′]H(u, x) � = � δ′ ηδ′[u2nxδ′]H(u, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='23, the right hand side is bounded by C1+2nΓ � 5 6n + 1 2 � ⊂ O � Γ � n − 7 12 �� , so �en = 2n−2 � r=0 � δ,δ′ ηδ∪δ′ � [u2n−rxδ] �T � u2e−W(u · ¯x) �� � [urxδ′]H(u, x) � + O � Γ � n − 7 12 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (26) Using the notation and statement of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='37, this becomes �en = 2n � s=0 Bn,s + O � Γ � n − 7 12 �� = Bn,0 + O � Γ � n − 7 12 �� where Bn,0 is given by the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (26) restricted to summands where r + |δ| − 2m2 = r+� k≥3 kmk(δ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The sum over r therefore trivializes and we only have to account for r = 0 and the sum over deranged partitions reduces to a sum over partitions that only have parts of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We may write this as, � m2≥0 � δ′ η[2m2]∪δ′ � [u2nxm2 2 x0 3x0 4 · · · ] �T � u2e−W(u·¯x)�� � [u0xδ′]H(u, x) � = � m2≥0 η[2m2 ][u2nxm2 2 x0 3x0 4 · · · ] �T � u2e−W(u·¯x)� , where we used the fact that [u0x∅]H(u, x) = 1 and [u0xδ′]H(u, x) = 0 if δ′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='21, [u2nxm2 2 x0 3x0 4 · · · ] �T � u2e−W(u·¯x)� = �χ 2n−2m2 2 �(2n − 2m2) µ(2) 4 m2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using µ(2) = −1 and η[2m2 ] = η2,m2 gives the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The odd forested graph complex As remarked in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3, the Euler characteristic e(Out(Fn)) is equal to the Euler char- acteristic of Kontsevich’s Lie graph complex, which is equal to the Euler characteristic of the forested graph complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In [22] Kontsevich defined an odd version1 of the Lie graph complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The two graph complexes differ only by the definition of the orientation of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that the (even) forested graph complex is generated by all even forested graphs (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3), which are forested graphs with no automorphisms α that induce an odd permutation αΦ : EΦ → EΦ on the forest edges EΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In other words, all automorphisms α of an even forested graph satisfy sign(αΦ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Every automorphism α of a graph G also induces an automorphism αH : H1(G, Z) → H1(G, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An odd forested graph is a forested graph all of whose automorphisms α satisfy sign(αΦ) det(αH) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The complex spanned by such odd forested graphs computes H∗(Out(Fn+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �Q), where �Q is the representation obtained by composing the canonical group homomorphism Out(Fn) → GLn(Z) with the determinant map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The techniques developed in this paper can be used almost verbatim to compute the associated Euler characteristic eodd(Out(Fn)) = � k(−1)kHk(Out(Fn), �Q), which is equal to the Euler characteristic of Kontsevich’s odd Lie graph complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 we define � [G,Φ] (−1)e(Φ) = eodd(Out(Fn+1)), where we sum over all isomorphism classes of connected odd forested graphs [G, Φ] of Euler characteristic χ(G) = −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As we did in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 we also define �eodd n = � [G,Φ] (−1)e(Φ), (27) where we sum over all isomorphism classes of possibly disconnected odd forested graphs of Euler characteristic χ(G) = −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5 works for odd forested graphs as well as for even ones, giving � n≥0 �eodd n ℏn = ∞ � n=1 � 1 1 − ℏn �eodd(Out(Fn+1)) and we can compute eodd(Out(Fn+1)) from the coefficients �eodd n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Recall that for a forested graph (G, Φ) and an automorphism α ∈ Aut(G, Φ), the number eα(Φ) denotes the number of cycles of the permutation on the forest edges, αΦ : EΦ → EΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For a forested graph (G, Φ), the sum � α∈Aut(G,Φ) det(αH)(−1)eα(Φ) is equal to (−1)e(Φ)|Aut(G, Φ)| if (G, Φ) is odd or 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The argument for Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4 applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Each automorphism α of a forested graph (G, Φ) also provides us with a permutation αEG\\Φ : EG \\ EΦ → EG \\ EΦ that permutes the non-forest edges, a set of permutations (αe)e∈EG\\EΦ of order 1 or 2 that might change the orientation of each non-forest edge, and a permutation αH0(Φ) that permutes the connected components of the forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 1In [28], Willwacher used the opposite notions of “even” and “odd” orientation of a graph, so that Kontsevich’s “odd” graph complexes are Willwacher’s “even” graph complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 37 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' For a connected forested graph (G, Φ) and an automorphism α : (G, Φ) → (G, Φ), det(αH) = sign(αH0(Φ))sign(αEG\\EΦ) � e∈EG\\EΦ sign(αe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Let ZE and ZV be the Z-vector spaces generated by the edge and vertex set of (G, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We have the exact sequence 0 → H1(G, Z) → ZE → ZV → H0(G, Z) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The spaces ZE and ZV come with a natural bilinear form, hence we can dualize the usual boundary operator ∂1 : ZE → ZV to ∂∗ 1 : ZV → ZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We get the isomorphism ZV ⊕H1(G, Z) → ZE⊕H0(G, Z) given by (v, c) �→ (c+∂∗ 1v, ∂0v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' An automorphism of G also gives automorphisms on all the vector spaces in this discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As determinant are multiplicative under direct sums, det(αZV ⊕H1(G,Z)) = det(αZV ) det(αH1(G,Z)) = det(αZE) det(αH0(G,Z)) = det(αZE⊕H0(G,Z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Since G is connected H0(G, Z) is one-dimensional and α cannot change its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore, det(αH0(G,Z)) = 1 and det(αZV ) det(αH1(G,Z)) = det(αZE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The vector space ZE can be decom- posed ZE = ZEΦ ⊕ Z(EG \\ EΦ) and α acts block-wise on both summands as it does not mix for- est and non-forested edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' It follows that det(αZV ) det(αH1(G,Z)) = det(αZEΦ) det(αZ(EG\\EΦ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' From Φ, we get the short exact sequence, 0 → ZEΦ → ZV → H0(Φ, Z) → 0, where we used the fact that a forest has no first homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Consequently, det(αZV ) = det(αZEΦ) det(αH0(Φ,Z)) and det(αZEΦ) det(αH0(Φ,Z)) det(αH1(G,Z)) = det(αZEΦ) det(αZ(EG\\EΦ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Ordering and directing the edges EG \\ EΦ gives a basis of Z(EG \\ EΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The orientation can be changed by switching two edges or reversing the direction of one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Hence, det(αZEΦ) = sign(αEG\\EΦ) � e∈EG\\EΦ sign(αe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Fixing an ordering of the connected components of Φ gives a basis of H0(Φ, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Therefore, det(αH0(Φ,Z)) is equal to the sign of the permutation that α induces on the components of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The fact that det X = ±1 for all X ∈ GLn(Z) gives the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Combining Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (27) results in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' �eodd n = � [G,Φ] 1 |Aut(G, Φ)| � α∈Aut(G,Φ) sign(αH0(Φ))sign(αEG\\EΦ) \uf8eb \uf8ed � e∈EG\\EΦ sign(αe) \uf8f6 \uf8f8 (−1)eα(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Here, we sum over all forested graphs [G, Φ] of Euler characteristic χ(G) = −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This statement is the odd version of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As in Section 3 we can produce a formula for �eodd by counting forests and matchings separately before combining both expressions to give a counting formula for forested graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In the odd case, we have to change a couple of signs in the derivation to accommodate the additional sign factors in the statement above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We define an odd version of our forest generating function F in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (15) using the same notation and by weighting each odd permutation of the connected components of the forest with a sign: Fodd(u, x) = � s≥0 1 s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (Φ,γ)∈AF[s] sign(γH0(Φ))(−1)eγ(Φ)xγus−2k(Φ) (28) where sign(γH0(Φ)) is the sign of the permutation induced by γ on the connected components of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' This factor will account for the sign(αH0(Φ)) term in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using the same argument as for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10, but accounting for a sign flip for each even cycle on the set of components or equivalently, by setting yk = (−1)k+1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' (14), results in THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 38 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Fodd(u, x) = exp \uf8eb \uf8ed� k≥1 (−1)k+1u−2k V((u · x)[k]) k \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' To account for the factor sign(αEG\\EΦ) �� e∈EG\\EΦ sign(αe) � in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3, we have to use a signed version of the matchings that we introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A permutation α ∈ S2n and a fixed-point free involution on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , 2n} such that α ◦ ι = ι ◦ α give rise to a permutation αι of the orbits of ι and a set of n permutations αe1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' , αen that permute the elements of each individual orbit of ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' We define ηodd α = � ι◦α=α◦ι sign(αι) n � k=1 sign(αek), where we sum over all such pairs ι and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Using these definitions with the argument for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3, we get Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � λ ηodd λ [u2nxλ]Fodd(u, x) = �eodd n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Following the argument in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3 for the derivation of a formula for ηλ, the cycle index series for the signed version of a matching of two points is given by Eodd(x) = 1 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � (φ,α)∈AE[2] sign(α)xα = 1 2 � x2 1 − x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3, we can use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 to get a generating function for the numbers ηodd α : Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � n≥0 1 (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' � α∈S2n ηodd α = exp \uf8eb \uf8ed� k≥1 (−1)k+1 2k � x2 k − x2k � \uf8f6 \uf8f8 , Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Set yk = (−1)k+1 after applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='2 as each even cycle of the permutation induced on the components is counted with a minus sign this way and the sign of a permutation is equal to (−1)# of even cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ Repeating the computation for Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5 while accounting for the changed signs we find, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' If λ = [1m12m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' nmn], then ηodd λ = n � k=1 ηodd k,mk, where ηodd k,2s = � ks(2s − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if k is odd �s r=0(−1)r�2s 2r � kr(2r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if k is even and ηodd k,2s+1 = � 0 if k is odd − �s r=0(−1)r�2s+1 2r � kr(2r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' if k is even .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' THE EULER CHARACTERISTIC OF THE MODULI SPACE OF GRAPHS 39 Via exactly the same procedure described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12, but substituting the odd versions of the respective series, we get an effective algorithm for computing the numbers eodd(Out(Fn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The first few values are listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The analytic argument in Section 4 also works in the odd case, as the relevant coefficients of F and Fodd agree, the values of ηα and ηodd α are equal for the trivial permutation and |ηodd α | ≤ ηα for all α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The modified signs only have a nontrivial consequence in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Instead of the statement of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13 we find that in the odd case n � m=0 (−1)m 2mm!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' ηodd 2,m = e 1 4 + O � 1 Ln � , after repeating the computation using the numbers from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The remaining proof is completely equivalent up to the substitution of the relevant number e− 1 4 → e 1 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Following through the argument again results in the odd version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The Euler characteristic eodd(Out(Fn)) has the leading asymptotic behaviour eodd(Out(Fn)) ∼ −e 1 4 �n e �n 1 (n log n)2 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Use Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='8, [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' A] and Stirling’s formula (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' □ References [1] Emil Artin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The Gamma Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Holt, Rinehart and Winston, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [2] Laurent Bartholdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' The rational homology of the outer automorphism group of F7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' New York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', 22:191–197, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [3] François Bergeron, Gilbert Labelle, and Pierre Leroux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Combinatorial species and tree-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Cam- bridge University Press, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [4] Marko Berghoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Feynman amplitudes on moduli spaces of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Poincare D Comb.' metadata={'source': 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+page_content=' Feynman diagrams and low-dimensional topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' In First European Congress of Math- ematics Paris, July 6–10, 1992, pages 97–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Springer, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [24] Sava Krstić and Karen Vogtmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Equivariant outer space and automorphisms of free-by-finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', 68(2):216–262, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [25] Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Integral Euler characteristic of Out(F11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Experi- mental Mathematics, 24(1):93–97, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [26] George Pólya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Kombinatorische anzahlbestimmungen für gruppen, graphen und chemische verbindungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Acta mathematica, 68:145–254, 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [27] Jos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Vermaseren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' New features of FORM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' preprint arXiv:math-ph/0010025, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' [28] Thomas Willwacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Kontsevich’s graph complex and the Grothendieck-Teichmüller Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=', 200(3):671–760, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Table of χ(Out(Fn)), e(Out(Fn)) and eodd(Out(Fn)) for n ≤ 15 n χ(Out(Fn)) e(Out(Fn)) eodd(Out(Fn)) 2 − 1 24 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='042 1 0 3 − 1 48 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='021 1 0 4 − 161 5760 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='028 2 −1 5 − 367 5760 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='064 1 0 6 − 120257 580608 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='21 2 −1 7 − 39793 45360 ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='88 1 −1 8 − 6389072441 1393459200 ≈ −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 993607187 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='34836480 ≈ −29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−38 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 5048071877071 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24524881920 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='≈ −206 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−124 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−275 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 9718190078959 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5748019200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='≈ −1691 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−1202 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−2224 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 375393773534736899347 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='24103053950976000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='≈ −15575 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−10738 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−20358 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 2495397080915203519 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='15692092416000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='≈ −159023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−112901 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−207321 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 1031156416543036906701911 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='578473294823424000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='≈ −1782548 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−1271148 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−2320136 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='− 6147011108414481406421 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='282457663488000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='≈ −21762593 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−15668391 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='−28287408 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='The numbers χ(Out(Fn)) were computed using [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='5], the numbers e(Out(Fn)) were computed as described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content='12 and the numbers eodd(Out(Fn)) similarly as dis- cussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' See [9] for the technical computer programming details of these computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' Michael Borinsky, ETH Zürich, Institute for Theoretical Studies, Clausiusstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} +page_content=' 47, 8092 Zürich, Switzerland Karen Vogtmann, University of Warwick, Mathematics Institute, Zeeman Building, Coventry CV4 7AL, United Kingdom' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQfLvvB/content/2301.01121v1.pdf'} diff --git a/ddFKT4oBgHgl3EQfqi7m/content/tmp_files/2301.11875v1.pdf.txt b/ddFKT4oBgHgl3EQfqi7m/content/tmp_files/2301.11875v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a56296745ee3aa8812e00270bc3f1f131b64e1d8 --- /dev/null +++ b/ddFKT4oBgHgl3EQfqi7m/content/tmp_files/2301.11875v1.pdf.txt @@ -0,0 +1,2865 @@ +Prepared for submission to JHEP +DESY 23-009 +UWThPh-2023-2 +Refining the GENEVA method for Higgs boson +production via gluon fusion +Simone Alioli,a Georgios Billis,a Alessandro Broggio,a,b Alessandro Gavardi,a,c Stefan +Kallweit,a Matthew A. Lim,c,d Giulia Marinelli,a Riccardo Nagara and Davide +Napoletanoa +aUniversit`a degli Studi di Milano-Bicocca & INFN, Piazza della Scienza 3, Milano 20126, Italy +bFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria +cDeutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany +dDepartment of Physics and Astronomy, University of Sussex, Sussex House, Brighton, BN1 9RH, +UK +E-mail: simone.alioli@unimib.it, georgios.billis@unimib.it, +alessandro.broggio@univie.ac.at, alessandro.gavardi@desy.de, +stefan.kallweit@unimib.it, m.a.lim@sussex.ac.uk, +g.marinelli10@campus.unimib.it, riccardo.nagar@unimib.it, +davide.napoletano@unimib.it +Abstract: We describe a number of improvements to the Geneva method for matching +NNLO calculations to parton shower programs. In particular, we detail changes to the +resummed calculation used in the matching procedure, including disentangling the cross +section dependence on factorisation and beam scales, and an improved treatment of timelike +logarithms. We also discuss modifications in the implementation of the splitting functions +which serve to make the resummed calculation differential in the higher multiplicity phase +space. These changes improve the stability of the numerical cancellation of the nonsingular +term at small values of the resolution parameter. As a case study, we consider the gluon- +initiated Higgs boson production process gg → H. We validate the NNLO accuracy of our +predictions against independent calculations, and compare our showered and hadronised +results with recent data taken at the ATLAS and CMS experiments in the diphoton decay +channel, finding good agreement. +arXiv:2301.11875v1 [hep-ph] 27 Jan 2023 + +Contents +1 +Introduction +1 +2 +Theoretical framework +3 +2.1 +The GENEVA method +3 +2.2 +Higgs boson production via gluon fusion +6 +2.3 +T0 resummation +7 +3 +Novel features of the GENEVA method +9 +3.1 +Improved treatment of splitting functions +9 +3.1.1 +Infrared limits +11 +3.1.2 +Soft limit of the 0 → 1 splitting +12 +3.1.3 +Numerical validation +13 +3.2 +Independent scale variations +15 +3.2.1 +Exposing the µF dependence of the singular cross section +16 +3.2.2 +Choice of the factorisation scale +17 +3.3 +Treatment of timelike logarithms +18 +4 +Validation of the gg → H process +20 +4.1 +Partonic results at NNLO +21 +4.2 +Interface with PYTHIA8 +21 +5 +Comparison with LHC data +24 +6 +Conclusions +29 +1 +Introduction +In recent years, the quest for precision at the Large Hadron Collider (LHC) has seen many +impressive milestones in the development of theoretical tools used to describe hadronic +collisions. Many processes are currently known at next-to-next-to-leading order (NNLO) +in perturbative QCD, and several 2 → 1 processes even at one order higher (N3LO). One +particular direction in which much fruitful progress has been made is in the matching +of higher order perturbative calculations to parton shower (PS) programs, resulting in +Monte Carlo event generators which combine the advantages of fixed-order calculations +with the flexibility of parton shower tools. This paradigm generally goes under the name +of NNLO+PS. +Several different methods which reach NNLO+PS accuracy have been proposed [1–7], +with most applications using a resummed calculation – either directly or via the Sudakov +factor in a shower Monte Carlo – in a suitable resolution variable alongside the fixed order to +– 1 – + +achieve the matching. Of these methods, the Geneva approach [4, 8] has the advantage of +being particularly flexible with regard to the framework used for the resummed calculation +and the choice of the resolution variable, while also exploiting the possibility of reaching +higher logarithmic accuracies in both direct QCD and soft-collinear effective theory (SCET) +formalisms. This has resulted in the application of the method to a number of colour singlet +processes [4, 9–15], as well as first steps towards implementations involving coloured final +states [16]. +In this work, we describe a number of improvements to the Geneva event generator, +which both extend the capabilities of the program and improve its numerical performance. +Specifically, we detail a new treatment of the splitting functions, which were first introduced +in the original Geneva implementation [8] and serve to make the resummed calculation +used in the matching procedure differential in the higher multiplicity phase space. The new +approach significantly increases the performance of the code in extreme soft and collinear +regions, where the cancellation of large logarithmic terms is extremely delicate. We also +implement a more rigorous treatment of the theoretical uncertainties by disentangling +the factorisation and renormalisation scale dependences in the cross section and allowing +their independent variations. This puts our uncertainty estimation on a robust and more +conservative theoretical footing, and will also prove important for the implementation of +processes featuring perturbatively generated heavy flavours in the initial state. Finally, +we also discuss the treatment of timelike logarithms in our calculation, the inclusion of +which has been shown to improve the perturbative convergence of colour singlet production +processes such as gg → H [17–19]. +In order to study the various improvements to the Geneva program, we have imple- +mented the gluon-initiated Higgs boson production process (gg → H). We use a resummed +calculation in the zero-jettiness resolution variable T0 obtained via SCET up to N3LL ac- +curacy. The process is interesting from both an experimental and a theoretical perspective +for a number of reasons. +Experimentally, the gluon-fusion production channel was of utmost importance for the +discovery of the Higgs boson [20, 21]. Nowadays Higgs physics remains a crucial aspect of +the LHC programme [22–34], and constraining the scalar boson’s properties and couplings +to probe the nature of the Higgs sector is a priority for Run 3 of LHC and beyond [35]. +From the theory side, many calculations work in the limit in which the top-quark mass +is considered to be large compared to other scales present in the process, the so-called +heavy-top limit (HTL). This significantly simplifies the computational complexity since +the top-quark loop coupling the Higgs boson to gluons is integrated out, resulting in an +effective ggH vertex and further effective vertices with more gluons and Higgs bosons. +Consequently, calculations including QCD corrections up to N3LO are now available in +this limit [36–43], including matching to resummed calculations up to N3LL′ accuracy in +transverse momentum [43–51] and in jet veto observables [52–55]. There has also been +a considerable amount of work on improving calculations beyond the HTL by including +quark mass effects [56–60]; this has culminated in a calculation of the exact top-quark mass +dependence at NNLO in QCD [61, 62]. Additionally, the fact that the perturbative series is +known to be poorly convergent has motivated the study of alternative scale choices which +– 2 – + +include π2 terms arising from kinematic logarithms at all orders. Finally, the simplicity of +this process in terms of its kinematics and matrix elements makes it a particularly suitable +testing ground for the improvements which we will detail in this work. +The rest of the paper is organised as follows. In sec. 2, we provide a brief recap of the +Geneva method and its application to gluon-induced Higgs production, before discussing +the new features which have been implemented in the program in sec. 3. In sec. 4 we +validate the NNLO accuracy of our calculation for the gg → H process and discuss the +matching to the parton shower provided by Pythia8 [63]. Finally, we compare our results +with the pp → H → γγ data collected at the ATLAS and CMS experiments [64, 65] in +sec. 5. We give our conclusions in sec. 6. +2 +Theoretical framework +In the following we lay out the theoretical framework we work in. We start by giving a +summary of the Geneva event generator formalism, which includes the matching procedure +of the fixed-order calculation to the resummed prediction. We then focus on the definition +of the process under study, i.e. Higgs boson production via gluon fusion, and on its zero- +jettiness resummation. +2.1 +The GENEVA method +The complete derivation of the Geneva method has been presented extensively in several +publications, e.g. in refs. [4, 9]. Here, we explicitly refrain from entering into the finer +details of the method, and we only briefly recall the general formulae, highlighting some +key features that are important for this process. +We use N-jettiness [66] to resolve the QCD emissions that can be associated with each +event produced by Geneva: T0 as the zero-jet resolution parameter, and T1 to separate +between one or more emissions. The partonic event space is then divided into three regions: +Φ0 for events with no extra emissions, Φ1 for one-jet events, and Φ2 for the remaining events +with two jets in the final state. These phase space regions are defined via two thresholds, +T cut +0 +and T cut +1 +. +The differential cross section for the production of events with no extra emissions is +given by +dσmc +0 +dΦ0 +(T cut +0 +) = dσNNLL′ +dΦ0 +(T cut +0 +) − dσNNLL′ +dΦ0 +(T cut +0 +) +���� +NNLO0 ++ (B0 + V0 + W0)(Φ0) + +� dΦ1 +dΦ0 +(B1 + V1)(Φ1) θ +� +T0(Φ1) < T cut +0 +� ++ +� dΦ2 +dΦ0 +B2(Φ2) θ +� +T0(Φ2) < T cut +0 +� +. +(2.1) +Here we use the primed counting for the resummation order as in e.g. ref. [52]. For the +– 3 – + +case of a single extra emission we have two contributions: that above T cut +0 +dσmc +1 +dΦ1 +(T0 > T cut +0 +; T cut +1 +) = +�� +dσNNLL′ +dΦ0dT0 +− dσNNLL′ +dΦ0dT0 +���� +NLO1 +� +P(Φ1) + (B1 + V C +1 )(Φ1) +� +× U1(Φ1, T cut +1 +) θ(T0 > T cut +0 +) ++ +� +� dΦ2 +dΦT +1 +B2(Φ2) θ +� +T0(Φ2) > T cut +0 +� +θ(T1 < T cut +1 +) +− dΦ2 +dΦC +1 +C2(Φ2) θ(T0 > T cut +0 +) +� +− B1(Φ1) U (1) +1 (Φ1, T cut +1 +) θ(T0 > T cut +0 +) , +(2.2) +and the nonsingular below T cut +0 +, arising from non-projectable configurations, +dσmc +1 +dΦ1 +(T0 ≤ T cut +0 +; T cut +1 +) = (B1 + V1)(Φ1) Θ +FKS +map(Φ1) θ(T0 < T cut +0 +) . +(2.3) +Similarly the case of two extra emissions also receives two contributions, +dσmc +≥2 +dΦ2 +(T0 > T cut +0 +, T1 > T cut +1 +) = +��dσNNLL′ +dΦ0dT0 +− dσNNLL′ +dΦ0dT0 +���� +NLO1 +� +P(�Φ1) ++ (B1 + V C +1 )(�Φ1) +� +U ′ +1(�Φ1, T1) θ(T0 > T cut +0 +) +����Φ1=ΦT +1 (Φ2) +× P(Φ2) θ(T1 > T cut +1 +) ++ +� +B2(Φ2) θ(T1 > T cut +1 +) − B1(ΦT +1 ) U (1)′ +1 +��Φ1, T1 +� +× P(Φ2) Θ(T1 > T cut +1 +) +� +θ +� +T0(Φ2) > T cut +0 +� +, +(2.4) +and +dσmc +≥2 +dΦ2 +(T0 > T cut +0 +, T1 ≤ T cut +1 +) = B2(Φ2) Θ +T +map(Φ2) θ(T1 < T cut +1 +) θ +� +T0(Φ2) > T cut +0 +� +, +(2.5) +above and below T cut +1 +, respectively. +In the formulae above, Bn, Vn and Wn are the 0-, 1- and 2-loop matrix elements for +n QCD partons in the final state (including parton densities); analogously, we denote by +NkLOn a quantity with n additional partons in the final state computed at NkLO accuracy. +Since it is necessary to evaluate the resummed and resummed-expanded terms on phase +space points resulting from a projection from a higher to a lower multiplicity, we introduce +a shorthand for such projected phase space points, �ΦN. We use the abbreviation +dΦM +dΦO +N += dΦM δ[�ΦN − ΦO +N(ΦM)] ΘO(ΦM) +(2.6) +to indicate an integration over the portion of the ΦM phase space which can be reached +from a ΦN point while keeping some observable O also fixed, with N < M. The ΘO(ΦM) +term additionally limits the integration to the phase space points belonging to the singular +– 4 – + +contribution for the given observable O. For example, when generating 1-body events we +use +dΦ2 +dΦT +1 +≡ dΦ2 δ[�Φ1 − ΦT +1 (Φ2)] ΘT (Φ2) , +(2.7) +where the 1 → 2 mapping has been constructed to preserve T0, i.e. +T0(ΦT +1 (Φ2)) = T0(Φ2) , +(2.8) +and ΘT (Φ2) guarantees that the Φ2 point is reached from a genuine QCD splitting of +the Φ1 point. The use of a T0-preserving mapping is necessary to ensure that the point- +wise singular T0 dependence is alike among all terms in eqs. (2.2) and (2.4) and that the +cancellation of said singular terms is guaranteed on an event-by-event basis. +The non-projectable regions of Φ1 and Φ2, on the other hand, are assigned to the cross +sections in eqs. (2.3) and (2.5). These events are entirely nonsingular in nature. We denote +the constraints due to the choice of map by Θmap, using the FKS map [67] for the Φ1 → �Φ0 +projection and, as mentioned above, a T0-preserving map for the Φ2 → �Φ1 projection. +The term V C +1 denotes the contributions of soft and collinear origins in a standard NLO +local subtraction, +V C +1 (Φ1) = V1(Φ1) + +� +dΦ2 +dΦC +1 +C2(Φ2) , +(2.9) +with C2 a singular approximant of B2; in practice we use the subtraction counterterms +which we integrate over the radiation variables dΦ2/dΦC +1 using the singular limit C of the +phase space mapping. +In the formulae involving one or two extra emissions, U1 is a next-to-leading-logarithmic +(NLL) Sudakov factor which resums large logarithms of T1, and U ′ +1 its derivative with +respect to T1; the O(αs) expansions of these quantities are denoted by U (1) +1 +and U (1)′ +1 +respectively. +We extend the differential dependence of the resummed terms from the N-jet to the +(N +1)-jet phase space using a normalised splitting probability P(ΦN+1) which satisfies +� +dΦN+1 +dΦNdTN +P(ΦN+1) = 1 . +(2.10) +The two extra variables are chosen to be an energy ratio z and an azimuthal angle φ. The +functional forms of the P(ΦN+1) are in principle only constrained by eq. (2.10). However, +in order to correctly model the soft-collinear limit behaviour, we find it useful to write them +in terms of the Altarelli-Parisi splitting kernels, weighted by parton distribution functions +(PDFs). +In previous implementations of the Geneva method, the splitting functions P(ΦN+1) +were computed using a “hit-or-miss” method based on precomputed upper bounds, which +did not require knowledge of an analytic expression for the integration limits of z and φ. +At the same time, however, this introduced some numerical instabilities. In this work, +we improve on this situation by including the exact integration limits and evaluate the +splitting functions directly for each phase space point, as detailed in sec. 3.1. +– 5 – + +2.2 +Higgs boson production via gluon fusion +We consider the production of a stable Higgs boson via the gluon fusion channel in proton- +proton scattering, pp → H + X, where X denotes any additional hadronic radiation in the +final state. At leading order (LO) in the strong coupling this results in a single contribution +gg → H at partonic level [68], while at next-to-leading order (NLO) (anti)quark-initiated +channels also start to contribute [69–71]. +For a stable Higgs boson it is phenomenologically reasonable to work in the HTL ef- +fective field theory (EFT), in which the contributions from the top-quark loops coupling +the Higgs boson to gluons have been integrated out. This EFT supplements the Standard +Model (SM) vertices with additional, effective couplings between gluons and Higgs bosons. +Introducing these effective vertices has the advantage of reducing the complexity of the +matrix element computations. The cross section dependence on the top-quark mass mt +can be partially restored by rescaling the HTL result by a factor equal to the ratio be- +tween the LO mt-exact result and that obtained in pure EFT. This is later referred to as +rescaled EFT (rEFT), and reproduces the exact mt dependence of the LO cross section +by construction. It is known to be a good approximation, for inclusive quantities, at least +up to NNLO [62]. The resulting approximation can instead be problematic for differential +distributions, for instance the transverse momentum of the Higgs boson when the accom- +panying radiation resolves the top-quark loop, i.e. when its transverse momentum is larger +than mt. For the case of a finite top-quark mass, the NNLO corrections have been recently +calculated for the inclusive cross section [61, 62], and those at NLO for Higgs boson pro- +duction in association with up to two hard jets [72, 73]. At this level of precision, however, +one also needs to take into account the interference between contributions including both +massive top and bottom quarks, which is known at NLO for the Higgs plus jet case [74, 75]. +Since the problem of including the quark mass effects for precise phenomenological studies +is largely independent of the matching of fixed-order and resummed calculations to par- +ton showers in the Geneva method, which is the topic of the present study, we leave the +investigation of these effects to future work. +In this work, the Higgs boson is always produced on shell with a mass mH =125.09 GeV. +When comparing with data in the fiducial regions of the ATLAS or CMS experiments, we +will consider Higgs boson decays. In this case we work in the narrow-width approximation, +which for the Higgs boson is particularly accurate since ΓH/mH ∼ O(10−5). The Higgs +decay products can always be added a posteriori due to the scalar nature of the boson, +which implies that they are isotropically distributed without spin correlations with the +initial state. For the rest of this work we will consider a collider energy of +√ +S = 13 TeV +and assume the following values for the SM parameters affecting our calculations: +GF = 1.16639 × 10−5, +mt = 173.1 GeV . +(2.11) +For the matrix elements in the HTL approximation we use the heftpphj and heftpphjj +libraries of OpenLoops2 [76–78], which we then rescale by the rEFT factor rEFT = +1.06545. +– 6 – + +2.3 +T0 resummation +The formulae presented in sec. 2.1 require the evaluation of the resummed spectrum and +cumulant in the resolution variable T0 up to at least NNLL′ accuracy. +Although the +Geneva method does not depend on any particular resummation formalism, in practice we +often find it convenient to exploit results derived via SCET [79–85]. Within this framework, +a factorisation theorem for the zero-jettiness was first derived in refs. [86, 87] for colour +singlet production. In the case of the gluon-fusion channel for Higgs production it reads +dσSCET +dΦ0dT0 += Hgg→H(Q2, µ) +� +dtadtb Bg(ta, xa, µ) Bg(tb, xb, µ) Sgg +� +T0 − ta + tb +Q +, µ +� +, (2.12) +where Hgg→H, Sgg, and Bg are the hard, soft and beam functions, respectively. +The process-specific hard function Hgg→H(Q2, µ) is defined as the square of the Wilson +coefficient that results from matching the QCD Hamiltonian to the SCET operators, and +encodes information about the Born and virtual squared matrix elements. It depends only +on the Higgs boson virtuality Q2. In this section and whenever we consider Higgs boson +production specifically, we set Q = mH; elsewhere, we consider Q to be a generic hard +scale. +Given that we work in the HTL approximation, we perform a two-step matching pro- +cedure: we first integrate out the hard degrees of freedom above the top-quark mass, and +subsequently match the resulting EFT onto SCET. The final hard function then arises +from the product of two Wilson coefficients, the first from the HTL approximation and the +second from the matching to SCET; we evaluate both at the same scale µ. In principle, +within this approach one could resum ln(mt/mH) contributions by renormalisation group +equation (RGE) evolution. However, given the values of the top quark and Higgs boson +masses, these logarithms are never large and, consequently, we include them only at fixed +order in the hard function. Alternatively, if one wants to include the full top-quark mass +effects, a single-step matching can be performed as in e.g. ref. [52] at NNLL. Extending this +to NNLL′ accuracy requires the three-loop hard function with the exact top-quark mass +dependence [61, 62]. +The beam functions Bg(t, x, µ) are the inclusive gluon beam functions [86], which +depend on the transverse virtualities ta,b of the initial-state partons that participate in the +hard interaction and on their momentum fractions xa,b. While they are nonperturbative +objects, for t ≫ ΛQCD they admit an operator product expansion (OPE), +Bi(t, x, µ) = +� +j +� 1 +x +dξ +ξ Iij +� +t, x +ξ , µ +� +fj(ξ, µ) +� +1 + O +�Λ2 +QCD +t +�� +≡ +� +j +[Iij ⊗x fj] (t, x, µ) +� +1 + O +�Λ2 +QCD +t +�� +, +(2.13) +where the Iij(t, z, µ) are matching coefficients that describe the collinear virtual and real +initial-state radiation (ISR) and the fj(ξ, µ) are the usual PDFs. For later use, we denote +the Mellin convolution via the symbol ⊗x. +– 7 – + +Finally, Sgg(k, µ) is the gluon hemisphere soft function for beam thrust. Like the beam +functions, Sgg(k, µ) is a nonperturbative object and for k ≫ ΛQCD it also satisfies an OPE, +where the LO matching coefficient is calculable in perturbation theory. Its perturbative +component depends only on the colour representation of the hard partons, and therefore +the gluon case can be derived from that of the quark channel via Casimir scaling. In our +calculation we neglect the nonperturbative part of the soft function. We then rely on the +hadronisation model of the parton shower to provide the missing contribution. +The functions in eq. (2.12) are all evaluated at a common scale µ and satisfy RGEs. +The scale dependence in each of these functions involves potentially large logarithms of +ratios of disparate scales, which may impact their perturbative convergence. In order to +reduce the effect of these large logarithms, we evaluate each function at its characteristic +(canonical) scale, i.e. µS = T0, µH = mH, and µB = √µSµH. Since the cross section needs +to be evaluated at a common scale µ, we use the RGEs to evolve each function to µ. In +doing so, we resum said logarithms at all orders in perturbation theory. The resummed +formula for the T0 spectrum is then given by (see e.g. ref. [9] for more details) +dσresum +dΦ0dT0 += Hgg→H(Q2, µH) UH(µH, µ) +× +� +dta dtb [Bg(ta, xa, µB) ⊗ UB(µB, µ)] [Bg(tb, xb, µB) ⊗ UB(µB, µ)] +× +� +Sgg(T0 − ta + tb +Q +, µS) ⊗ US(µS, µ) +� +, +(2.14) +where we denote the standard convolutions between the different functions and the RGE +evolution factors via the ⊗ symbol. +In order to achieve NNLL′ accuracy in the T0 resummation, each of the hard, soft and +beam function boundary terms must be known at 2-loop order. For the beam function they +were calculated at 2-loops in ref. [88], and in fact they are known up to 3-loop order [89]. +Our implementation of the gluon beam function relies on an interface to scetlib [43, 90, +91], a library which provides ingredients for resummed calculations in SCET. The soft +function has been known at 2-loops for some time [92, 93], and recent work has aimed to +push this calculation to the 3-loop order [94–96]. The hard function has appeared several +times in the literature, see e.g. refs. [52, 97], and is known analytically with full top-quark +mass dependence at NNLO [61]. +In addition, the anomalous dimensions and the beta +function which enter the evolution factors and the fixed-order expansion of eq. (2.14) must +be known at 2-loop (noncusp [52]) and 3-loop (cusp [98–100], β(αs) [101, 102]) order. By +including them at one order higher [52, 103, 104], one can achieve resummation at N3LL. +The resummation of T0 for the case of Higgs boson production via gluon fusion has +already been studied in ref. [52] up to NNLL accuracy. In the present work, we extend this +calculation to NNLL′ and N3LL. For the determination of the canonical scales we employ +the T0-dependent profile functions described e.g. in sec. 3 of ref. [9] with {x0, x1, x2, x3} = +{1.5 GeV/mH, 0.2, 0.35, 0.5}. The use of such T0 dependent scales is known to cause a +difference between the integrated spectrum and the cumulant, which is formally of higher +order. This is a result of the noncommutativity of the scale setting and the integration +– 8 – + +steps. In previous Geneva implementations, this problem has been alleviated by explicitly +adding higher order terms to restore the cumulant cross section (see eq. (45) of ref. [9]). +This can be done either by using a ‘brute-force’ approach, in which the integrated spectrum +is simply replaced by the cumulant, or by smoothly transitioning from one to the other +as a function of T0. In all Geneva implementations thus far we have followed the latter +approach, which has the advantage of preserving the T0 spectrum in its peak region. +In the case of gg → H production at 13 TeV, the difference between the integrated +spectrum and the cumulant amounts to ∼ 18% of the total cross section. Given the size of +these corrections, we found the previously adopted solution to be insufficient to completely +solve the mismatch. In particular, our smooth fix modifies the T0 spectrum in the region +between ∼ 10 and ∼ 30 GeV by too large an amount, moving the central value of the +first outside the uncertainty bands of the second. We therefore revert to the brute-force +approach, and only require the preservation of the resummed cumulant cross section by +fixing κ(T0) = 1 (see eq. (45) of ref. [9]) such that the spectrum is exactly equal to the +derivative of the cumulant. +3 +Novel features of the GENEVA method +In this section we discuss the new improvements that have been incorporated in the Geneva +method. Here we focus on their impact on the gg → H process, however we note that they +can be straightforwardly generalised to several other processes (and indeed have already +been tested for Drell-Yan, double Higgs [15], and t¯t production [16]). +3.1 +Improved treatment of splitting functions +The N-jettiness spectra computed through resummation techniques cannot be directly used +for generating events with N + 1 final-state partons, since they do not carry a dependence +on the full ΦN+1 configurations, but only on TN and the projected ΦN configurations with +N final-state partons. For this reason, a splitting function P(ΦN+1) was introduced in +ref. [8] in order to make the resummed calculation fully differential in the higher order +phase space. +In general, the N → N + 1 splitting function P(ΦN+1) is defined such that for every +integrable function g(ΦN, TN) +� +dΦN+1 P(ΦN+1) g(ΦN, TN) = +� +dΦN dTN g(ΦN, TN) . +(3.1) +If the function g(ΦN, TN) is the TN spectrum, then multiplying it by the P functions makes +it differential over the dΦN+1 phase space without affecting the distributions of observables +that only depend on ΦN and TN. +In order to provide an explicit expression for P, we write the phase space of the ΦN+1 +configurations with a valid ΦN projection as the product of dΦN, dTN and the phase space +parametrised by two additional radiation variables z and φ. In this way the integral over +– 9 – + +the projectable ΦN+1 configurations at fixed ΦN and TN can be expressed as +� +dΦN+1 +dΦN dTN += +N+2 +� +k=1 +� zmax +k +(ΦN,TN) +zmin +k +(ΦN,TN) +dz Jk(ΦN, TN, z) +� φmax +k +(ΦN,TN,z) +φmin +k +(ΦN,TN,z) +dφ, +(3.2) +where the index k runs over the N + 2 possible emitter partons (mothers) of the ΦN +configurations. For each mother k and its associated mapping, we assume that the Jacobian +Jk(ΦN, TN, z) = +dΦN+1 +dΦN dTN dz dφ +���� +k +(3.3) +does not depend on φ. This is true for all the mappings considered in this paper. The +integral over the ΦN+1 configurations summed over the nreal partonic subprocesses with +N + 1 final-state partons for a generic function gβ(ΦN+1) can now be written as +nreal +� +β=1 +� +dΦN+1 gβ(ΦN+1) = +nreal +� +β=1 +� +unproj. ΦN+1 +dΦN+1 gβ(ΦN+1) + +(3.4) +nBorn +� +α=1 +� +dΦN dTN +N+2 +� +k=1 +� zmax +k +zmin +k +dz Jk(ΦN, TN, z) +� φmax +k +φmin +k +dφ +nsplit +k� +j=1 +gk→i+j +α +(ΦN, TN, z, φ) , +where nBorn is the number of subprocesses with N final-state partons, and nsplit +k +the number +of possible QCD splittings k → i+j, with i the emitted parton and j the sister. The function +gk→i+j +α +(ΦN, TN, z, φ) on the right-hand side is equal to gβ(ΦN+1) expressed in terms of the +underlying Born process index α and the splitting indices k and j. For ease of notation, +the full dependence of the z and φ integration limits on the phase space variables is not +shown. The unprojectable ΦN+1 configurations are those for which either the two closest +partons do not represent a valid QCD splitting, the ΦN configuration obtained from the +projection is not kinematically allowed, or the flavour configuration of the ΦN is invalid. +In order to fulfil the condition presented in eq. (3.1), we choose splitting functions +P(ΦN+1) that depend on the mother and sister indices and vanish in the unprojectable +ΦN+1 configurations: +P(ΦN+1) = +� +0 +if ΦN+1 is unprojectable, +Pkj(ΦN, TN, z, φ) +if ΦN → ΦN+1 via the k → i + j splitting. +(3.5) +The Pkj must then satisfy the equation +N+2 +� +k=1 +� zmax +k +(ΦN,TN) +zmin +k +(ΦN,TN) +dz Jk(ΦN, TN, z) +� φmax +k +(ΦN,TN,z) +φmin +k +(ΦN,TN,z) +dφ +nsplit +k� +j=1 +Pkj(ΦN, TN, z, φ) = 1 +(3.6) +– 10 – + +for all values of ΦN and TN. Without loss of generality, in the projectable ΦN+1 configu- +rations we can express them as +Pkj(ΦN, TN, z, φ) = +(3.7) +fkj(ΦN, TN, z, φ) +N+2 +� +k′=1 +� zmax +k′ +(ΦN,TN) +zmin +k′ (ΦN,TN) +dz′Jk′� +ΦN, TN, z′� � φmax +k′ (ΦN,TN,z′) +φmin +k′ (ΦN,TN,z′) +dφ′ +nsplit +k′ +� +j′=1 +fk′j′� +ΦN, TN, z′, φ′� +, +where fkj is a generic function that we specify later. If we choose it to be independent of +φ, the above expression simplifies to +Pkj(ΦN, TN, z) = +(3.8) +fkj(ΦN, TN, z) +N+2 +� +k′=1 +� zmax +k′ +(ΦN,TN) +zmin +k′ (ΦN,TN) +dz′ Jk′� +ΦN, TN, z′� +∆φk′� +ΦN, TN, z′� +nsplit +k′ +� +j′=1 +fk′j′� +ΦN, TN, z′� +, +where ∆φk(ΦN, TN, z) = φmax +k +(ΦN, TN, z) − φmin +k (ΦN, TN, z). +In order to perform the integral in the denominator of eq. (3.8), we compute the +integration limits on z and φ and the Jacobian Jk both for the 0 → 1 and 1 → 2 splitting +mappings for each ΦN+1 configuration. In the previous Geneva implementation of the +splitting functions the computation of the integration limits was avoided by precomputing +their upper bounds and then using a “hit-or-miss” integration method. +We highlight +that, whenever the constraints on z and φ are in the form of an inequality involving +both the variables, we only compute an overestimate of the true integration limits on z +analytically. We then determine the true limits numerically by imposing the condition +∆φk(ΦN, TN, z) > 0. +3.1.1 +Infrared limits +For this section we introduce the acronyms ISRA (initial-state radiation A), ISRB (initial- +state radiation B), and FSR (final-state radiation) to indicate the N + 2 possible mothers +we have to deal with: the parton from the first (A) and second (B) beam, respectively, and +the final-state partons. We furthermore label ISRA and ISRB collectively as ISR. +The exact form of the function fkj in eq. (3.7) can significantly affect the efficiency +of the Monte Carlo event generator in the region of small TN > T cut +N . In this region, the +logarithmically enhanced terms appearing in the fixed-order calculation have to cancel those +coming from the resummed-expanded contributions. For this reason the main criterion we +follow in the choice of fkj is to achieve a good approximation of the behaviour of the +associated matrix element in the infrared limit when TN → 0. +For simplicity, in practical applications we choose not to include the azimuthal depen- +dence in the form of the fkj functions, using eq. (3.8). We define +fkj(ΦN, TN, z) = +� +� +� +� +� +αs(µR) fA +a (xa, µF ) fB +b (xb, µF ) z ˆPjk(z) +if k is ISR, +αs(µR) fA +a (xa, µF ) fB +b (xb, µF ) ˆPkj(z) +if k is FSR, +(3.9) +– 11 – + +where a and b are the initial-state partons, αs(µR) is the strong coupling evaluated at +the renormalisation scale µR, and fH +i (xi, µF ) is the PDF of the parton i in the hadron H +evaluated at longitudinal momentum fraction xi and factorisation scale µF . The renormal- +isation and factorisation scales are fixed to µR = µF = Q, where Q is the virtuality of the +colour singlet system. The ˆPkj are the unregulated Altarelli-Parisi splitting functions +ˆPqq(z) = CF +1 + z2 +1 − z , +ˆPqg(z) = TF +� +z2 + (1 − z)2� +, +ˆPgq(z) = CF +1 + (1 − z)2 +z +, +ˆPgg(z) = 2CA +� +z +1 − z + 1 − z +z ++ z (1 − z) +� +. +(3.10) +We highlight that for the 0 → 1 splitting, connecting events with no extra partons to +events with one extra parton, the PDFs are evaluated at the exact momentum fractions +xa(z) and xb(z) of the real emission phase space Φ1 rather than their infrared limits. This +has proven to be necessary to obtain an accurate description also in the tail of the colour +singlet transverse momentum distribution. We note that in this case we also reproduce the +correct soft limit, as shown in sec. 3.1.2. For the 1 → 2 splitting the true xa and xb also +depend on φ. In this case they are approximated by dropping this additional dependence, +which still represents an improvement with respect to the strict collinear limit. +3.1.2 +Soft limit of the 0 → 1 splitting +In the following we show that the expression of fkj introduced in eq. (3.9) correctly repro- +duces both the singular soft and collinear limits at O(αs) in the 0 → 1 splitting. +In the case of colour singlet production in hadron-hadron collisions, let us consider the +k → i + j splitting connecting the Born matrix element B0 and the real matrix element +B1 (in both cases excluding parton densities). This can be expressed in terms of the FKS +variables ξ = 2 E/√s = 1 − Q2/s and y = cos θ. Here, s is the partonic centre-of-mass +energy squared, and E and θ are the energy of the emitted parton and the angle between +the emitted and the right-moving incoming parton in the partonic centre-of-mass frame, +respectively. +In the soft limit of the emitted particle i, we have +lim +ξ→0 B1 = 64παs(µR) +Q2 +Ck +ξ2 (1 − y2) B0, +(3.11) +where Ck = CF for the quark-initiated processes and Ck = CA for the gluon-initiated. +In the azimuthally averaged collinear limit between particles i and j, we have +lim +y→±1 B1 = 16παs(µR) +Q2 +1 − ξ +ξ (1 ∓ y) +ˆPjk(1 − ξ) B0 , +(3.12) +where y → 1 and y → −1 represent the collinear limits with respect to incoming parton a +and b respectively. If the colour singlet production process is quark-initiated or has only +scalar particles in the final state, the above expressions also hold prior to averaging over +the azimuthal angle. +– 12 – + +We consider a configuration with one final-state parton with momentum p, where +T0 = ˆp± , +z = Q/(Q + ˆp∓) . +(3.13) +Here ˆp is obtained by longitudinally boosting p from the laboratory frame to the frame +where the colour singlet has zero rapidity, and ˆp± = ˆp0 ∓ ˆp3. We have chosen z such that +in the collinear limit it reduces to the energy fraction of the emitter with respect to the +sister, while providing the correct scaling also for the single soft limit. +In order to show that the singular limits in eq. (3.9) reproduce the above results, we +rewrite T0 and z in terms of the FKS variables ξ and y, and then compare the ensuing +expression to eqs. (3.11) and (3.12). They read +T0 = +Q ξ +2√1 − ξ (1 ∓ y) +� +2 − ξ (1 ± y) +2 − ξ (1 ∓ y) , +z = +� +1 + ξ (1 ± y) +2√1 − ξ +� +2 − ξ (1 ∓ y) +2 − ξ (1 ± y) +�−1 +. +(3.14) +Therefore in the infrared singular limit one obtains +T0 → Q ξ +2 (1 ∓ y) , +(3.15) +z → 1 − ξ +2 (1 ± y) +in the soft limit, +(3.16) +z → 1 − ξ +in the collinear limit. +(3.17) +Multiplying the NLL singular T0 spectrum expanded at O(αs) by the splitting func- +tions, in the infrared limit we find +Pkj(Φ0, T0, z) dσNLL +dΦ0dT0 +���� +O(αs) +→ 8παs(µR) +Q T0 +z ˆPjk(z) fa(xa, µF ) fb(xb, µF ) B0(Φ0) , +(3.18) +up to power corrections. By using the above expressions for T0 and z, it can be shown +that this reproduces both the soft and collinear limits given in eqs. (3.11) and (3.12). We +remark that with the choice of z given in eq. (3.13) the soft limit can be entirely captured +by using the Altarelli-Parisi splitting collinear kernels. The validity of eq. (3.18) can be +understood to be a consequence of the fact that the noncusp soft anomalous dimension is +zero at one loop order, resulting in the lack of a single logarithmic contribution to the T0 +spectrum coming from the soft function. +3.1.3 +Numerical validation +In this section we present the effects of the improved splitting function Pimpr implemen- +tation described above in the case of Higgs boson production via gluon fusion, setting +Q = mH; we focus on the pH +T and the T0 spectrum. We compare the results of a fixed- +order calculation with those obtained by truncating the resummation formula in eq. (2.14) +– 13 – + +0 +25 +50 +75 +100 +dσ / dT0 +[pb/GeV] +pp → H + X +Geneva NLO+NLL′ +√ +S = 13 TeV +rEFT +fixed order +singular × Pimpr +singular × Porig +10−4 +10−3 +10−2 +10−1 +1 +10 +102 +T0 +[GeV] +−1 +0 +1 +2 +3 +dσNS / dT0 +[pb/GeV] +nonsingular (Pimpr) +nonsingular (Porig) +0 +25 +50 +75 +100 +125 +dσ / dpH +T +[pb/GeV] +pp → H + X +Geneva NLO+NLL′ +√ +S = 13 TeV +rEFT +fixed order +singular × Pimpr +singular × Porig +10−2 +10−1 +1 +10 +102 +pH +T +[GeV] +−2 +0 +2 +4 +6 +dσNS / dpH +T +[pb/GeV] +nonsingular (Pimpr) +nonsingular (Porig) +Figure 1: +Comparison of the fixed-order, singular, and nonsingular distributions at +NLO+NLL′, both for T0 (left) and pH +T (right). +We show the singular and nonsingular +distributions both for the original and improved versions of the splitting function imple- +mentation in Geneva. +multiplied by the splitting function to the same order. We do so for the results at LO1 +compared with the NLL′ resummed-expanded in fig. 1, and for those at NLO1 compared +with the NNLL′ resummed-expanded in fig. 2. We also show the nonsingular contribution, +defined as the difference between these fixed-order and resummed-expanded pieces. In all +plots we also show the results obtained with the original implementation Porig of the split- +ting function in eq. (3.8), which was based on a hit-or-miss method using upper bounds +tabulated on a grid. +We begin the discussion with the results for the T0 distribution. As expected, the +improved implementation gives identical results to the original, both at LO1 and NLO1. +This is a consequence of the fact that T0 is preserved by the splitting, by construction. We +observe that at extremely low values of T0 the presence of technical cuts in the fixed-order +calculation affects the convergence to the singular predictions in both approaches. When +instead considering the LO1 results for the pH +T distribution, we notice how the improved +implementation of the splitting functions correctly captures the logarithmic behaviour of +the matrix element at fixed order. This can be seen by the fact that the improved non- +singular distribution converges to zero, contrary to the original case which converges to a +finite value. Similarly, an improvement is also visible for the NLO1 case. Here, however, +the new splitting function Pimpr is not able to exactly reproduce the complete logarith- +– 14 – + +−300 +−200 +−100 +0 +100 +dσ / dT0 +[pb/GeV] +pp → H + X +Geneva NNLO+NNLL′ +√ +S = 13 TeV +rEFT +fixed order +singular × Pimpr +singular × Porig +10−2 +10−1 +1 +10 +102 +T0 +[GeV] +−4 +−2 +0 +2 +4 +6 +dσNS / dT0 +[pb/GeV] +nonsingular (Pimpr) +nonsingular (Porig) +−1500 +−1250 +−1000 +−750 +−500 +−250 +0 +dσ / dpH +T +[pb/GeV] +pp → H + X +Geneva NNLO+NNLL′ +√ +S = 13 TeV +rEFT +fixed order +singular × Pimpr +singular × Porig +10−1 +1 +10 +102 +pH +T +[GeV] +−200 +−150 +−100 +−50 +0 +50 +100 +150 +dσNS / dpH +T +[pb/GeV] +nonsingular (Pimpr) +nonsingular (Porig) +Figure 2: +Comparison of the fixed-order, singular, and nonsingular distributions at +NNLO+NNLL′, both for T0 (left) and pH +T (right). +We show the singular and nonsin- +gular distributions both for the original and improved versions of the splitting function +implementation in Geneva. +mic behaviour of the NLO1 result, as it appears to miss a single logarithmic contribution +∼ 1/pH +T . This is implied by the fact that the improved nonsingular contribution converges +to a nonzero constant at low values of pH +T . This must however be compared with the orig- +inal approach, Porig, where the divergent behaviour of the nonsingular plot suggests that +that implementation also fails to capture the logarithmic structure up to ∼ ln2(pH +T )/pH +T . +3.2 +Independent scale variations +In traditional implementations of fixed-order QCD calculations, a differentiation is made +between the factorisation scale µF and the renormalisation scale µR. The former is associ- +ated with the scale of collinear factorisation, while the latter is introduced in dimensional +regularisation in order to render the strong coupling dimensionless. +To date, implementations of Geneva have assumed these scales to be equal. Doing +so facilitated the matching to the resummed calculation, where a sole “nonsingular” scale +µNS appears as the endpoint of the RGE running, typically taken to be a hard scale Q of +the problem. The two scales were then varied in a correlated fashion (“diagonal” in the +{µR, µF } space) when probing the higher order uncertainties. This approach, however, +can hinder a complete and thorough uncertainty estimation as it neglects those variations +which are off-diagonal, i.e. where µR and µF are varied independently. +In this section +we provide an improved and robust uncertainty estimation within the Geneva framework +– 15 – + +by exposing the µF dependence of the singular cross section that eventually allows for +off-diagonal scale variations, and discuss the choice of µF in the infrared region. +3.2.1 +Exposing the µF dependence of the singular cross section +The collinear beam functions Bi entering the T0 factorisation in eq. (2.12) satisfy the OPE +in eq. (2.13). In resummed predictions, they are evaluated at a scale µ = µB where µB +minimises the singular logarithmic structure of Bi, whereas at fixed order µ = µR = µF = +Q, where for example Q = mH for on-shell Higgs boson production. +In order to expose the µF dependence of the beam functions, we rewrite eq. (2.13) as +Bi(t, x, µ) = +� +j +Iij(t, x, µ) ⊗x fj(x, µ) += +� +j,k +Iik(t, x, µ) ⊗x Ukj(x, µ, µF ) ⊗x fj(x, µF ) +≡ +� +j +ˆIij(t, x, µ, µF ) ⊗x fj(x, µF ) , +(3.19) +where we dropped the power corrections. Here we evolve the PDFs from µF to µ using the +evolution kernel Uij(x, µ, µF ) that results from the solution of the DGLAP equations, +µ d +dµfi(x, µ) = 2 +� +j +Pij(x, µ) ⊗x fj(x, µ) , +(3.20) +and we follow the conventions of ref. [90] for the perturbative expansion of the splitting +kernels Pij(x, µ).1 Although the µF dependence in eq. (3.19) cancels exactly between the +PDFs and the evolution kernel, as soon as ˆIij is truncated at a given order, a residual µF +dependence appears in the beam function, +Bi(t, x, µ) �→ Bi(t, x, µ, µF ) . +(3.21) +In order to manifest this dependence explicitly, we note that the matching coefficients +and the evolution kernel in eq. (3.19) admit perturbative expansions in the strong coupling +constant, +ˆIij(t, x, µ, µF ) = δ(t)1ij(x) + +∞ +� +n=1 +ˆI(n) +ij (t, x, µ, µF ) +�αs(µ) +4π +�n +, +(3.22) +Uij(x, µ, µF ) = 1ij(x) + +∞ +� +n=1 +U(n) +ij (x, µ, µF ) +�αs(µ) +4π +�n +(3.23) +where 1ij(x) ≡ δijδ(1 − x). We first obtain closed-form expressions for the Uij, which +can be achieved either by directly solving eq. (3.20) or by using the solutions of the RGE +1The splitting functions used here are multiplied by a factor of 2 with respect to the unregulated ones +defined in eq. (3.10). +– 16 – + +satisfied by Iij (see eq. (2.17) of ref. [90]). We have +µ d +dµU−1 +ij (x, µ, µF ) ⊗x fj(x, µ) = −U−1 +ij (x, µ, µF ) ⊗x µ d +dµfj(x, µ) += −2 U−1 +ik (x, µ, µF ) ⊗x Pkj(x, µ) ⊗x fj(x, µ) +⇒ +µ d +dµU−1 +ij (x, µ, µF ) = −2 U−1 +ik (x, µ, µF ) ⊗x Pkj(x, µ) , +(3.24) +where U−1 +ij +denotes the inverse of Uij. Here and in the following, repeated flavour indices +are implicitly summed over. We note that eq. (3.24) is exactly the same as eq. (2.17) of +ref. [90] if we set γB = γν = 0. It is therefore straightforward to use its solution, which up +to O(α2 +s) reads +U−1 (0) +ij +(x, µ, µF ) = 1ij(x) , +(3.25) +U−1 (1) +ij +(x, µ, µF ) = −2 L P (0) +ij (x) , +(3.26) +U−1 (2) +ij +(x, µ, µF ) = 2 L2� +P (0) +ik (x) ⊗x P (0) +kj (x) − β0P (0) +ij (x) +� +− 2 L P (1) +ij (x) , +(3.27) +where we have abbreviated L = ln(µ/µF ). Solving the closure equation of the evolution +kernels U−1 +ik (x, µ, µF ) ⊗x Ukj(x, µ, µF ) = 1ij(x) at each order in perturbation theory yields +U(1) +ij (x, µ, µF ) = 2 L P (0) +ij (x) , +(3.28) +U(2) +ij (x, µ, µF ) = L2� +2β0P (0) +ij (x) + 2P (0) +ik (x) ⊗x P (0) +kj (x) +� ++ 2 L P (1) +ij (x) . +(3.29) +Substituting eqs. (3.28), (3.29) in eq. (3.19), we arrive at the explicit results for the µF - +dependent matching coefficients. They read +ˆI(1) +ij (t, x, µ, µF ) = I(1) +ij (t, x, µ) + 2 δ(t) L P (0) +ij (x) , +(3.30) +ˆI(2) +ij (t, x, µ, µF ) = I(2) +ij (t, x, µ) + L1(t, µ2) +� +2 L Γi +0 P (0) +ij (x) +� ++ L0(t, µ2) +� +−γi +B 0 L P (0) +ij (x) + 2 L P (0) +ik (x) ⊗x P (0) +kj (x) +� ++ δ(t) +� +L2� +2β0P (0) +ij (x) + 2P (0) +ik (x) ⊗x P (0) +kj (x) +� ++ 2 L +� +P (1) +ij (x) + I(1) +ik (x) ⊗x P (0) +kj (x) +�� +, +(3.31) +where the expressions for the (µF -independent) matching coefficients I(n) +ij (t, x, µ), the +anomalous dimensions γi +B 0 and Γi +0, and the plus distributions Ln can be found in ref. [90]. +3.2.2 +Choice of the factorisation scale +The choice of the factorisation scale µF is in principle subject to different requirements +in the fixed-order and in the resummation region. In order to minimise the size of the +logarithms L = ln(µ/µF ) in eqs. (3.30) and (3.31), in the resummation region we demand +that µF ∼ µB, i.e. the beam scale. On the other hand, in the fixed-order region, a natural +scale setting is µF ∼ µR ∼ µNS so that the fixed-order perturbative convergence is not +– 17 – + +jeopardised. However, given that the beam function profile scale µB(T0) flows to µNS in +the fixed-order region, +µB +T0→Q +−−−−→ Q ≡ µNS , +(3.32) +choosing µF = µB satisfies both conditions. +When considering the scale variations for the estimation of the theoretical uncertainty, +the definition of the profile scales is extended to +µS(T0/Q, α) = κR µNS frun(T0/Q) fα +vary(T0/Q) , +(3.33) +µB(T0/Q, α, β) = κR µNS +� +frun(T0/Q) fα +vary(T0/Q) +�1/2−β , +(3.34) +µF (T0/Q, αf, β) = κF µNS +� +frun(T0/Q) fαf +vary(T0/Q) +�1/2−β , +(3.35) +where the central predictions are obtained setting κR = κF = 1, α = αf = β = 0. The +function frun is defined in ref. [9] and the function fvary in ref. [105]. We note that a different +parameter is used in the exponent of the function fvary for µB and µF in eqs. (3.34) and +(3.35) above, while we use the same β parameter for both. This is justified because the +β-variations are introduced in order to disentangle the variations of µB and µS; no ratios +of µF /µS appear in the singular cross section, therefore there is no need for an independent +β parameter in µF . +We also extend the fixed-order uncertainty ∆FO to include the off-diagonal µF and +µR = κR µNS scale variations, resulting from the envelope of a 7-point scale variation +(κR, κF ) = {(1, 1), (2, 2), (1/2, 1/2), (1, 2), (1, 1/2), (2, 1), (1/2, 1)} , +(3.36) +where we have excluded the cases when µF and µR are varied in opposite directions. The +resummation uncertainty ∆res estimated by the usual profile scale and transition point +variations [105] (already present in Geneva) is extended by considering in addition the +variations of the parameters α, αf, and β appearing in eq. (3.35). Explicitly, we extend +the original procedure to include the additional parameter variations: +(β, α, αf) = {(0, 0, 0), (1/6, 0, 0), (−1/6, 0, 0), +(3.37) +(0, 1, 1), (0, −1, −1), (0, 1, 0), (0, −1, 0), (0, 0, 1), (0, 0, −1)} . +These variations are enveloped with the variations of the transition points xi, and are +summed in quadrature with the fixed-order variations in eq. (3.36) to obtain the final +theoretical uncertainty. +3.3 +Treatment of timelike logarithms +Radiative corrections in colour singlet production processes such as Drell-Yan or gluon- +initiated Higgs production contain Sudakov logarithms of the form αn +s lnm(−q2/µ2 +R) , m ≤ +2n, where qµ is the momentum of the colour singlet system. +They primarily arise in +the calculation of the corresponding form factor, and their coefficients are linked to the +structure of its infrared singularities [106]. +For such processes qµ is a timelike vector, +– 18 – + +i.e. q2 > 0, and the scale choice µ2 +R = q2 results in the Sudakov logarithms developing +an imaginary part, since lnm(−1) = (±iπ)m. The presence of such ‘timelike’ logarithms +at each order might negatively affect the perturbative convergence of the cross section, +where they result in additional terms proportional to powers of π2. The severity of this +effect is process specific; for Drell-Yan or gluon-initiated Higgs production it has been +explicitly studied for both exclusive [45, 46, 52, 54, 87, 105, 107–109] and inclusive [19, 43] +observables. A way to mitigate the impact of said contributions is to choose µR such that +timelike logarithms are eliminated, i.e. to evaluate the form factor at the complex scale +µR = −i|q| = −i Q [110–113]. +In factorised singular cross sections, the square of the form factor naturally appears +as the hard function H(Q2, µH) of the process. Following the above discussion, the hard +function can be evaluated at the complex scale µH = −i Q. This choice implies a nontrivial +renormalisation group evolution between µH and the real scale Q, in the form of a rotation +in the complex µ plane. This procedure is also referred to as ‘timelike resummation’. It +has been applied to a multitude of exclusive and inclusive processes, including the case of +Higgs boson production, see e.g. refs. [18, 19, 43, 52, 87, 114]. +In addition, it has been shown that not only the singular, but also the nonsingu- +lar [52, 54] and the total [19] cross section can benefit from this prescription, in terms of an +improved perturbative convergence and a reduced ensuing scale uncertainty. This can be +better understood when one considers that by integrating the timelike resummed exclusive +cross section, one obtains the corresponding resummed inclusive prediction. For the non- +singular contribution, a factorisation formula remains in general unknown, meaning that +it cannot be directly evaluated at the complex scale µH – it still, however, contains the +form factor plagued by timelike logarithms. In this case, the procedure for the treatment +of these logarithms involves re-expanding the nonsingular contribution, extracting from it +the hard function evaluated at Q, and replacing it with that evaluated at the scale µH, as +detailed in ref. [19]. +In our implementation we perform these steps at the same order as the T0 resummation, +both in the singular and the nonsingular terms. This implies that the improved perturbative +convergence following the choice of a complex-valued scale µH = −i Q does not only apply +to the singular T0 spectrum but also to the inclusive predictions. +In order to study the uncertainty associated with this choice of scale, we follow the pre- +scription introduced in ref. [19], designed to probe the structure of the timelike logarithms. +The uncertainty ∆ϕ is estimated by the envelope of the phase variations +µH = Q e−iϕ , +ϕ ∈ [π/4, 3π/4] , +(3.38) +while the central value predictions correspond to ϕ = π/2. Since there is no dynamical +parameter governing the choice of the scale µH, the timelike resummation is performed +throughout the T0 spectrum, i.e. even when T0 resummation is off. We therefore consider +the uncertainty resulting from variations in eq. (3.38) as an independent source and add it +to the other uncertainties in quadrature. Thus, for inclusive predictions we have +∆2 +incl = ∆2 +FO + ∆2 +ϕ , +(3.39) +– 19 – + +0 +5 +10 +15 +dσ / dyH +[pb] +pp → H + X +√ +S = 13 TeV +rEFT +µH = mH +µH = −i mH +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +yH +−0.2 +−0.1 +0.0 +0.1 +0.2 +ratio − 1 +10−3 +10−2 +10−1 +100 +dσ / dT0 +[pb/GeV] +pp → H + X +√ +S = 13 TeV +rEFT +µH = mH +µH = −i mH +1 +10 +20 +30 +40 +50 +100 +150 +T0 +[GeV] +−0.4 +−0.2 +0.0 +0.2 +0.4 +ratio − 1 +Figure 3: Comparison of the yH (left) and T0 (right) distributions with different choices +of µH. +whereas for exclusive predictions we use +∆2 +excl = ∆2 +FO + ∆2 +res + ∆2 +ϕ . +(3.40) +In the Geneva implementation of the gg → H process we use the hardfunc module +from scetlib [91] for the hard function evaluation and evolution in the complex plane. +Since for this process we set Q = mH, we pick +µH = −i mH . +(3.41) +With this choice, we observe a difference in the total cross section result with respect to the +µH = mH case that can be substantial despite being formally of higher order. The effects +of the complex choice of scale µH on differential observables are illustrated in fig. 3, where +we compare predictions at NNLO+NNLL′ for the T0 and yH distributions with µH = mH +and µH = −i mH. In this and the following figures, the theoretical uncertainty is shown +as a shaded band, while the Monte Carlo integration errors are shown as thin vertical +bars. For the Higgs boson rapidity distribution, we observe an increase of around 10% +that is almost independent of yH, and a reduction in the uncertainty band as expected. +The T0 spectrum shows a larger effect, especially in the tail of the distribution, where our +prediction is entirely driven by the fixed-order result. Nonetheless, we observe a reduction +in the uncertainty band particularly in the peak and transition regions of the spectrum, +between 5 and 45 GeV. +4 +Validation of the gg → H process +In this section we validate our predictions. We first compare our partonic NNLO results +with two independent calculations, and then discuss the interface to the Pythia8 shower. +– 20 – + +Geneva +ggHiggs +Matrix +σNNLO, rEFT +gg→H +[pb] +42.33+4.39 +−4.34 +42.35+4.55 +−4.41 +42.33+4.54 +−4.40 +Table 1: Comparison of the Geneva, ggHiggs, and Matrix results for the gg → H +inclusive cross section. The results are obtained at NNLO in the HTL approximation, and +rescaled with the rEFT factor. +4.1 +Partonic results at NNLO +Here we validate the NNLO accuracy of the total cross section obtained with Geneva +and that of the only differential inclusive quantity available, the Higgs boson rapidity. +We compare the total cross section with the independent calculations implemented in +ggHiggs [60, 115–118] and Matrix [119], and the rapidity distribution with Matrix only. +The Matrix predictions are based on the qT -subtraction approach and are extrapolated +towards the zero qT -cut value. We set the input parameters of our calculations as described +in sec. 2.2, and we choose the central factorisation and renormalisation scales equal to each +other and to the Higgs boson mass, µR = µF = mH. We set our resolution cutoffs to +T cut +0 += T cut +1 += 1 GeV. We employ the PDF set PDF4LHC15 nnlo 100 from LHAPDF [120], +and take the value of αs(mZ) from the same set, so that αs(mH) = 0.11263. +In table 1 we report the values of the inclusive gg → H cross section and the as- +sociated 7-point scale variations calculated at NNLO and rescaled with the rEFT factor +using Geneva, ggHiggs, and Matrix. We observe excellent agreement between the three +predictions; by choosing T cut +0 += 1 GeV, the neglected power-suppressed terms in Geneva +are at the permille level and amount to an acceptable ∼ 0.02 pb error for the central value. +In fig. 4 we compare the Higgs rapidity spectrum obtained with Geneva with the +NNLO result provided by Matrix, including the 7-point scale variations. We observe very +good agreement both in the central values and in the envelope of the scale variations, up to +large values of |yH|. The symmetry of the pp collider allows us to show only the absolute +value of yH, and thus further reduce the Monte Carlo uncertainty. +4.2 +Interface with PYTHIA8 +In this section we briefly recap the main features of the interface used in Geneva to match +the partonic results to the Pythia8 [121] parton shower. As this is not the main focus of +this work, however, we refer the interested reader to ref. [4] for a detailed discussion and +ref. [15] for additional details on the accuracy of the matched calculation. Given that so far +we have constructed partonic results with NNLL′ accuracy in the resolution variable T0, +we wish to preserve this resummed accuracy after the parton shower as far as is possible. +At the same time, for all other observables we need to guarantee that the accuracy of the +parton shower is preserved. This is a nontrivial condition: since the ordering variable of the +Pythia8 parton shower is the relative transverse momentum while the resolution variable +we use is the N-jettiness, the shower can in principle produce emissions which double-count +regions of the phase space. +– 21 – + +0 +5 +10 +15 +20 +25 +dσ / d|yH| +[pb] +pp → H + X +√ +S = 13 TeV +rEFT +µH = mH +MATRIX +Geneva +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +|yH| +−0.1 +0.0 +0.1 +ratio − 1 +Figure 4: Comparison of Geneva and Matrix at NNLO for the yH distribution. +To avoid this issue, we perform the matching employing the following prescription. We +set the starting scale of the parton shower by taking the maximum relative k⊥ determined +by the lower scale of the resummation. The latter is defined on an event-by-event basis +and corresponds to either T c +N ≡ T cut +0 +, T cut +1 +or T1 (Φ2), depending on whether the relative +partonic configuration has N = 0, 1 or 2 jets, respectively. We then let the shower run +down to the internal minimum p⊥, which produces a certain number of emissions k. Lastly, +we check that the resulting event fulfils the condition +TN(ΦN+k) ≤ T c +N , +(4.1) +which ensures that both accuracies are correctly preserved. +In fig. 5 we show the effect of the Pythia8 shower on the pH +T and yH partonic distri- +butions. For the results presented in this section we use the default Pythia8 parameters +for the shower and the hadronisation model. The rapidity distribution, being an inclusive +observable, is exactly preserved by the shower, as expected. The Higgs transverse momen- +tum is an exclusive observable, and the shower can therefore have a significant impact on +its shape: in this case we see an effect of ∼ 15% in the pH +T < 15 GeV bin, and smaller +effects ≲ 5% in the rest of the spectrum, especially in the tail of the distribution. After +hadronisation, we find that most of these discrepancies are reduced. +The parton shower and hadronisation effects on the T0 distribution are displayed in +fig. 6. As mentioned above, our matching procedure to the Pythia8 shower is designed +with the aim that the T0 logarithmic accuracy is not spoiled. We explicitly check this +in the left panel, where we compare the T0 distribution at NNLL′ before and after the +parton shower matching with the partonic prediction at N3LL. Note that for this process, +which is gluon-initiated, one expects that the parton shower effects are larger than for +quark-initiated processes, e.g. because of the larger Casimir factors. Nonetheless, in the +peak region T0 < 25 GeV, we find that the showered distribution lies in between the +– 22 – + +10−3 +10−2 +10−1 +100 +dσ / dpH +T +[pb/GeV] +pp → H + X +√ +S = 13 TeV +rEFT +Geneva +Geneva+Pythia8 +Geneva+Pythia8 (hadronised) +0 +50 +100 +150 +200 +250 +300 +pH +T +[GeV] +−0.2 +−0.1 +0.0 +0.1 +0.2 +ratio − 1 +0 +5 +10 +15 +dσ / dyH +[pb] +pp → H + X +√ +S = 13 TeV +rEFT +Geneva +Geneva+Pythia8 +Geneva+Pythia8 (hadronised) +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +yH +−0.1 +0.0 +0.1 +ratio − 1 +Figure 5: Comparison of the partonic, showered, and hadronised results for the pT +H (left) +and yH (right) distributions. +−1.0 +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +dσ / dT0 +[pb/GeV] +pp → H + X +√ +S = 13 TeV +rEFT +Geneva (NNLL′+NNLO) +Geneva (NNLL′+NNLO)+Pythia8 +Geneva (N3LL+NNLO) +0 +5 +10 +15 +20 +25 +T0 +[GeV] +−0.2 +−0.1 +0.0 +0.1 +0.2 +ratio − 1 +10−3 +10−2 +10−1 +100 +dσ / dT0 +[pb/GeV] +pp → H + X +√ +S = 13 TeV +rEFT +Geneva +Geneva+Pythia8 +Geneva+Pythia8 (hadronised) +1 +10 +20 +30 +40 +50 +100 +150 +T0 +[GeV] +−0.2 +0.0 +0.2 +ratio − 1 +Figure 6: Effects of the parton shower on the T0 spectrum: comparison of the partonic, +showered, and N3LL-resummed distribution (left), and comparison of the partonic, show- +ered, and hadronised results (right). +central NNLL′ and N3LL curves, and within the overlap of the two uncertainty bands. We +therefore conclude that the quantitative effects of the shower are on par with (or smaller +than) the effects of the next logarithmic order in the resummation. +The hadronisation effects on the T0 distribution are displayed in the right panel of +fig. 6. As expected for this observable, we observe O(1) effects in the peak region, which +decrease for larger values of T0. +In the region around T0 ≈ mH/2, which corresponds +– 23 – + +1 +10 +102 +σfid +[fb] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH += 0 += 1 += 2 +≥ 3 +Njets +−0.4 +−0.2 +0.0 +0.2 +0.4 +ratio − 1 +0 +10 +20 +30 +40 +50 +60 +70 +80 +dσ / d|yH| +[fb] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +|yH| +−0.50 +−0.25 +0.00 +0.25 +0.50 +ratio − 1 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +dσ / dpH +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0 +20 +50 +100 +200 +500 +pH +T +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +dσ / dpj1 +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0 +30 +60 +90 +120 +350 +pj1 +T +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +Figure 7: Comparison of the ATLAS data [64] with the Geneva+Pythia8 results at +13 TeV. We show the fiducial cross sections for different values of Njets (top left), as well +as the distributions of |yH| (top right), pH +T (bottom left), and pj1 +T (bottom right). +to the point at which the T0 resummation is switched off, we find a more pronounced +discrepancy between the Geneva partonic and showered results. We have verified that +this is an artefact related to our choice of setting the T0 spectrum equal to the derivative +of the cumulant as explained at the end of sec. 2.3. +5 +Comparison with LHC data +We compare the predictions obtained with Geneva with the latest experimental results for +the Higgs boson inclusive and differential cross sections in the H → γγ decay channel. The +results are provided both by the ATLAS [64] and CMS [65] experiments, and are obtained +– 24 – + +10−3 +10−2 +10−1 +100 +dσ / dpH +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +Njets =0 +0 < +τ j +C +GeV < 15 +15 < +τ j +C +GeV < 25 +25 < +τ j +C +GeV < 40 +40 < +τ j +C +GeV < 400 +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0-350 +0-100 +100-350 +0-120 +120-350 +0-200 +200-350 +0-250 +250-350 +pH +T +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +10−3 +10−2 +10−1 +100 +dσ / dpH +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +0 < |yH| < 0.5 +0.5 < |yH| < 1.0 +1.0 < |yH| < 1.5 +1.5 < |yH| < 2.5 +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0-45 +45-120 +120-350 +0-45 +45-120 +120-350 +0-45 +45-120 +120-350 +0-45 +45-120 +120-350 +pH +T +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +dσ / dτ j1 +C +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0 +5 +15 +25 +40 +τ j1 +C +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +dσ / dpHj +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +ATLAS, 139 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH+bbH+tH +0 +30 +60 +120 +pHj +T +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +Figure 8: Comparison of the ATLAS data [64] with the Geneva+Pythia8 results at +13 TeV. We show the pH +T distributions in bins of τ j1 +C (top left) and of |yH| (top right), as +well as the τ j1 +C (bottom left) and the pHj +T +(bottom right) distributions. +from the LHC data at a centre-of-mass energy of 13 TeV using 139 fb−1 and 137 fb−1 of +proton-proton collision data, respectively. +In the ATLAS measurement [64], the fiducial phase space is identified by requiring +the existence of two isolated photons with pγ +T > 25 GeV in the final state. +Photons +are considered isolated if the transverse energy of charged particles with pT > 1 GeV +within a cone of radius Riso = 0.2 around the photon direction does not exceed 5% of +the photon’s transverse momentum. +The two isolated photons must additionally have +transverse momenta larger than 35% and 25% of the diphoton invariant mass, for the +leading and subleading photons respectively. The invariant mass of the diphoton system +– 25 – + +0.1 +1 +10 +102 +σfid +[fb] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH += 0 += 1 += 2 += 3 +≥ 4 +Njets +−0.4 +−0.2 +0.0 +0.2 +0.4 +ratio − 1 +0 +10 +20 +30 +40 +50 +60 +70 +80 +dσ / d|yH| +[fb] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +|yH| +−0.50 +−0.25 +0.00 +0.25 +0.50 +ratio − 1 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +dσ / dpH +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +σ(pH +T >450 GeV) +100 GeV +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +0 +20 +50 +100 +200 +500 +pH +T +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +dσ / dpj1 +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +σ(pj1 +T >200 GeV) +50 GeV +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +50 +75 +100 +125 +150 +175 +200 +225 +250 +pj1 +T +[GeV] +−1 +0 +1 +ratio − 1 +Figure 9: Comparison of the CMS data [65] with the Geneva+Pythia8 results at 13 TeV. +We show the fiducial cross sections for different values of Njets (top left), as well as the +distributions of |yH| (top right), pH +T (bottom left), and pj1 +T (bottom right). +must be in the range 105 GeV < mγγ < 160 GeV. Moreover, photons are required to +have pseudorapidity |ηγ| < 1.37 or 1.52 < |ηγ| < 2.37. For this measurement, jets are +defined using the anti-kT algorithm with radius R = 0.4, and must have pj +T > 30 GeV and +|yj| < 4.4. Jets must also be separated from photons with pγ +T > 15 GeV by a distance +∆Rγj > 0.4. +Similarly, in the CMS measurement [65], the fiducial region is defined by having two +isolated photons in the final state. In this case, photons are isolated if the transverse energy +of all particles inside a cone of radius Riso = 0.3 is less than 10 GeV. The transverse +momenta of the leading (subleading) isolated photon must satisfy pγ +T > 35 (25) GeV, and +– 26 – + +10−3 +10−2 +10−1 +100 +101 +dσ / dτ j1 +C +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +σ(τ j1 +C >80 GeV) +30 GeV +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +0 +20 +40 +60 +80 +100 +τ j1 +C +[GeV] +−0.5 +0.0 +0.5 +ratio − 1 +10−1 +100 +101 +102 +103 +dσ / d|φ∗ +η| +[fb] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +σ(|φ∗ +η|>4) +1.5 +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +0 +0.2 +0.5 +1 +2 +5 +|φ∗ +η| +−0.5 +0.0 +0.5 +ratio − 1 +10−2 +10−1 +100 +101 +dσ / dpH +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +Njets = 0 +σ(pH +T >60 GeV) +15 GeV +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +0 +10 +20 +30 +40 +50 +60 +70 +pH +T +[GeV] +−1 +0 +1 +ratio − 1 +10−2 +10−1 +100 +101 +dσ / dpH +T +[fb/GeV] +pp → (H → γγ) + X +√ +S = 13 TeV +rEFT +Njets = 1 +σ(pH +T >170 GeV) +70 GeV +CMS, 137 fb−1 +Geneva+Pythia8 (ggH) + XH +XH =VBF+VH+ttH +0 +50 +100 +150 +200 +pH +T +[GeV] +−1 +0 +1 +ratio − 1 +Figure 10: Comparison of the CMS data [65] with the Geneva+Pythia8 results at +13 TeV. We show the τ j1 +C (top left) and |φ∗ +η| (top right) distributions, as well as the pH +T +distributions for events with Njets = 0 (bottom left) and Njets = 1 (bottom right). +amount to at least 1/3 (1/4) of the reconstructed Higgs invariant mass. In turn, the Higgs +invariant mass must lie between 100 and 180 GeV. Photons must also satisfy |ηγ| < 2.5. +Also in this case, jets are constructed using the anti-kT algorithm with R = 0.4, and are +required to have pj +T > 30 GeV. Jets with |ηj| < 2.5 are used for observables with one extra +jet or to count the number of jets, while a looser cut |ηj| < 4.7 is applied for observables +requiring at least two jets in the final state. +Due to the lack of availability of these analyses in the Rivet [122] framework, we +have implemented the ATLAS and CMS analyses within the Geneva code. The H → γγ +decay is inserted by the Pythia8 particle decays handler on top of the events produced +– 27 – + +by Geneva. Its kinematics are treated at leading order in QCD, and we set the branching +ratio to BR(H → γγ) = 2.27 × 10−3, i.e. the value reported in ref. [123] and calculated +with HDECAY [124]. +The Geneva prediction for the gluon-fusion production channel +is obtained at NNLO+NNLL′ +T0+NLLT1, and setting the scale of the hard function µH = +−i mH. We use matrix elements computed in the infinite-top-mass limit and rescaled in +the rEFT scheme. We set T cut +0 += T cut +1 += 1 GeV. We use the PDF set PDF4LHC15 NNLO, +and take the value of αs(mZ) from there. +The partonic prediction is matched to the +Pythia8 QCD+QED shower, including multiparton interaction (MPI) contributions. We +use the AZNLO tune [125] for the ATLAS comparison, and the CP5 tune [126] for CMS. +Showered events are then hadronised using the default Pythia8 Lund string model [127, +128]. In order to obtain a meaningful comparison with the experimental data, we include +the contributions from other Higgs boson production modes (labelled overall as XH) by +summing them to the Geneva results for the gluon-fusion channel alone.2 For ATLAS +these include vector-boson fusion (VBF), Higgsstrahlung (V H), and associated production +with t¯t, b¯b, and t, all computed at NLO accuracy in QCD. For CMS these only include +contributions from VBF, V H, and t¯tH. +The outcome of the comparison with the experimental results is shown in figs. 7 and +8 for the ATLAS data, and in figs. 9 and 10 for the CMS data. For the ATLAS data, we +show the pH +T , Njets, |yH|, pj1 +T , pHj +T , and τ j1 +C distributions, as well as the pH +T spectra in bins +of τ j1 +C and in bins of |yH|. For the CMS data, we show the pH +T , Njets, |yH|, pj1 +T , τ j1 +C , and +|φ∗ +η| distributions, as well as the pH +T spectra in different jet multiplicity bins (N = 0 and +N = 1). The definitions of τ j +C and φ∗ +η are given by +τ j +C = +mj +T +2 cosh(yj − yH) , +φ∗ +η = tan +�φacop +2 +� +sin θ∗ +η , +(5.1) +where φacop = π − |∆φγγ| and sin θ∗ +η = [cosh(∆ηγγ/2)]−1 [129]. +With the ATLAS fiducial cuts, we obtain a total fiducial cross section of 58.8+1.5 +−3.0 fb, +to be compared to the experimental finding of 67 ± 6 fb. In the CMS fiducial region, we +obtain a total cross section of 66.6+1.6 +−3.3 fb, which is compatible with the measurement of +73.4+6.1 +−5.9 fb. In both cases our predictions agree with the measured results within roughly +one standard deviation. We note that our results are not rescaled to the total N3LO gluon- +fusion cross section, contrary to the theoretical predictions used in the ATLAS publication +for their comparison. +Regarding the distributions, we find overall good agreement between the Geneva +predictions and the measurements. For the ATLAS data we find slight deviations in the pH +T +peak and a more marked discrepancy in the tail of the distribution. The latter corresponds +to the region where the HTL approximation is less accurate. We also find slight deviations +in the |yH| spectrum. The deviations in both spectra are consistent with those obtained +using other calculations, as shown in ref. [64]. Similarly, for the CMS data our results +2The values of the XH distributions are taken from the plots in ATLAS and CMS publications. +– 28 – + +underestimate the bins corresponding to the pH +T peak, again in a similar fashion to other +predictions [65]. Large deviations are also found in the first bin of the pH +T distribution with +Njets = 1, once again in agreement with other theoretical predictions. +6 +Conclusions +We have described a number of improvements to the Geneva method, which are particu- +larly useful for processes featuring large higher order corrections such as the production of +colour singlet final states via gluon fusion. Specifically, we detailed a new implementation +of the splitting functions which serve to make the resummed calculation fully differential +in higher multiplicity phase spaces. This results in an improved behaviour of the nonsin- +gular cross section as a function of the colour singlet transverse momentum in the infrared +limit. In addition, following earlier work [43] we have introduced a separation between +the beam scale in our SCET-based resummed calculation µB and the scale associated with +collinear factorisation µF which appears in the fixed-order calculation. This allowed us to +achieve a more robust estimate of the theoretical uncertainties associated with our calcula- +tion in the fixed-order region, including uncorrelated variations of the renormalisation and +factorisation scales. Finally, we have addressed the issue of large contributions from π2 +terms originating from timelike logarithms in 2 → 1 processes, by enabling the choice of a +complex-valued hard scale µH. We studied the associated resummation of said logarithms +in our fully-differential calculation, and showed that, as previously noted in the literature, +the perturbative convergence can thus be improved. +Throughout this work, we have used the gluon-initiated Higgs production process to +study the effects of our improvements. We have constructed an NNLO+PS event generator +for the process, including the resummation of the zero-jettiness variable up to N3LL accu- +racy. The availability of recent experimental results [64, 65] for this process also allowed us +to make a detailed comparison of our final, showered events with data. We stress, however, +that the issues which we addressed in this work have a more general applicability, and we +anticipate that the future implementations of processes in Geneva will make use of these +developments. +In this study, we have consistently worked in a heavy-top limit in which the top- +quark has been integrated out of the SM Lagrangian, resulting in an effective gluon-Higgs +coupling. Given the advancement in recent years towards including the exact quark mass +dependence at NNLO [61, 62], it would be desirable to incorporate this progress into a +Geneva event generator at NNLO+PS. We leave this issue to future work. +Acknowledgements +We thank L. Rottoli for his collaboration in the early stages of this project. We are also +grateful to A. Cueto, M. Donega, M. Malberti, and S. Pigazzini for their help with the +comparison of Geneva with the ATLAS and CMS results. We thank F. Tackmann for pro- +viding us with a preliminary version of scetlib and for useful comments to the manuscript. +This project has received funding from the European Research Council (ERC) under the +– 29 – + +European Union’s Horizon 2020 research and innovation programme (Grant agreements +No. 714788 REINVENT and 101002090 COLORFREE) The work of SA and GB is sup- +ported by MIUR through the FARE grant R18ZRBEAFC. SA also acknowledges funding +from Fondazione Cariplo and Regione Lombardia, grant 2017-2070. MAL is supported +by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy – +EXC 2121 “Quantum Universe” – 390833306, and also by the UKRI guarantee scheme +for the Marie Sk�lodowska-Curie postdoctoral fellowship, grant ref. EP/X021416/1. 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J. +C 71 (2011) 1600 [1009.1580]. +– 37 – + diff --git a/ddFKT4oBgHgl3EQfqi7m/content/tmp_files/load_file.txt b/ddFKT4oBgHgl3EQfqi7m/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4e4d8959664273dbd52486d95cf0611b0cb72e58 --- /dev/null +++ b/ddFKT4oBgHgl3EQfqi7m/content/tmp_files/load_file.txt @@ -0,0 +1,2008 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf,len=2007 +page_content='Prepared for submission to JHEP DESY 23-009 UWThPh-2023-2 Refining the GENEVA method for Higgs boson production via gluon fusion Simone Alioli,a Georgios Billis,a Alessandro Broggio,a,b Alessandro Gavardi,a,c Stefan Kallweit,a Matthew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Lim,c,d Giulia Marinelli,a Riccardo Nagara and Davide Napoletanoa aUniversit`a degli Studi di Milano-Bicocca & INFN, Piazza della Scienza 3, Milano 20126, Italy bFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria cDeutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany dDepartment of Physics and Astronomy, University of Sussex, Sussex House, Brighton, BN1 9RH, UK E-mail: simone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='alioli@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='it, georgios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='billis@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='it, alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='broggio@univie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='at, alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='gavardi@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='de, stefan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='kallweit@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='it, m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='lim@sussex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='uk, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='marinelli10@campus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='it, riccardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nagar@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='it, davide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='napoletano@unimib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='it Abstract: We describe a number of improvements to the Geneva method for matching NNLO calculations to parton shower programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In particular, we detail changes to the resummed calculation used in the matching procedure, including disentangling the cross section dependence on factorisation and beam scales, and an improved treatment of timelike logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We also discuss modifications in the implementation of the splitting functions which serve to make the resummed calculation differential in the higher multiplicity phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' These changes improve the stability of the numerical cancellation of the nonsingular term at small values of the resolution parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' As a case study, we consider the gluon- initiated Higgs boson production process gg → H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We validate the NNLO accuracy of our predictions against independent calculations, and compare our showered and hadronised results with recent data taken at the ATLAS and CMS experiments in the diphoton decay channel, finding good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='11875v1 [hep-ph] 27 Jan 2023 Contents 1 Introduction 1 2 Theoretical framework 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 The GENEVA method 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Higgs boson production via gluon fusion 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 T0 resummation 7 3 Novel features of the GENEVA method 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Improved treatment of splitting functions 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Infrared limits 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Soft limit of the 0 → 1 splitting 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 Numerical validation 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Independent scale variations 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Exposing the µF dependence of the singular cross section 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Choice of the factorisation scale 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 Treatment of timelike logarithms 18 4 Validation of the gg → H process 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Partonic results at NNLO 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Interface with PYTHIA8 21 5 Comparison with LHC data 24 6 Conclusions 29 1 Introduction In recent years, the quest for precision at the Large Hadron Collider (LHC) has seen many impressive milestones in the development of theoretical tools used to describe hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Many processes are currently known at next-to-next-to-leading order (NNLO) in perturbative QCD, and several 2 → 1 processes even at one order higher (N3LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' One particular direction in which much fruitful progress has been made is in the matching of higher order perturbative calculations to parton shower (PS) programs, resulting in Monte Carlo event generators which combine the advantages of fixed-order calculations with the flexibility of parton shower tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This paradigm generally goes under the name of NNLO+PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Several different methods which reach NNLO+PS accuracy have been proposed [1–7], with most applications using a resummed calculation – either directly or via the Sudakov factor in a shower Monte Carlo – in a suitable resolution variable alongside the fixed order to – 1 – achieve the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Of these methods, the Geneva approach [4, 8] has the advantage of being particularly flexible with regard to the framework used for the resummed calculation and the choice of the resolution variable, while also exploiting the possibility of reaching higher logarithmic accuracies in both direct QCD and soft-collinear effective theory (SCET) formalisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This has resulted in the application of the method to a number of colour singlet processes [4, 9–15], as well as first steps towards implementations involving coloured final states [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this work, we describe a number of improvements to the Geneva event generator, which both extend the capabilities of the program and improve its numerical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Specifically, we detail a new treatment of the splitting functions, which were first introduced in the original Geneva implementation [8] and serve to make the resummed calculation used in the matching procedure differential in the higher multiplicity phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The new approach significantly increases the performance of the code in extreme soft and collinear regions, where the cancellation of large logarithmic terms is extremely delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We also implement a more rigorous treatment of the theoretical uncertainties by disentangling the factorisation and renormalisation scale dependences in the cross section and allowing their independent variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This puts our uncertainty estimation on a robust and more conservative theoretical footing, and will also prove important for the implementation of processes featuring perturbatively generated heavy flavours in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Finally, we also discuss the treatment of timelike logarithms in our calculation, the inclusion of which has been shown to improve the perturbative convergence of colour singlet production processes such as gg → H [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to study the various improvements to the Geneva program, we have imple- mented the gluon-initiated Higgs boson production process (gg → H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We use a resummed calculation in the zero-jettiness resolution variable T0 obtained via SCET up to N3LL ac- curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The process is interesting from both an experimental and a theoretical perspective for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Experimentally, the gluon-fusion production channel was of utmost importance for the discovery of the Higgs boson [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Nowadays Higgs physics remains a crucial aspect of the LHC programme [22–34], and constraining the scalar boson’s properties and couplings to probe the nature of the Higgs sector is a priority for Run 3 of LHC and beyond [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' From the theory side, many calculations work in the limit in which the top-quark mass is considered to be large compared to other scales present in the process, the so-called heavy-top limit (HTL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This significantly simplifies the computational complexity since the top-quark loop coupling the Higgs boson to gluons is integrated out, resulting in an effective ggH vertex and further effective vertices with more gluons and Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Consequently, calculations including QCD corrections up to N3LO are now available in this limit [36–43], including matching to resummed calculations up to N3LL′ accuracy in transverse momentum [43–51] and in jet veto observables [52–55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' There has also been a considerable amount of work on improving calculations beyond the HTL by including quark mass effects [56–60];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' this has culminated in a calculation of the exact top-quark mass dependence at NNLO in QCD [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Additionally, the fact that the perturbative series is known to be poorly convergent has motivated the study of alternative scale choices which – 2 – include π2 terms arising from kinematic logarithms at all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Finally, the simplicity of this process in terms of its kinematics and matrix elements makes it a particularly suitable testing ground for the improvements which we will detail in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The rest of the paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2, we provide a brief recap of the Geneva method and its application to gluon-induced Higgs production, before discussing the new features which have been implemented in the program in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 4 we validate the NNLO accuracy of our calculation for the gg → H process and discuss the matching to the parton shower provided by Pythia8 [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Finally, we compare our results with the pp → H → γγ data collected at the ATLAS and CMS experiments [64, 65] in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We give our conclusions in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2 Theoretical framework In the following we lay out the theoretical framework we work in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We start by giving a summary of the Geneva event generator formalism, which includes the matching procedure of the fixed-order calculation to the resummed prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We then focus on the definition of the process under study, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Higgs boson production via gluon fusion, and on its zero- jettiness resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 The GENEVA method The complete derivation of the Geneva method has been presented extensively in several publications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [4, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Here, we explicitly refrain from entering into the finer details of the method, and we only briefly recall the general formulae, highlighting some key features that are important for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We use N-jettiness [66] to resolve the QCD emissions that can be associated with each event produced by Geneva: T0 as the zero-jet resolution parameter, and T1 to separate between one or more emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The partonic event space is then divided into three regions: Φ0 for events with no extra emissions, Φ1 for one-jet events, and Φ2 for the remaining events with two jets in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' These phase space regions are defined via two thresholds, T cut 0 and T cut 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The differential cross section for the production of events with no extra emissions is given by dσmc 0 dΦ0 (T cut 0 ) = dσNNLL′ dΦ0 (T cut 0 ) − dσNNLL′ dΦ0 (T cut 0 ) ���� NNLO0 + (B0 + V0 + W0)(Φ0) + � dΦ1 dΦ0 (B1 + V1)(Φ1) θ � T0(Φ1) < T cut 0 � + � dΦ2 dΦ0 B2(Φ2) θ � T0(Φ2) < T cut 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1) Here we use the primed counting for the resummation order as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the – 3 – case of a single extra emission we have two contributions: that above T cut 0 dσmc 1 dΦ1 (T0 > T cut 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' T cut 1 ) = �� dσNNLL′ dΦ0dT0 − dσNNLL′ dΦ0dT0 ���� NLO1 � P(Φ1) + (B1 + V C 1 )(Φ1) � × U1(Φ1, T cut 1 ) θ(T0 > T cut 0 ) + � � dΦ2 dΦT 1 B2(Φ2) θ � T0(Φ2) > T cut 0 � θ(T1 < T cut 1 ) − dΦ2 dΦC 1 C2(Φ2) θ(T0 > T cut 0 ) � − B1(Φ1) U (1) 1 (Φ1, T cut 1 ) θ(T0 > T cut 0 ) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2) and the nonsingular below T cut 0 , arising from non-projectable configurations, dσmc 1 dΦ1 (T0 ≤ T cut 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' T cut 1 ) = (B1 + V1)(Φ1) Θ FKS map(Φ1) θ(T0 < T cut 0 ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3) Similarly the case of two extra emissions also receives two contributions, dσmc ≥2 dΦ2 (T0 > T cut 0 , T1 > T cut 1 ) = ��dσNNLL′ dΦ0dT0 − dσNNLL′ dΦ0dT0 ���� NLO1 � P(�Φ1) + (B1 + V C 1 )(�Φ1) � U ′ 1(�Φ1, T1) θ(T0 > T cut 0 ) ����Φ1=ΦT 1 (Φ2) × P(Φ2) θ(T1 > T cut 1 ) + � B2(Φ2) θ(T1 > T cut 1 ) − B1(ΦT 1 ) U (1)′ 1 ��Φ1, T1 � × P(Φ2) Θ(T1 > T cut 1 ) � θ � T0(Φ2) > T cut 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4) and dσmc ≥2 dΦ2 (T0 > T cut 0 , T1 ≤ T cut 1 ) = B2(Φ2) Θ T map(Φ2) θ(T1 < T cut 1 ) θ � T0(Φ2) > T cut 0 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5) above and below T cut 1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the formulae above, Bn, Vn and Wn are the 0-, 1- and 2-loop matrix elements for n QCD partons in the final state (including parton densities);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' analogously, we denote by NkLOn a quantity with n additional partons in the final state computed at NkLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Since it is necessary to evaluate the resummed and resummed-expanded terms on phase space points resulting from a projection from a higher to a lower multiplicity, we introduce a shorthand for such projected phase space points, �ΦN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We use the abbreviation dΦM dΦO N = dΦM δ[�ΦN − ΦO N(ΦM)] ΘO(ΦM) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6) to indicate an integration over the portion of the ΦM phase space which can be reached from a ΦN point while keeping some observable O also fixed, with N < M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The ΘO(ΦM) term additionally limits the integration to the phase space points belonging to the singular – 4 – contribution for the given observable O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For example, when generating 1-body events we use dΦ2 dΦT 1 ≡ dΦ2 δ[�Φ1 − ΦT 1 (Φ2)] ΘT (Φ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='7) where the 1 → 2 mapping has been constructed to preserve T0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' T0(ΦT 1 (Φ2)) = T0(Φ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8) and ΘT (Φ2) guarantees that the Φ2 point is reached from a genuine QCD splitting of the Φ1 point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The use of a T0-preserving mapping is necessary to ensure that the point- wise singular T0 dependence is alike among all terms in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4) and that the cancellation of said singular terms is guaranteed on an event-by-event basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The non-projectable regions of Φ1 and Φ2, on the other hand, are assigned to the cross sections in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' These events are entirely nonsingular in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We denote the constraints due to the choice of map by Θmap, using the FKS map [67] for the Φ1 → �Φ0 projection and, as mentioned above, a T0-preserving map for the Φ2 → �Φ1 projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The term V C 1 denotes the contributions of soft and collinear origins in a standard NLO local subtraction, V C 1 (Φ1) = V1(Φ1) + � dΦ2 dΦC 1 C2(Φ2) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='9) with C2 a singular approximant of B2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' in practice we use the subtraction counterterms which we integrate over the radiation variables dΦ2/dΦC 1 using the singular limit C of the phase space mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the formulae involving one or two extra emissions, U1 is a next-to-leading-logarithmic (NLL) Sudakov factor which resums large logarithms of T1, and U ′ 1 its derivative with respect to T1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' the O(αs) expansions of these quantities are denoted by U (1) 1 and U (1)′ 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We extend the differential dependence of the resummed terms from the N-jet to the (N +1)-jet phase space using a normalised splitting probability P(ΦN+1) which satisfies � dΦN+1 dΦNdTN P(ΦN+1) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10) The two extra variables are chosen to be an energy ratio z and an azimuthal angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The functional forms of the P(ΦN+1) are in principle only constrained by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' However, in order to correctly model the soft-collinear limit behaviour, we find it useful to write them in terms of the Altarelli-Parisi splitting kernels, weighted by parton distribution functions (PDFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In previous implementations of the Geneva method, the splitting functions P(ΦN+1) were computed using a “hit-or-miss” method based on precomputed upper bounds, which did not require knowledge of an analytic expression for the integration limits of z and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' At the same time, however, this introduced some numerical instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this work, we improve on this situation by including the exact integration limits and evaluate the splitting functions directly for each phase space point, as detailed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 5 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Higgs boson production via gluon fusion We consider the production of a stable Higgs boson via the gluon fusion channel in proton- proton scattering, pp → H + X, where X denotes any additional hadronic radiation in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' At leading order (LO) in the strong coupling this results in a single contribution gg → H at partonic level [68], while at next-to-leading order (NLO) (anti)quark-initiated channels also start to contribute [69–71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For a stable Higgs boson it is phenomenologically reasonable to work in the HTL ef- fective field theory (EFT), in which the contributions from the top-quark loops coupling the Higgs boson to gluons have been integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This EFT supplements the Standard Model (SM) vertices with additional, effective couplings between gluons and Higgs bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Introducing these effective vertices has the advantage of reducing the complexity of the matrix element computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The cross section dependence on the top-quark mass mt can be partially restored by rescaling the HTL result by a factor equal to the ratio be- tween the LO mt-exact result and that obtained in pure EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is later referred to as rescaled EFT (rEFT), and reproduces the exact mt dependence of the LO cross section by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' It is known to be a good approximation, for inclusive quantities, at least up to NNLO [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The resulting approximation can instead be problematic for differential distributions, for instance the transverse momentum of the Higgs boson when the accom- panying radiation resolves the top-quark loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' when its transverse momentum is larger than mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the case of a finite top-quark mass, the NNLO corrections have been recently calculated for the inclusive cross section [61, 62], and those at NLO for Higgs boson pro- duction in association with up to two hard jets [72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' At this level of precision, however, one also needs to take into account the interference between contributions including both massive top and bottom quarks, which is known at NLO for the Higgs plus jet case [74, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Since the problem of including the quark mass effects for precise phenomenological studies is largely independent of the matching of fixed-order and resummed calculations to par- ton showers in the Geneva method, which is the topic of the present study, we leave the investigation of these effects to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this work, the Higgs boson is always produced on shell with a mass mH =125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='09 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' When comparing with data in the fiducial regions of the ATLAS or CMS experiments, we will consider Higgs boson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this case we work in the narrow-width approximation, which for the Higgs boson is particularly accurate since ΓH/mH ∼ O(10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The Higgs decay products can always be added a posteriori due to the scalar nature of the boson, which implies that they are isotropically distributed without spin correlations with the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the rest of this work we will consider a collider energy of √ S = 13 TeV and assume the following values for the SM parameters affecting our calculations: GF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='16639 × 10−5, mt = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='11) For the matrix elements in the HTL approximation we use the heftpphj and heftpphjj libraries of OpenLoops2 [76–78], which we then rescale by the rEFT factor rEFT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='06545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 6 – 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 T0 resummation The formulae presented in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 require the evaluation of the resummed spectrum and cumulant in the resolution variable T0 up to at least NNLL′ accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Although the Geneva method does not depend on any particular resummation formalism, in practice we often find it convenient to exploit results derived via SCET [79–85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Within this framework, a factorisation theorem for the zero-jettiness was first derived in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [86, 87] for colour singlet production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the case of the gluon-fusion channel for Higgs production it reads dσSCET dΦ0dT0 = Hgg→H(Q2, µ) � dtadtb Bg(ta, xa, µ) Bg(tb, xb, µ) Sgg � T0 − ta + tb Q , µ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='12) where Hgg→H, Sgg, and Bg are the hard, soft and beam functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The process-specific hard function Hgg→H(Q2, µ) is defined as the square of the Wilson coefficient that results from matching the QCD Hamiltonian to the SCET operators, and encodes information about the Born and virtual squared matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' It depends only on the Higgs boson virtuality Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this section and whenever we consider Higgs boson production specifically, we set Q = mH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' elsewhere, we consider Q to be a generic hard scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Given that we work in the HTL approximation, we perform a two-step matching pro- cedure: we first integrate out the hard degrees of freedom above the top-quark mass, and subsequently match the resulting EFT onto SCET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The final hard function then arises from the product of two Wilson coefficients, the first from the HTL approximation and the second from the matching to SCET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' we evaluate both at the same scale µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In principle, within this approach one could resum ln(mt/mH) contributions by renormalisation group equation (RGE) evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' However, given the values of the top quark and Higgs boson masses, these logarithms are never large and, consequently, we include them only at fixed order in the hard function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Alternatively, if one wants to include the full top-quark mass effects, a single-step matching can be performed as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [52] at NNLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Extending this to NNLL′ accuracy requires the three-loop hard function with the exact top-quark mass dependence [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The beam functions Bg(t, x, µ) are the inclusive gluon beam functions [86], which depend on the transverse virtualities ta,b of the initial-state partons that participate in the hard interaction and on their momentum fractions xa,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' While they are nonperturbative objects, for t ≫ ΛQCD they admit an operator product expansion (OPE), Bi(t, x, µ) = � j � 1 x dξ ξ Iij � t, x ξ , µ � fj(ξ, µ) � 1 + O �Λ2 QCD t �� ≡ � j [Iij ⊗x fj] (t, x, µ) � 1 + O �Λ2 QCD t �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='13) where the Iij(t, z, µ) are matching coefficients that describe the collinear virtual and real initial-state radiation (ISR) and the fj(ξ, µ) are the usual PDFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For later use, we denote the Mellin convolution via the symbol ⊗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 7 – Finally, Sgg(k, µ) is the gluon hemisphere soft function for beam thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Like the beam functions, Sgg(k, µ) is a nonperturbative object and for k ≫ ΛQCD it also satisfies an OPE, where the LO matching coefficient is calculable in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Its perturbative component depends only on the colour representation of the hard partons, and therefore the gluon case can be derived from that of the quark channel via Casimir scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In our calculation we neglect the nonperturbative part of the soft function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We then rely on the hadronisation model of the parton shower to provide the missing contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The functions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='12) are all evaluated at a common scale µ and satisfy RGEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The scale dependence in each of these functions involves potentially large logarithms of ratios of disparate scales, which may impact their perturbative convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to reduce the effect of these large logarithms, we evaluate each function at its characteristic (canonical) scale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' µS = T0, µH = mH, and µB = √µSµH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Since the cross section needs to be evaluated at a common scale µ, we use the RGEs to evolve each function to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In doing so, we resum said logarithms at all orders in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The resummed formula for the T0 spectrum is then given by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [9] for more details) dσresum dΦ0dT0 = Hgg→H(Q2, µH) UH(µH, µ) × � dta dtb [Bg(ta, xa, µB) ⊗ UB(µB, µ)] [Bg(tb, xb, µB) ⊗ UB(µB, µ)] × � Sgg(T0 − ta + tb Q , µS) ⊗ US(µS, µ) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='14) where we denote the standard convolutions between the different functions and the RGE evolution factors via the ⊗ symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to achieve NNLL′ accuracy in the T0 resummation, each of the hard, soft and beam function boundary terms must be known at 2-loop order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the beam function they were calculated at 2-loops in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [88], and in fact they are known up to 3-loop order [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Our implementation of the gluon beam function relies on an interface to scetlib [43, 90, 91], a library which provides ingredients for resummed calculations in SCET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The soft function has been known at 2-loops for some time [92, 93], and recent work has aimed to push this calculation to the 3-loop order [94–96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The hard function has appeared several times in the literature, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [52, 97], and is known analytically with full top-quark mass dependence at NNLO [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In addition, the anomalous dimensions and the beta function which enter the evolution factors and the fixed-order expansion of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='14) must be known at 2-loop (noncusp [52]) and 3-loop (cusp [98–100], β(αs) [101, 102]) order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' By including them at one order higher [52, 103, 104], one can achieve resummation at N3LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The resummation of T0 for the case of Higgs boson production via gluon fusion has already been studied in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [52] up to NNLL accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the present work, we extend this calculation to NNLL′ and N3LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the determination of the canonical scales we employ the T0-dependent profile functions described e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [9] with {x0, x1, x2, x3} = {1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 GeV/mH, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='35, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The use of such T0 dependent scales is known to cause a difference between the integrated spectrum and the cumulant, which is formally of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is a result of the noncommutativity of the scale setting and the integration – 8 – steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In previous Geneva implementations, this problem has been alleviated by explicitly adding higher order terms to restore the cumulant cross section (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (45) of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This can be done either by using a ‘brute-force’ approach, in which the integrated spectrum is simply replaced by the cumulant, or by smoothly transitioning from one to the other as a function of T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In all Geneva implementations thus far we have followed the latter approach, which has the advantage of preserving the T0 spectrum in its peak region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the case of gg → H production at 13 TeV, the difference between the integrated spectrum and the cumulant amounts to ∼ 18% of the total cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Given the size of these corrections, we found the previously adopted solution to be insufficient to completely solve the mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In particular, our smooth fix modifies the T0 spectrum in the region between ∼ 10 and ∼ 30 GeV by too large an amount, moving the central value of the first outside the uncertainty bands of the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We therefore revert to the brute-force approach, and only require the preservation of the resummed cumulant cross section by fixing κ(T0) = 1 (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (45) of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [9]) such that the spectrum is exactly equal to the derivative of the cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3 Novel features of the GENEVA method In this section we discuss the new improvements that have been incorporated in the Geneva method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Here we focus on their impact on the gg → H process, however we note that they can be straightforwardly generalised to several other processes (and indeed have already been tested for Drell-Yan, double Higgs [15], and t¯t production [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Improved treatment of splitting functions The N-jettiness spectra computed through resummation techniques cannot be directly used for generating events with N + 1 final-state partons, since they do not carry a dependence on the full ΦN+1 configurations, but only on TN and the projected ΦN configurations with N final-state partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For this reason, a splitting function P(ΦN+1) was introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [8] in order to make the resummed calculation fully differential in the higher order phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In general, the N → N + 1 splitting function P(ΦN+1) is defined such that for every integrable function g(ΦN, TN) � dΦN+1 P(ΦN+1) g(ΦN, TN) = � dΦN dTN g(ΦN, TN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1) If the function g(ΦN, TN) is the TN spectrum, then multiplying it by the P functions makes it differential over the dΦN+1 phase space without affecting the distributions of observables that only depend on ΦN and TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to provide an explicit expression for P, we write the phase space of the ΦN+1 configurations with a valid ΦN projection as the product of dΦN, dTN and the phase space parametrised by two additional radiation variables z and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this way the integral over – 9 – the projectable ΦN+1 configurations at fixed ΦN and TN can be expressed as � dΦN+1 dΦN dTN = N+2 � k=1 � zmax k (ΦN,TN) zmin k (ΦN,TN) dz Jk(ΦN, TN, z) � φmax k (ΦN,TN,z) φmin k (ΦN,TN,z) dφ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2) where the index k runs over the N + 2 possible emitter partons (mothers) of the ΦN configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For each mother k and its associated mapping, we assume that the Jacobian Jk(ΦN, TN, z) = dΦN+1 dΦN dTN dz dφ ���� k (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3) does not depend on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is true for all the mappings considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The integral over the ΦN+1 configurations summed over the nreal partonic subprocesses with N + 1 final-state partons for a generic function gβ(ΦN+1) can now be written as nreal � β=1 � dΦN+1 gβ(ΦN+1) = nreal � β=1 � unproj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' ΦN+1 dΦN+1 gβ(ΦN+1) + (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4) nBorn � α=1 � dΦN dTN N+2 � k=1 � zmax k zmin k dz Jk(ΦN, TN, z) � φmax k φmin k dφ nsplit k� j=1 gk→i+j α (ΦN, TN, z, φ) , where nBorn is the number of subprocesses with N final-state partons, and nsplit k the number of possible QCD splittings k → i+j, with i the emitted parton and j the sister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The function gk→i+j α (ΦN, TN, z, φ) on the right-hand side is equal to gβ(ΦN+1) expressed in terms of the underlying Born process index α and the splitting indices k and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For ease of notation, the full dependence of the z and φ integration limits on the phase space variables is not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The unprojectable ΦN+1 configurations are those for which either the two closest partons do not represent a valid QCD splitting, the ΦN configuration obtained from the projection is not kinematically allowed, or the flavour configuration of the ΦN is invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to fulfil the condition presented in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1), we choose splitting functions P(ΦN+1) that depend on the mother and sister indices and vanish in the unprojectable ΦN+1 configurations: P(ΦN+1) = � 0 if ΦN+1 is unprojectable, Pkj(ΦN, TN, z, φ) if ΦN → ΦN+1 via the k → i + j splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5) The Pkj must then satisfy the equation N+2 � k=1 � zmax k (ΦN,TN) zmin k (ΦN,TN) dz Jk(ΦN, TN, z) � φmax k (ΦN,TN,z) φmin k (ΦN,TN,z) dφ nsplit k� j=1 Pkj(ΦN, TN, z, φ) = 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6) – 10 – for all values of ΦN and TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Without loss of generality, in the projectable ΦN+1 configu- rations we can express them as Pkj(ΦN, TN, z, φ) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='7) fkj(ΦN, TN, z, φ) N+2 � k′=1 � zmax k′ (ΦN,TN) zmin k′ (ΦN,TN) dz′Jk′� ΦN, TN, z′� � φmax k′ (ΦN,TN,z′) φmin k′ (ΦN,TN,z′) dφ′ nsplit k′ � j′=1 fk′j′� ΦN, TN, z′, φ′� , where fkj is a generic function that we specify later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' If we choose it to be independent of φ, the above expression simplifies to Pkj(ΦN, TN, z) = (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8) fkj(ΦN, TN, z) N+2 � k′=1 � zmax k′ (ΦN,TN) zmin k′ (ΦN,TN) dz′ Jk′� ΦN, TN, z′� ∆φk′� ΦN, TN, z′� nsplit k′ � j′=1 fk′j′� ΦN, TN, z′� , where ∆φk(ΦN, TN, z) = φmax k (ΦN, TN, z) − φmin k (ΦN, TN, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to perform the integral in the denominator of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8), we compute the integration limits on z and φ and the Jacobian Jk both for the 0 → 1 and 1 → 2 splitting mappings for each ΦN+1 configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the previous Geneva implementation of the splitting functions the computation of the integration limits was avoided by precomputing their upper bounds and then using a “hit-or-miss” integration method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We highlight that, whenever the constraints on z and φ are in the form of an inequality involving both the variables, we only compute an overestimate of the true integration limits on z analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We then determine the true limits numerically by imposing the condition ∆φk(ΦN, TN, z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Infrared limits For this section we introduce the acronyms ISRA (initial-state radiation A), ISRB (initial- state radiation B), and FSR (final-state radiation) to indicate the N + 2 possible mothers we have to deal with: the parton from the first (A) and second (B) beam, respectively, and the final-state partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We furthermore label ISRA and ISRB collectively as ISR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The exact form of the function fkj in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='7) can significantly affect the efficiency of the Monte Carlo event generator in the region of small TN > T cut N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this region, the logarithmically enhanced terms appearing in the fixed-order calculation have to cancel those coming from the resummed-expanded contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For this reason the main criterion we follow in the choice of fkj is to achieve a good approximation of the behaviour of the associated matrix element in the infrared limit when TN → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For simplicity, in practical applications we choose not to include the azimuthal depen- dence in the form of the fkj functions, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We define fkj(ΦN, TN, z) = � � � � � αs(µR) fA a (xa, µF ) fB b (xb, µF ) z ˆPjk(z) if k is ISR, αs(µR) fA a (xa, µF ) fB b (xb, µF ) ˆPkj(z) if k is FSR, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='9) – 11 – where a and b are the initial-state partons, αs(µR) is the strong coupling evaluated at the renormalisation scale µR, and fH i (xi, µF ) is the PDF of the parton i in the hadron H evaluated at longitudinal momentum fraction xi and factorisation scale µF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The renormal- isation and factorisation scales are fixed to µR = µF = Q, where Q is the virtuality of the colour singlet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The ˆPkj are the unregulated Altarelli-Parisi splitting functions ˆPqq(z) = CF 1 + z2 1 − z , ˆPqg(z) = TF � z2 + (1 − z)2� , ˆPgq(z) = CF 1 + (1 − z)2 z , ˆPgg(z) = 2CA � z 1 − z + 1 − z z + z (1 − z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10) We highlight that for the 0 → 1 splitting, connecting events with no extra partons to events with one extra parton, the PDFs are evaluated at the exact momentum fractions xa(z) and xb(z) of the real emission phase space Φ1 rather than their infrared limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This has proven to be necessary to obtain an accurate description also in the tail of the colour singlet transverse momentum distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We note that in this case we also reproduce the correct soft limit, as shown in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the 1 → 2 splitting the true xa and xb also depend on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this case they are approximated by dropping this additional dependence, which still represents an improvement with respect to the strict collinear limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Soft limit of the 0 → 1 splitting In the following we show that the expression of fkj introduced in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='9) correctly repro- duces both the singular soft and collinear limits at O(αs) in the 0 → 1 splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the case of colour singlet production in hadron-hadron collisions, let us consider the k → i + j splitting connecting the Born matrix element B0 and the real matrix element B1 (in both cases excluding parton densities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This can be expressed in terms of the FKS variables ξ = 2 E/√s = 1 − Q2/s and y = cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Here, s is the partonic centre-of-mass energy squared, and E and θ are the energy of the emitted parton and the angle between the emitted and the right-moving incoming parton in the partonic centre-of-mass frame, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the soft limit of the emitted particle i, we have lim ξ→0 B1 = 64παs(µR) Q2 Ck ξ2 (1 − y2) B0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='11) where Ck = CF for the quark-initiated processes and Ck = CA for the gluon-initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the azimuthally averaged collinear limit between particles i and j, we have lim y→±1 B1 = 16παs(µR) Q2 1 − ξ ξ (1 ∓ y) ˆPjk(1 − ξ) B0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='12) where y → 1 and y → −1 represent the collinear limits with respect to incoming parton a and b respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' If the colour singlet production process is quark-initiated or has only scalar particles in the final state, the above expressions also hold prior to averaging over the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 12 – We consider a configuration with one final-state parton with momentum p, where T0 = ˆp± , z = Q/(Q + ˆp∓) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='13) Here ˆp is obtained by longitudinally boosting p from the laboratory frame to the frame where the colour singlet has zero rapidity, and ˆp± = ˆp0 ∓ ˆp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We have chosen z such that in the collinear limit it reduces to the energy fraction of the emitter with respect to the sister, while providing the correct scaling also for the single soft limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to show that the singular limits in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='9) reproduce the above results, we rewrite T0 and z in terms of the FKS variables ξ and y, and then compare the ensuing expression to eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' They read T0 = Q ξ 2√1 − ξ (1 ∓ y) � 2 − ξ (1 ± y) 2 − ξ (1 ∓ y) , z = � 1 + ξ (1 ± y) 2√1 − ξ � 2 − ξ (1 ∓ y) 2 − ξ (1 ± y) �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='14) Therefore in the infrared singular limit one obtains T0 → Q ξ 2 (1 ∓ y) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='15) z → 1 − ξ 2 (1 ± y) in the soft limit, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='16) z → 1 − ξ in the collinear limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='17) Multiplying the NLL singular T0 spectrum expanded at O(αs) by the splitting func- tions, in the infrared limit we find Pkj(Φ0, T0, z) dσNLL dΦ0dT0 ���� O(αs) → 8παs(µR) Q T0 z ˆPjk(z) fa(xa, µF ) fb(xb, µF ) B0(Φ0) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='18) up to power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' By using the above expressions for T0 and z, it can be shown that this reproduces both the soft and collinear limits given in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We remark that with the choice of z given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='13) the soft limit can be entirely captured by using the Altarelli-Parisi splitting collinear kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The validity of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='18) can be understood to be a consequence of the fact that the noncusp soft anomalous dimension is zero at one loop order, resulting in the lack of a single logarithmic contribution to the T0 spectrum coming from the soft function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 Numerical validation In this section we present the effects of the improved splitting function Pimpr implemen- tation described above in the case of Higgs boson production via gluon fusion, setting Q = mH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' we focus on the pH T and the T0 spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We compare the results of a fixed- order calculation with those obtained by truncating the resummation formula in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='– 13 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσ / dT0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='pp → H + X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Geneva NLO+NLL′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='S = 13 TeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='rEFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='fixed order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Pimpr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Porig ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσNS / dT0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Pimpr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Porig) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσ / dpH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='pp → H + X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Geneva NLO+NLL′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='S = 13 TeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='rEFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='fixed order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Pimpr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Porig ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='pH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσNS / dpH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Pimpr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Porig) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Figure 1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Comparison of the fixed-order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' singular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' and nonsingular distributions at NLO+NLL′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' both for T0 (left) and pH T (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We show the singular and nonsingular distributions both for the original and improved versions of the splitting function imple- mentation in Geneva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' multiplied by the splitting function to the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We do so for the results at LO1 compared with the NLL′ resummed-expanded in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 1, and for those at NLO1 compared with the NNLL′ resummed-expanded in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We also show the nonsingular contribution, defined as the difference between these fixed-order and resummed-expanded pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In all plots we also show the results obtained with the original implementation Porig of the split- ting function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8), which was based on a hit-or-miss method using upper bounds tabulated on a grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We begin the discussion with the results for the T0 distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' As expected, the improved implementation gives identical results to the original, both at LO1 and NLO1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is a consequence of the fact that T0 is preserved by the splitting, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We observe that at extremely low values of T0 the presence of technical cuts in the fixed-order calculation affects the convergence to the singular predictions in both approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' When instead considering the LO1 results for the pH T distribution, we notice how the improved implementation of the splitting functions correctly captures the logarithmic behaviour of the matrix element at fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This can be seen by the fact that the improved non- singular distribution converges to zero, contrary to the original case which converges to a finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Similarly, an improvement is also visible for the NLO1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='the new splitting function Pimpr is not able to exactly reproduce the complete logarith- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='– 14 – ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσ / dT0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='pp → H + X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Geneva NNLO+NNLL′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='S = 13 TeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='rEFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='fixed order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Pimpr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Porig ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσNS / dT0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Pimpr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Porig) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−1250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσ / dpH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='pp → H + X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Geneva NNLO+NNLL′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='S = 13 TeV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='rEFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='fixed order ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Pimpr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='singular × Porig ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='pH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='−50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='dσNS / dpH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='[pb/GeV] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Pimpr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='nonsingular (Porig) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Figure 2: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='Comparison of the fixed-order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' singular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' and nonsingular distributions at NNLO+NNLL′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' both for T0 (left) and pH T (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We show the singular and nonsin- gular distributions both for the original and improved versions of the splitting function implementation in Geneva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' mic behaviour of the NLO1 result, as it appears to miss a single logarithmic contribution ∼ 1/pH T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is implied by the fact that the improved nonsingular contribution converges to a nonzero constant at low values of pH T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This must however be compared with the orig- inal approach, Porig, where the divergent behaviour of the nonsingular plot suggests that that implementation also fails to capture the logarithmic structure up to ∼ ln2(pH T )/pH T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Independent scale variations In traditional implementations of fixed-order QCD calculations, a differentiation is made between the factorisation scale µF and the renormalisation scale µR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The former is associ- ated with the scale of collinear factorisation, while the latter is introduced in dimensional regularisation in order to render the strong coupling dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' To date, implementations of Geneva have assumed these scales to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Doing so facilitated the matching to the resummed calculation, where a sole “nonsingular” scale µNS appears as the endpoint of the RGE running, typically taken to be a hard scale Q of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The two scales were then varied in a correlated fashion (“diagonal” in the {µR, µF } space) when probing the higher order uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This approach, however, can hinder a complete and thorough uncertainty estimation as it neglects those variations which are off-diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' where µR and µF are varied independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this section we provide an improved and robust uncertainty estimation within the Geneva framework – 15 – by exposing the µF dependence of the singular cross section that eventually allows for off-diagonal scale variations, and discuss the choice of µF in the infrared region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Exposing the µF dependence of the singular cross section The collinear beam functions Bi entering the T0 factorisation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='12) satisfy the OPE in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In resummed predictions, they are evaluated at a scale µ = µB where µB minimises the singular logarithmic structure of Bi, whereas at fixed order µ = µR = µF = Q, where for example Q = mH for on-shell Higgs boson production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to expose the µF dependence of the beam functions, we rewrite eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='13) as Bi(t, x, µ) = � j Iij(t, x, µ) ⊗x fj(x, µ) = � j,k Iik(t, x, µ) ⊗x Ukj(x, µ, µF ) ⊗x fj(x, µF ) ≡ � j ˆIij(t, x, µ, µF ) ⊗x fj(x, µF ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='19) where we dropped the power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Here we evolve the PDFs from µF to µ using the evolution kernel Uij(x, µ, µF ) that results from the solution of the DGLAP equations, µ d dµfi(x, µ) = 2 � j Pij(x, µ) ⊗x fj(x, µ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='20) and we follow the conventions of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [90] for the perturbative expansion of the splitting kernels Pij(x, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Although the µF dependence in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='19) cancels exactly between the PDFs and the evolution kernel, as soon as ˆIij is truncated at a given order, a residual µF dependence appears in the beam function, Bi(t, x, µ) �→ Bi(t, x, µ, µF ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='21) In order to manifest this dependence explicitly, we note that the matching coefficients and the evolution kernel in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='19) admit perturbative expansions in the strong coupling constant, ˆIij(t, x, µ, µF ) = δ(t)1ij(x) + ∞ � n=1 ˆI(n) ij (t, x, µ, µF ) �αs(µ) 4π �n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='22) Uij(x, µ, µF ) = 1ij(x) + ∞ � n=1 U(n) ij (x, µ, µF ) �αs(µ) 4π �n (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='23) where 1ij(x) ≡ δijδ(1 − x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We first obtain closed-form expressions for the Uij, which can be achieved either by directly solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='20) or by using the solutions of the RGE 1The splitting functions used here are multiplied by a factor of 2 with respect to the unregulated ones defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 16 – satisfied by Iij (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='17) of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [90]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We have µ d dµU−1 ij (x, µ, µF ) ⊗x fj(x, µ) = −U−1 ij (x, µ, µF ) ⊗x µ d dµfj(x, µ) = −2 U−1 ik (x, µ, µF ) ⊗x Pkj(x, µ) ⊗x fj(x, µ) ⇒ µ d dµU−1 ij (x, µ, µF ) = −2 U−1 ik (x, µ, µF ) ⊗x Pkj(x, µ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='24) where U−1 ij denotes the inverse of Uij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Here and in the following, repeated flavour indices are implicitly summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We note that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='24) is exactly the same as eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='17) of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [90] if we set γB = γν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' It is therefore straightforward to use its solution, which up to O(α2 s) reads U−1 (0) ij (x, µ, µF ) = 1ij(x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25) U−1 (1) ij (x, µ, µF ) = −2 L P (0) ij (x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='26) U−1 (2) ij (x, µ, µF ) = 2 L2� P (0) ik (x) ⊗x P (0) kj (x) − β0P (0) ij (x) � − 2 L P (1) ij (x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='27) where we have abbreviated L = ln(µ/µF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Solving the closure equation of the evolution kernels U−1 ik (x, µ, µF ) ⊗x Ukj(x, µ, µF ) = 1ij(x) at each order in perturbation theory yields U(1) ij (x, µ, µF ) = 2 L P (0) ij (x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='28) U(2) ij (x, µ, µF ) = L2� 2β0P (0) ij (x) + 2P (0) ik (x) ⊗x P (0) kj (x) � + 2 L P (1) ij (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='29) Substituting eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='28), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='29) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='19), we arrive at the explicit results for the µF - dependent matching coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' They read ˆI(1) ij (t, x, µ, µF ) = I(1) ij (t, x, µ) + 2 δ(t) L P (0) ij (x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='30) ˆI(2) ij (t, x, µ, µF ) = I(2) ij (t, x, µ) + L1(t, µ2) � 2 L Γi 0 P (0) ij (x) � + L0(t, µ2) � −γi B 0 L P (0) ij (x) + 2 L P (0) ik (x) ⊗x P (0) kj (x) � + δ(t) � L2� 2β0P (0) ij (x) + 2P (0) ik (x) ⊗x P (0) kj (x) � + 2 L � P (1) ij (x) + I(1) ik (x) ⊗x P (0) kj (x) �� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='31) where the expressions for the (µF -independent) matching coefficients I(n) ij (t, x, µ), the anomalous dimensions γi B 0 and Γi 0, and the plus distributions Ln can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Choice of the factorisation scale The choice of the factorisation scale µF is in principle subject to different requirements in the fixed-order and in the resummation region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to minimise the size of the logarithms L = ln(µ/µF ) in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='30) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='31), in the resummation region we demand that µF ∼ µB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' the beam scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' On the other hand, in the fixed-order region, a natural scale setting is µF ∼ µR ∼ µNS so that the fixed-order perturbative convergence is not – 17 – jeopardised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' However, given that the beam function profile scale µB(T0) flows to µNS in the fixed-order region, µB T0→Q −−−−→ Q ≡ µNS , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='32) choosing µF = µB satisfies both conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' When considering the scale variations for the estimation of the theoretical uncertainty, the definition of the profile scales is extended to µS(T0/Q, α) = κR µNS frun(T0/Q) fα vary(T0/Q) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='33) µB(T0/Q, α, β) = κR µNS � frun(T0/Q) fα vary(T0/Q) �1/2−β , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='34) µF (T0/Q, αf, β) = κF µNS � frun(T0/Q) fαf vary(T0/Q) �1/2−β , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='35) where the central predictions are obtained setting κR = κF = 1, α = αf = β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The function frun is defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [9] and the function fvary in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We note that a different parameter is used in the exponent of the function fvary for µB and µF in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='35) above, while we use the same β parameter for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is justified because the β-variations are introduced in order to disentangle the variations of µB and µS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' no ratios of µF /µS appear in the singular cross section, therefore there is no need for an independent β parameter in µF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We also extend the fixed-order uncertainty ∆FO to include the off-diagonal µF and µR = κR µNS scale variations, resulting from the envelope of a 7-point scale variation (κR, κF ) = {(1, 1), (2, 2), (1/2, 1/2), (1, 2), (1, 1/2), (2, 1), (1/2, 1)} , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='36) where we have excluded the cases when µF and µR are varied in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The resummation uncertainty ∆res estimated by the usual profile scale and transition point variations [105] (already present in Geneva) is extended by considering in addition the variations of the parameters α, αf, and β appearing in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Explicitly, we extend the original procedure to include the additional parameter variations: (β, α, αf) = {(0, 0, 0), (1/6, 0, 0), (−1/6, 0, 0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='37) (0, 1, 1), (0, −1, −1), (0, 1, 0), (0, −1, 0), (0, 0, 1), (0, 0, −1)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' These variations are enveloped with the variations of the transition points xi, and are summed in quadrature with the fixed-order variations in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='36) to obtain the final theoretical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 Treatment of timelike logarithms Radiative corrections in colour singlet production processes such as Drell-Yan or gluon- initiated Higgs production contain Sudakov logarithms of the form αn s lnm(−q2/µ2 R) , m ≤ 2n, where qµ is the momentum of the colour singlet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' They primarily arise in the calculation of the corresponding form factor, and their coefficients are linked to the structure of its infrared singularities [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For such processes qµ is a timelike vector, – 18 – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' q2 > 0, and the scale choice µ2 R = q2 results in the Sudakov logarithms developing an imaginary part, since lnm(−1) = (±iπ)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The presence of such ‘timelike’ logarithms at each order might negatively affect the perturbative convergence of the cross section, where they result in additional terms proportional to powers of π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The severity of this effect is process specific;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' for Drell-Yan or gluon-initiated Higgs production it has been explicitly studied for both exclusive [45, 46, 52, 54, 87, 105, 107–109] and inclusive [19, 43] observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' A way to mitigate the impact of said contributions is to choose µR such that timelike logarithms are eliminated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' to evaluate the form factor at the complex scale µR = −i|q| = −i Q [110–113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In factorised singular cross sections, the square of the form factor naturally appears as the hard function H(Q2, µH) of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Following the above discussion, the hard function can be evaluated at the complex scale µH = −i Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This choice implies a nontrivial renormalisation group evolution between µH and the real scale Q, in the form of a rotation in the complex µ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This procedure is also referred to as ‘timelike resummation’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' It has been applied to a multitude of exclusive and inclusive processes, including the case of Higgs boson production, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [18, 19, 43, 52, 87, 114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In addition, it has been shown that not only the singular, but also the nonsingu- lar [52, 54] and the total [19] cross section can benefit from this prescription, in terms of an improved perturbative convergence and a reduced ensuing scale uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This can be better understood when one considers that by integrating the timelike resummed exclusive cross section, one obtains the corresponding resummed inclusive prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the non- singular contribution, a factorisation formula remains in general unknown, meaning that it cannot be directly evaluated at the complex scale µH – it still, however, contains the form factor plagued by timelike logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this case, the procedure for the treatment of these logarithms involves re-expanding the nonsingular contribution, extracting from it the hard function evaluated at Q, and replacing it with that evaluated at the scale µH, as detailed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In our implementation we perform these steps at the same order as the T0 resummation, both in the singular and the nonsingular terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This implies that the improved perturbative convergence following the choice of a complex-valued scale µH = −i Q does not only apply to the singular T0 spectrum but also to the inclusive predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to study the uncertainty associated with this choice of scale, we follow the pre- scription introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [19], designed to probe the structure of the timelike logarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The uncertainty ∆ϕ is estimated by the envelope of the phase variations µH = Q e−iϕ , ϕ ∈ [π/4, 3π/4] , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='38) while the central value predictions correspond to ϕ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Since there is no dynamical parameter governing the choice of the scale µH, the timelike resummation is performed throughout the T0 spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' even when T0 resummation is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We therefore consider the uncertainty resulting from variations in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='38) as an independent source and add it to the other uncertainties in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Thus, for inclusive predictions we have ∆2 incl = ∆2 FO + ∆2 ϕ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='39) – 19 – 0 5 10 15 dσ / dyH [pb] pp → H + X √ S = 13 TeV rEFT µH = mH µH = −i mH −4 −3 −2 −1 0 1 2 3 4 yH −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ratio − 1 10−3 10−2 10−1 100 dσ / dT0 [pb/GeV] pp → H + X √ S = 13 TeV rEFT µH = mH µH = −i mH 1 10 20 30 40 50 100 150 T0 [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 ratio − 1 Figure 3: Comparison of the yH (left) and T0 (right) distributions with different choices of µH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' whereas for exclusive predictions we use ∆2 excl = ∆2 FO + ∆2 res + ∆2 ϕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='40) In the Geneva implementation of the gg → H process we use the hardfunc module from scetlib [91] for the hard function evaluation and evolution in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Since for this process we set Q = mH, we pick µH = −i mH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='41) With this choice, we observe a difference in the total cross section result with respect to the µH = mH case that can be substantial despite being formally of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The effects of the complex choice of scale µH on differential observables are illustrated in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 3, where we compare predictions at NNLO+NNLL′ for the T0 and yH distributions with µH = mH and µH = −i mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this and the following figures, the theoretical uncertainty is shown as a shaded band, while the Monte Carlo integration errors are shown as thin vertical bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the Higgs boson rapidity distribution, we observe an increase of around 10% that is almost independent of yH, and a reduction in the uncertainty band as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The T0 spectrum shows a larger effect, especially in the tail of the distribution, where our prediction is entirely driven by the fixed-order result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Nonetheless, we observe a reduction in the uncertainty band particularly in the peak and transition regions of the spectrum, between 5 and 45 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 4 Validation of the gg → H process In this section we validate our predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We first compare our partonic NNLO results with two independent calculations, and then discuss the interface to the Pythia8 shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 20 – Geneva ggHiggs Matrix σNNLO, rEFT gg→H [pb] 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='33+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='39 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='34 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='35+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='55 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='41 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='33+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='54 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='40 Table 1: Comparison of the Geneva, ggHiggs, and Matrix results for the gg → H inclusive cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The results are obtained at NNLO in the HTL approximation, and rescaled with the rEFT factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 Partonic results at NNLO Here we validate the NNLO accuracy of the total cross section obtained with Geneva and that of the only differential inclusive quantity available, the Higgs boson rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We compare the total cross section with the independent calculations implemented in ggHiggs [60, 115–118] and Matrix [119], and the rapidity distribution with Matrix only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The Matrix predictions are based on the qT -subtraction approach and are extrapolated towards the zero qT -cut value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We set the input parameters of our calculations as described in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2, and we choose the central factorisation and renormalisation scales equal to each other and to the Higgs boson mass, µR = µF = mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We set our resolution cutoffs to T cut 0 = T cut 1 = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We employ the PDF set PDF4LHC15 nnlo 100 from LHAPDF [120], and take the value of αs(mZ) from the same set, so that αs(mH) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='11263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In table 1 we report the values of the inclusive gg → H cross section and the as- sociated 7-point scale variations calculated at NNLO and rescaled with the rEFT factor using Geneva, ggHiggs, and Matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We observe excellent agreement between the three predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' by choosing T cut 0 = 1 GeV, the neglected power-suppressed terms in Geneva are at the permille level and amount to an acceptable ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='02 pb error for the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 4 we compare the Higgs rapidity spectrum obtained with Geneva with the NNLO result provided by Matrix, including the 7-point scale variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We observe very good agreement both in the central values and in the envelope of the scale variations, up to large values of |yH|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The symmetry of the pp collider allows us to show only the absolute value of yH, and thus further reduce the Monte Carlo uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 Interface with PYTHIA8 In this section we briefly recap the main features of the interface used in Geneva to match the partonic results to the Pythia8 [121] parton shower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' As this is not the main focus of this work, however, we refer the interested reader to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [4] for a detailed discussion and ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [15] for additional details on the accuracy of the matched calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Given that so far we have constructed partonic results with NNLL′ accuracy in the resolution variable T0, we wish to preserve this resummed accuracy after the parton shower as far as is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' At the same time, for all other observables we need to guarantee that the accuracy of the parton shower is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This is a nontrivial condition: since the ordering variable of the Pythia8 parton shower is the relative transverse momentum while the resolution variable we use is the N-jettiness, the shower can in principle produce emissions which double-count regions of the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 21 – 0 5 10 15 20 25 dσ / d|yH| [pb] pp → H + X √ S = 13 TeV rEFT µH = mH MATRIX Geneva 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 |yH| −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ratio − 1 Figure 4: Comparison of Geneva and Matrix at NNLO for the yH distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' To avoid this issue, we perform the matching employing the following prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We set the starting scale of the parton shower by taking the maximum relative k⊥ determined by the lower scale of the resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The latter is defined on an event-by-event basis and corresponds to either T c N ≡ T cut 0 , T cut 1 or T1 (Φ2), depending on whether the relative partonic configuration has N = 0, 1 or 2 jets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We then let the shower run down to the internal minimum p⊥, which produces a certain number of emissions k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Lastly, we check that the resulting event fulfils the condition TN(ΦN+k) ≤ T c N , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1) which ensures that both accuracies are correctly preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 5 we show the effect of the Pythia8 shower on the pH T and yH partonic distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the results presented in this section we use the default Pythia8 parameters for the shower and the hadronisation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The rapidity distribution, being an inclusive observable, is exactly preserved by the shower, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The Higgs transverse momen- tum is an exclusive observable, and the shower can therefore have a significant impact on its shape: in this case we see an effect of ∼ 15% in the pH T < 15 GeV bin, and smaller effects ≲ 5% in the rest of the spectrum, especially in the tail of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' After hadronisation, we find that most of these discrepancies are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The parton shower and hadronisation effects on the T0 distribution are displayed in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' As mentioned above, our matching procedure to the Pythia8 shower is designed with the aim that the T0 logarithmic accuracy is not spoiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We explicitly check this in the left panel, where we compare the T0 distribution at NNLL′ before and after the parton shower matching with the partonic prediction at N3LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Note that for this process, which is gluon-initiated, one expects that the parton shower effects are larger than for quark-initiated processes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' because of the larger Casimir factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Nonetheless, in the peak region T0 < 25 GeV, we find that the showered distribution lies in between the – 22 – 10−3 10−2 10−1 100 dσ / dpH T [pb/GeV] pp → H + X √ S = 13 TeV rEFT Geneva Geneva+Pythia8 Geneva+Pythia8 (hadronised) 0 50 100 150 200 250 300 pH T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ratio − 1 0 5 10 15 dσ / dyH [pb] pp → H + X √ S = 13 TeV rEFT Geneva Geneva+Pythia8 Geneva+Pythia8 (hadronised) −4 −3 −2 −1 0 1 2 3 4 yH −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 ratio − 1 Figure 5: Comparison of the partonic, showered, and hadronised results for the pT H (left) and yH (right) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 dσ / dT0 [pb/GeV] pp → H + X √ S = 13 TeV rEFT Geneva (NNLL′+NNLO) Geneva (NNLL′+NNLO)+Pythia8 Geneva (N3LL+NNLO) 0 5 10 15 20 25 T0 [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ratio − 1 10−3 10−2 10−1 100 dσ / dT0 [pb/GeV] pp → H + X √ S = 13 TeV rEFT Geneva Geneva+Pythia8 Geneva+Pythia8 (hadronised) 1 10 20 30 40 50 100 150 T0 [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 ratio − 1 Figure 6: Effects of the parton shower on the T0 spectrum: comparison of the partonic, showered, and N3LL-resummed distribution (left), and comparison of the partonic, show- ered, and hadronised results (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' central NNLL′ and N3LL curves, and within the overlap of the two uncertainty bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We therefore conclude that the quantitative effects of the shower are on par with (or smaller than) the effects of the next logarithmic order in the resummation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The hadronisation effects on the T0 distribution are displayed in the right panel of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' As expected for this observable, we observe O(1) effects in the peak region, which decrease for larger values of T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the region around T0 ≈ mH/2, which corresponds – 23 – 1 10 102 σfid [fb] pp → (H → γγ) + X √ S = 13 TeV rEFT ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH = 0 = 1 = 2 ≥ 3 Njets −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 ratio − 1 0 10 20 30 40 50 60 70 80 dσ / d|yH| [fb] pp → (H → γγ) + X √ S = 13 TeV rEFT ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 |yH| −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 ratio − 1 10−6 10−5 10−4 10−3 10−2 10−1 100 dσ / dpH T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0 20 50 100 200 500 pH T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 10−6 10−5 10−4 10−3 10−2 10−1 100 dσ / dpj1 T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0 30 60 90 120 350 pj1 T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 Figure 7: Comparison of the ATLAS data [64] with the Geneva+Pythia8 results at 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We show the fiducial cross sections for different values of Njets (top left), as well as the distributions of |yH| (top right), pH T (bottom left), and pj1 T (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' to the point at which the T0 resummation is switched off, we find a more pronounced discrepancy between the Geneva partonic and showered results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We have verified that this is an artefact related to our choice of setting the T0 spectrum equal to the derivative of the cumulant as explained at the end of sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 5 Comparison with LHC data We compare the predictions obtained with Geneva with the latest experimental results for the Higgs boson inclusive and differential cross sections in the H → γγ decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The results are provided both by the ATLAS [64] and CMS [65] experiments, and are obtained – 24 – 10−3 10−2 10−1 100 dσ / dpH T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT Njets =0 0 < τ j C GeV < 15 15 < τ j C GeV < 25 25 < τ j C GeV < 40 40 < τ j C GeV < 400 ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0-350 0-100 100-350 0-120 120-350 0-200 200-350 0-250 250-350 pH T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 10−3 10−2 10−1 100 dσ / dpH T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT 0 < |yH| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 < |yH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 < |yH| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 < |yH| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0-45 45-120 120-350 0-45 45-120 120-350 0-45 45-120 120-350 0-45 45-120 120-350 pH T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 dσ / dτ j1 C [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0 5 15 25 40 τ j1 C [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6 dσ / dpHj T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT ATLAS, 139 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH+bbH+tH 0 30 60 120 pHj T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 Figure 8: Comparison of the ATLAS data [64] with the Geneva+Pythia8 results at 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We show the pH T distributions in bins of τ j1 C (top left) and of |yH| (top right), as well as the τ j1 C (bottom left) and the pHj T (bottom right) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' from the LHC data at a centre-of-mass energy of 13 TeV using 139 fb−1 and 137 fb−1 of proton-proton collision data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the ATLAS measurement [64], the fiducial phase space is identified by requiring the existence of two isolated photons with pγ T > 25 GeV in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Photons are considered isolated if the transverse energy of charged particles with pT > 1 GeV within a cone of radius Riso = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 around the photon direction does not exceed 5% of the photon’s transverse momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The two isolated photons must additionally have transverse momenta larger than 35% and 25% of the diphoton invariant mass, for the leading and subleading photons respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The invariant mass of the diphoton system – 25 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 1 10 102 σfid [fb] pp → (H → γγ) + X √ S = 13 TeV rEFT CMS, 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH = 0 = 1 = 2 = 3 ≥ 4 Njets −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 ratio − 1 0 10 20 30 40 50 60 70 80 dσ / d|yH| [fb] pp → (H → γγ) + X √ S = 13 TeV rEFT CMS, 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 |yH| −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='50 ratio − 1 10−5 10−4 10−3 10−2 10−1 100 dσ / dpH T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT σ(pH T >450 GeV) 100 GeV CMS, 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 0 20 50 100 200 500 pH T [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4 dσ / dpj1 T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT σ(pj1 T >200 GeV) 50 GeV CMS, 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 50 75 100 125 150 175 200 225 250 pj1 T [GeV] −1 0 1 ratio − 1 Figure 9: Comparison of the CMS data [65] with the Geneva+Pythia8 results at 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We show the fiducial cross sections for different values of Njets (top left), as well as the distributions of |yH| (top right), pH T (bottom left), and pj1 T (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' must be in the range 105 GeV < mγγ < 160 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Moreover, photons are required to have pseudorapidity |ηγ| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='37 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='52 < |ηγ| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For this measurement, jets are defined using the anti-kT algorithm with radius R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4, and must have pj T > 30 GeV and |yj| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Jets must also be separated from photons with pγ T > 15 GeV by a distance ∆Rγj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Similarly, in the CMS measurement [65], the fiducial region is defined by having two isolated photons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this case, photons are isolated if the transverse energy of all particles inside a cone of radius Riso = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 is less than 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The transverse momenta of the leading (subleading) isolated photon must satisfy pγ T > 35 (25) GeV, and – 26 – 10−3 10−2 10−1 100 101 dσ / dτ j1 C [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT σ(τ j1 C >80 GeV) 30 GeV CMS, 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 0 20 40 60 80 100 τ j1 C [GeV] −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 10−1 100 101 102 103 dσ / d|φ∗ η| [fb] pp → (H → γγ) + X √ S = 13 TeV rEFT σ(|φ∗ η|>4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 CMS, 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 1 2 5 |φ∗ η| −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 ratio − 1 10−2 10−1 100 101 dσ / dpH T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT Njets = 0 σ(pH T >60 GeV) 15 GeV CMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 0 10 20 30 40 50 60 70 pH T [GeV] −1 0 1 ratio − 1 10−2 10−1 100 101 dσ / dpH T [fb/GeV] pp → (H → γγ) + X √ S = 13 TeV rEFT Njets = 1 σ(pH T >170 GeV) 70 GeV CMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 137 fb−1 Geneva+Pythia8 (ggH) + XH XH =VBF+VH+ttH 0 50 100 150 200 pH T [GeV] −1 0 1 ratio − 1 Figure 10: Comparison of the CMS data [65] with the Geneva+Pythia8 results at 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We show the τ j1 C (top left) and |φ∗ η| (top right) distributions, as well as the pH T distributions for events with Njets = 0 (bottom left) and Njets = 1 (bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' amount to at least 1/3 (1/4) of the reconstructed Higgs invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In turn, the Higgs invariant mass must lie between 100 and 180 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Photons must also satisfy |ηγ| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Also in this case, jets are constructed using the anti-kT algorithm with R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4, and are required to have pj T > 30 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Jets with |ηj| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 are used for observables with one extra jet or to count the number of jets, while a looser cut |ηj| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='7 is applied for observables requiring at least two jets in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Due to the lack of availability of these analyses in the Rivet [122] framework, we have implemented the ATLAS and CMS analyses within the Geneva code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The H → γγ decay is inserted by the Pythia8 particle decays handler on top of the events produced – 27 – by Geneva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Its kinematics are treated at leading order in QCD, and we set the branching ratio to BR(H → γγ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='27 × 10−3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' the value reported in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [123] and calculated with HDECAY [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The Geneva prediction for the gluon-fusion production channel is obtained at NNLO+NNLL′ T0+NLLT1, and setting the scale of the hard function µH = −i mH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We use matrix elements computed in the infinite-top-mass limit and rescaled in the rEFT scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We set T cut 0 = T cut 1 = 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We use the PDF set PDF4LHC15 NNLO, and take the value of αs(mZ) from there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The partonic prediction is matched to the Pythia8 QCD+QED shower, including multiparton interaction (MPI) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We use the AZNLO tune [125] for the ATLAS comparison, and the CP5 tune [126] for CMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Showered events are then hadronised using the default Pythia8 Lund string model [127, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In order to obtain a meaningful comparison with the experimental data, we include the contributions from other Higgs boson production modes (labelled overall as XH) by summing them to the Geneva results for the gluon-fusion channel alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='2 For ATLAS these include vector-boson fusion (VBF), Higgsstrahlung (V H), and associated production with t¯t, b¯b, and t, all computed at NLO accuracy in QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For CMS these only include contributions from VBF, V H, and t¯tH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The outcome of the comparison with the experimental results is shown in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 7 and 8 for the ATLAS data, and in figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 9 and 10 for the CMS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the ATLAS data, we show the pH T , Njets, |yH|, pj1 T , pHj T , and τ j1 C distributions, as well as the pH T spectra in bins of τ j1 C and in bins of |yH|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the CMS data, we show the pH T , Njets, |yH|, pj1 T , τ j1 C , and |φ∗ η| distributions, as well as the pH T spectra in different jet multiplicity bins (N = 0 and N = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The definitions of τ j C and φ∗ η are given by τ j C = mj T 2 cosh(yj − yH) , φ∗ η = tan �φacop 2 � sin θ∗ η , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1) where φacop = π − |∆φγγ| and sin θ∗ η = [cosh(∆ηγγ/2)]−1 [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' With the ATLAS fiducial cuts, we obtain a total fiducial cross section of 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='5 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='0 fb, to be compared to the experimental finding of 67 ± 6 fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In the CMS fiducial region, we obtain a total cross section of 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='6 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='3 fb, which is compatible with the measurement of 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='4+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='1 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='9 fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In both cases our predictions agree with the measured results within roughly one standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We note that our results are not rescaled to the total N3LO gluon- fusion cross section, contrary to the theoretical predictions used in the ATLAS publication for their comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Regarding the distributions, we find overall good agreement between the Geneva predictions and the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' For the ATLAS data we find slight deviations in the pH T peak and a more marked discrepancy in the tail of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The latter corresponds to the region where the HTL approximation is less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We also find slight deviations in the |yH| spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The deviations in both spectra are consistent with those obtained using other calculations, as shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Similarly, for the CMS data our results 2The values of the XH distributions are taken from the plots in ATLAS and CMS publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' – 28 – underestimate the bins corresponding to the pH T peak, again in a similar fashion to other predictions [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Large deviations are also found in the first bin of the pH T distribution with Njets = 1, once again in agreement with other theoretical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 6 Conclusions We have described a number of improvements to the Geneva method, which are particu- larly useful for processes featuring large higher order corrections such as the production of colour singlet final states via gluon fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Specifically, we detailed a new implementation of the splitting functions which serve to make the resummed calculation fully differential in higher multiplicity phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This results in an improved behaviour of the nonsin- gular cross section as a function of the colour singlet transverse momentum in the infrared limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In addition, following earlier work [43] we have introduced a separation between the beam scale in our SCET-based resummed calculation µB and the scale associated with collinear factorisation µF which appears in the fixed-order calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This allowed us to achieve a more robust estimate of the theoretical uncertainties associated with our calcula- tion in the fixed-order region, including uncorrelated variations of the renormalisation and factorisation scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Finally, we have addressed the issue of large contributions from π2 terms originating from timelike logarithms in 2 → 1 processes, by enabling the choice of a complex-valued hard scale µH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We studied the associated resummation of said logarithms in our fully-differential calculation, and showed that, as previously noted in the literature, the perturbative convergence can thus be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Throughout this work, we have used the gluon-initiated Higgs production process to study the effects of our improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We have constructed an NNLO+PS event generator for the process, including the resummation of the zero-jettiness variable up to N3LL accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' The availability of recent experimental results [64, 65] for this process also allowed us to make a detailed comparison of our final, showered events with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We stress, however, that the issues which we addressed in this work have a more general applicability, and we anticipate that the future implementations of processes in Geneva will make use of these developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' In this study, we have consistently worked in a heavy-top limit in which the top- quark has been integrated out of the SM Lagrangian, resulting in an effective gluon-Higgs coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Given the advancement in recent years towards including the exact quark mass dependence at NNLO [61, 62], it would be desirable to incorporate this progress into a Geneva event generator at NNLO+PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We leave this issue to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Acknowledgements We thank L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Rottoli for his collaboration in the early stages of this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We are also grateful to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Cueto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Donega, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Malberti, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Pigazzini for their help with the comparison of Geneva with the ATLAS and CMS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We thank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Tackmann for pro- viding us with a preliminary version of scetlib and for useful comments to the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' This project has received funding from the European Research Council (ERC) under the – 29 – European Union’s Horizon 2020 research and innovation programme (Grant agreements No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' 714788 REINVENT and 101002090 COLORFREE) The work of SA and GB is sup- ported by MIUR through the FARE grant R18ZRBEAFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' SA also acknowledges funding from Fondazione Cariplo and Regione Lombardia, grant 2017-2070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' MAL is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306, and also by the UKRI guarantee scheme for the Marie Sk�lodowska-Curie postdoctoral fellowship, grant ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' EP/X021416/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' We acknowledge the CINECA and the National Energy Research Scientific Computing Center (NERSC), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Department of Energy Office of Science User Facility operated under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' DEAC02-05CH11231, for the availability of the high performance computing resources needed for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} +page_content=' Hamilton, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddFKT4oBgHgl3EQfqi7m/content/2301.11875v1.pdf'} 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Edge ideals of finite simple graphs G on n vertices are the ideals I(G) of the +polynomial ring S in n variables generated by the quadratic monomials associated with +the edges of G. In this paper, we consider the possible pairs of dimensions and depths +of S/I(G) for connected graphs with a fixed number of vertices. We discuss such pairs +in the case where dimension is relatively large. As a corollary, we completely determine +the pairs for connected graphs with small number of vertices. We also study the possible +pairs for connected chordal graphs. +1. Introduction +Monomial ideals are one of the most well-studied objects in the area of combinatorial +commutative algebra. Especially, edge ideals of graphs are of particular interest in this +area. On the other hand, dimensions and depths are fundamental invariants on graded +(or local) rings. The goal of this paper is the investigation of the behaviors of pairs of +dimensions and depths for the quotient rings by edge ideals. +Throughout the present paper, we only treat finite simple graphs, so we omit “finite +simple”. Let Graph(n) denote the set of all connected graphs on the vertex set [n] := +{1, 2, . . . , n}. Let S = k[x1, . . . , xn] be a polynomial ring in n variables over a field k. +The edge ideal I(G) of G ∈ Graph(n) is defined by I(G) = (xixj : {i, j} ∈ E(G)) ⊂ S. +Invariants on edge ideals have been well studied in combinatorial commutative algebra. +Especially, the graded Betti numbers of the edge ideals of graphs have been considered +in various viewpoints. Since the (Castelnuovo–Mumford) regularity and the projective +dimension of edge ideals can be obtained from the information on Betti numbers, we can +also obtain some characterizations or bounds on them. For example, see [3, 4, 6, 7, 8, 9, 10]. +Note that the depths of edge ideals and the projective dimensions are equivalent invariants +in some sense (see Subsection 2.3). For more information on edge ideals, see, e.g., [5, +Section 9]. +In this paper, we focus on the relationship between dimensions and depths of edge ideals +for connected graphs. The study of their relationship is initiated in [6, Section 2] as far as +the authors know. Let us briefly recall what is obtained there. Given a positive integer n, +let +Graphdepth,dim(n) := {(depth S/I(G), dim S/I(G)) : G ∈ Graph(n)} , +where dim G (resp. depth G) denotes the dimension (resp. depth) of S/I(G). On the +other hand, let +C∗(n) := +� +(a, b) ∈ Z2 : 1 ≤ a ≤ b ≤ n − 1, a ≤ b + 1 − +� +b +n − b +�� +. +2020 Mathematics Subject Classification. +Primary: 13D02, Secondary: 13F55, 05E40, 05C70. +Key words and phrases. Edge ideals, dimension, depth, projective dimension. +1 + +Then the following is proved: +Theorem 1.1 ([6, Theorem 2.8]). Let n ≥ 2. Then we have C∗(n) ⊂ Graphdepth,dim(n). +By taking this theorem into account, the following question naturally arises: +Question 1.2. Does the equality C∗(n) = Graphdepth,dim(n) hold? +In [6, Example 2.2], it is mentioned that Graphdepth,dim(n) is computed by using com- +puter. As a consequence of the computations, we can verify that C∗(n) = Graphdepth,dim(n) +holds when n ≤ 9. +In this paper, we develop more theoretical considerations. The first main result of this +paper is the following: +Theorem 1.3 (See Remark 2.3, Propositions 3.2 and 3.3). Let (a, b) ∈ Graphdepth,dim(n). +If b ≥ n − 3, then (a, b) ∈ C∗(n). +This theorem enables us to show the equality C∗(n) = Graphdepth,dim(n) when n is +relatively small as follows: +Corollary 1.4. The equality C∗(n) = Graphdepth,dim(n) holds if n ≤ 12. +For a graph G, we say that G is chordal if any cycle in G of length at least 4 has a +chord. Let +Graphchordal +depth,dim(n) = {(depth S/I(G), dim S/I(G)) : G ∈ Graph(n), G is chordal} . +Namely, we restrict Graphdepth,dim(n) into chordal graphs. +The second main result of this paper is the following: +Theorem 1.5. Let n ≥ 2. Then we have C∗(n) = Graphchordal +depth,dim(n). +Note that in [6, Section 3], the possible pairs of dimensions and depths of the edge ideals +of Cameron–Walker graphs are determined. See [6, Theorem 3.15]. We can observe that +such pairs for Cameron–Walker graphs are quite small compared with C∗(n). +A brief structure of this paper is as follows. In Section 2, we recall some fundamental +materials used for the proofs of Theorems 1.3 and 1.5. In Section 3, we prove Theorem 1.3 +and Corollary 1.4. In Section 4, we prove Theorem 1.5. +Acknowledgements. The first named author is partially supported by JSPS Grant- +in-Aid for Scientists Research (C) JP20K03513. The second named author is partially +supported by Grant-in-Aid for JSPS Fellows JP21J21603. +2. Preliminaries +In this section, we prepare some terminologies, notions and statements for the proofs +of our main theorems. +2 + +2.1. Terminologies on graphs. Let G be a graph on the vertex set V (G) with the edge +set E(G). +• We call a subset W of V (G) an independent set (resp. clique) if {v, w} ̸∈ E(G) +(resp. {v, w} ∈ E(G)) for any v, w ∈ W. Let +α(G) = max{|W| : W is an independent set of G}, +called the independence number of G. It is well known that +dim S/I(G) = α(G). +• Let M ⊂ E(G) be a subset of the edge set. We say that M is an induced matching +of G if M satisfies that +– e ∩ e′ = ∅ for any e, e′ ∈ M with e ̸= e′; and +– there is no e′′ ∈ E(G) such that e ∩ e′′ ̸= ∅ and e′ ∩ e′′ ̸= ∅. +• For a vertex v of G, let N(v) = {w ∈ V : {v, w} ∈ E(G)}, let N[v] = N(v) ∪ {v} +and let deg v = |N(v)|. +• For a vertex v of G, let G \ v denote the induced subgraph of G by V (G) \ {v}. +Similarly, for U ⊂ V (G), let G\U denote the induced subgraph of G by V (G)\U. +• We say that C ⊂ V (G) is a vertex cover if C ∩ e ̸= ∅ for any e ∈ E(G). We call a +vertex cover C minimal if C \ {v} is not a vertex cover of G for any v ∈ C. +• A perfect elimination ordering of G is an ordering vn, . . . , v1 of the vertices of G +such that {v1, . . . , vk−1} ∩ N(vk) forms a clique for each k = 2, . . . , n. It is known +that G is chordal if and only if G has a perfect elimination ordering ([2]). +In this paper, given a graph G, we use the notation dim G, depth G and pdimG instead +of dim S/I(G), depth S/I(G) and pdim S/I(G) for short, respectively, where pdim denotes +the projective dimension. +2.2. Bouquets and projective dimensions of S/I(G). We recall the terminology and +the notation used in [9]. The graph B on the vertex set V (B) = {w, z1, . . . , zd} with the +edge set {{w, zi} : i = 1, . . . , d} is called a bouquet. Namely, a bouquet is nothing but a +star graph. A vertex w is called a root, the vertices zi are called flowers and the edges +{w, zi} are called stems of B. These words were originally used in [10, Definition 1.7]. +Given a graph G, we call B a bouquet of G if B is a (not necessarily induced) subgraph +of G. Let B = {B1, . . . , Bp} be a set of bouquets of G. We define the following: +F(B) := +p� +i=1 +{z ∈ V (Bi) : z is a flower of Bi}; +R(B) := +p� +i=1 +{z ∈ V (Bi) : z is a root of Bi}; +S(B) := +p� +i=1 +{s ∈ E(Bi) : s is a stem of Bi}. +Let G be a graph and let B = {B1, . . . , Bp} be a set of bouquets of G. Assume that +V (Bi) ∩ V (Bj) = ∅ holds for any 1 ≤ i < j ≤ p. +• We say that B is strongly disjoint if there are stems si ∈ E(Bi) for each i such that +{s1, . . . , sp} forms an induced matching of G. +• We say that B is semi-strongly disjoint if R(B) forms an independent set. +3 + +Note that if B is strongly disjoint, since any stem in Bi is incident to the root of Bi, we +see that any two roots should not be adjacent. Namely, B becomes semi-strongly disjoint. +Theorem 2.1 ([9, Theorem 3.1]). Given any graph G, we have +pdim G ≥ max{|F(B)| : B is a strongly disjoint set of bouquets of G}. +By Theorem 2.1, we see that given a graph G, we have +deg v ≤ pdim G for any u ∈ V (G) +(2.1) +since N(u) ∪ {u} forms a (set of a single) bouquet with a root u. +Theorem 2.2 ([9, Theorem 5.3, Corollary 5.6]). Let G be chordal. Then we have +pdim G = max{|F(B)| : B is a semi-strongly disjoint set of bouquets of G} += max{|C| : C is a minimal vertex cover of G}. +2.3. Auslander–Buchsbaum formula. Let G be a graph on [n]. Then S/I(G) is an +S-module. Since S is a polynomial ring, pdim G < ∞ holds. Hence, by the Auslander– +Buchsbaum formula (see, e.g., [1, Theorem 1.3.3]), we have +pdim G + depth G = n. +We will often use this in the sequel. +2.4. On the sets C∗(n) and C−(n). Let us recall the set C−(n) defined in [6, Section +2]. For n ≥ 2, let +C−(n) := {(1, n − 1)} ∪ +� +(a, b) ∈ Z2 : 1 ≤ a ≤ b, a ≤ +�n +2 +� +, b ≤ n − 2 +� +. +Then, by the proof of [6, Theorem 2.9], we see that C−(n) ⊂ C∗(n). The difference of +C−(n) and C∗(n) will be mentioned below. See Example 3.5. +Remark 2.3 (cf. [6, Proposition 1.3]). Let G ∈ Graph(n) and assume that α(G) = n − 1. +Then G itself must be a bouquet. In particular, G is chordal. Thus, by Theorem 2.2, +we have pdimG = n − 1. +Hence, depth G = 1. +These imply that if (a, n − 1) ∈ +Graphdepth,dim(n), then a = 1. In particular, (1, n − 1) ∈ C−(n) ⊂ C∗(n). +Remark 2.4. Given integers a, b with 1 ≤ a ≤ b ≤ n − 1, assume that a + b ≤ n. Then +(a, b) ∈ C− automatically holds. In fact, if b = n − 1, then a = 1; if b ≤ n − 2, since +2a ≤ a + b ≤ n holds, we have a ≤ ⌊n/2⌋. +3. In the case of graphs with large independence number +In this section, we consider the inclusion Graphdepth,dim(n) ⊂ C∗(n) for some special +cases. In particular, we discuss when (a, b) ∈ Graphdepth,dim(n) belongs to C∗(n). +First, we discuss the case where a = b, i.e., the corresponding graph is Cohen–Macaulay. +Proposition 3.1 (cf. [3, Theorem 3.3]). Assume that (b, b) ∈ Graphdepth,dim(n). Then +(b, b) ∈ C∗(n). +Proof. We see that (b, b) ∈ C∗(n) holds if and only if ⌈b/(n−b)⌉ ≤ 1 holds. Here, it follows +from [3, Theorem 3.3] that if G is Cohen–Macaulay, then b ≤ n/2. This implies that when +G is Cohen–Macaulay with b = dim G, we have b ≤ n/2, i.e., (b, b) ∈ C∗(n). +□ +4 + +Next, we discuss the case where b is relatively large. Note that we always have 1 ≤ b ≤ +n − 1 and the case b = n − 1 has been already discussed in Remark 2.3. +Proposition 3.2. Let n ≥ 4. If (a, n − 2) ̸∈ C∗(n), then (a, n − 2) ̸∈ Graphdepth,dim(n). +Proof. Let +f(a, b) := b + 1 − +� +b +n − b +� +− a. +Note that for integers a, b with 1 ≤ a ≤ b ≤ n − 1, f(a, b) ≥ 0 holds if and only if +(a, b) ∈ C∗(n). +In what follows, we lead a contradiction if there exists a graph G ∈ Graph(n) with +(depth G, dim G) = (a, n − 2) such that f(a, n − 2) < 0. +Since f(a, n − 2) = ⌊n/2⌋ − a, we have ⌊n/2⌋ < depth G = n − pdimG, i.e., pdimG < +⌈n/2⌉. On the other hand, since dim G = n − 2, there is an independent set W of G with +|W| = n − 2. Let v1, v2 be the remaining vertices, i.e., {v1, v2} = V (G) \ W. Since G is +connected and W is an independent set, we see that +n − 2 ≤ +� +w∈W +deg w = |N(v1) ∩ W| + |N(v2) ∩ W| += +� +deg v1 + deg v2 − 2, +if {v1, v2} ∈ E(G), +deg v1 + deg v2, +if {v1, v2} ̸∈ E(G). +On the other hand, since pdimG < ⌈n/2⌉, we see that deg vi < ⌈n/2⌉ for i = 1, 2 by (2.1). +If {v1, v2} ∈ E(G), then we see that +n − 2 ≤ deg v1 + deg v2 − 2 ≤ n − 1 +2 ++ n − 1 +2 +− 2 = n − 3, +a contradiction. Hence, {v1, v2} ̸∈ E(G). +Suppose that n is even. Then we see that +n − 2 ≤ deg v1 + deg v2 ≤ n − 2 +2 ++ n − 2 +2 += n − 2, +i.e., we have deg w = 1 for each w ∈ W. Since {v1, v2} ̸∈ E(G), we see that there is +no path in G connecting v1 and v2, a contradiction to the connectedness of G. Hence, n +should be odd. Moreover, there should be at least one vertex w0 in W with deg w0 ≥ 2. +Since � +w∈W deg w ≥ n − 1 and deg vi ≤ (n − 1)/2 hold, we conclude that G should look +like as follows: +v1 +v2 +w0 +Note that deg v1 = deg v2 = (n − 1)/2. In particular, G is chordal. By taking a set of +bouquets consisting of two bouquets whose roots are v1 and v2, we see from Theorem 2.2 +that +�n +2 +� +> pdim G ≥ n − 2, +5 + +a contradiction to n ≥ 4. +□ +Proposition 3.3. Let n ≥ 8. If (a, n − 3) ̸∈ C∗(n), then (a, n − 3) ̸∈ Graphdepth,dim(n). +Proof. Let f(a, b) be the same thing as above. Then f(a, n − 3) = ⌊2n/3⌋ − a − 1. +Suppose that there exists a graph G ∈ Graph(n) with (depth G, dim G) = (a, n − 3) +such that f(a, n − 3) < 0. Since dim G = n − 3, there is an independent set W with +|W| = n − 3. Let v1, v2, v3 be the remaining vertices. By (2.1), we have deg vi ≤ n − a < +n − ⌊2n/3⌋ + 1 = ⌈n/3⌉ + 1, i.e., +deg vi ≤ +�n +3 +� +for i = 1, 2, 3. +(3.1) +Similarly, for any strongly disjoint set B of bouquets of G, we have +|F(B)| ≤ +�n +3 +� +. +(3.2) +Here, we have the following four possibilities on the adjacencies of v1, v2, v3: +(i) +v1 +v2 +v3 +v1 +(ii) +v2 +v3 +(iii) +v1 +v2 +v3 +(iv) +v1 +v2 +v3 +We divide our discussions into these four cases. +(i) In this case, we see from (3.1) that +n − 3 ≤ +� +w∈W +deg w = +3 +� +i=1 +(deg vi − 2) ≤ 3 +��n +3 +� +− 2 +� +≤ 3 +�n + 2 +3 +− 2 +� += n − 4, +a contradiction. +(ii) By the similar computation to (i), we have that +n − 3 ≤ +� +w∈W +deg w = +3 +� +i=1 +deg vi − 4 ≤ 3 +�n +3 +� +− 4 = + + + + + +n − 4 +if n ≡ 0 +(mod 3), +n − 3 +if n ≡ 2 +(mod 3), +n − 2 +if n ≡ 1 +(mod 3). +Thus, we get the following possibilities: +• n ≡ 2 (mod 3) and deg w = 1 for each w ∈ W; +• n ≡ 1 (mod 3) and deg w = 1 for each w ∈ W; +• n ≡ 1 (mod 3), deg w0 = 2 for some w0 ∈ W and deg w = 1 for each w ∈ W \{w0}. +In the first two cases (resp. the third case), G should look like the left-most one (resp. +the central one or the right-most one) of the following figure: +v1 +v2 +v3 +v1 +v2 +v3 +w0 +v1 +v2 +v3 +w0 +6 + +Note that we have |N(vi) ∩ W| ≥ ⌈n/3⌉ − 2 > 1 for i = 1, 3 and |(N(v1) ∪ N(v3)) ∩ W| ≥ +2⌈n/3⌉ − 3 in any cases. Hence, we can always take a strongly disjoint set of bouquets B +consisting of two bouquets whose roots are v1 and v3 satisfying |F(B)| ≥ 2⌈n/3⌉ − 3 + 1. +(Note that “+1” comes from a stem {v1, v2}.) By (3.2), we obtain that +�n +3 +� +≥ 2 +�n +3 +� +− 2 ⇐⇒ 2 ≥ +�n +3 +� +⇐⇒ 6 ≥ n, +a contradiction. +(iii) Take bouquets B1, B2, B3 of G whose roots are v1, v2, v3, respectively, such that +V (Bi) ∩ V (Bj) = ∅ (i ̸= j) and F({B1, B2, B3}) = W hold. +• Assume that N(v3) \ N(v2) ̸= ∅. +Let w ∈ N(v3) \ N(v2). +Then, by taking a +strongly disjoint set B′ of bouquets consisting of B′ +2 and a bouquet {v3, w}, where +B′ +2 is constructed by attaching a new flower v1 to B2, we see from (3.2) that +�n +3 +� +≥ |F(B′)| = |F(B2)| + 2 = n − 3 − |F({B1, B3})| + 2 ≥ n − 1 − +�n +3 +� +. +Hence, we obtain the inequality +�n +3 +� +≥ n − 1 − +�n +3 +� +, +i.e., n ≤ 7, a contradiction. +• Assume that N(v3) \ N(v2) = ∅, i.e., N(v3) ⊂ N(v2). Take w ∈ N(v3) and let +B′ be a new bouquet with its root w and flowers v2, v3. Then {B1, B′} becomes +strongly disjoint. Thus, by (3.2), we have +�n +3 +� +≥ |F({B1, B′})| = |F(B1)| + 2 ≥ n − 3 − |F({B2})| + 2 ≥ n − 1 − +�n +3 +� +, +a contradiction. +(iv) Take bouquets B1, B2, B3 of G whose roots are v1, v2, v3, respectively, such that +V (Bi) ∩ V (Bj) = ∅ (i ̸= j) and F({B1, B2, B3}) = W hold. If the situation (iv) happens, +then there are j, k such that {Bj, Bk} is strongly disjoint. +In fact, if not, then either +F(Bj) ⊂ F(Bk) or F(Bk) ⊂ F(Bj) holds for each 1 ≤ j < k ≤ 3. This implies that +we have F(B1) ⊂ F(B2) ⊂ F(B3) after properly changing the indices of B1, B2, B3. In +particular, F(B3) = W. +Thus, n − 3 = deg v3 ≤ ⌈n/3⌉ ≤ (n + 2)/3, i.e., n ≤ 5, a +contradiction. +Let, say, B = {B1, B2} be a strongly disjoint set of bouquets of G. +• Suppose that |F(B)| ≥ ⌊2n/3⌋ − 1. Then we see that +�n +3 +� +≥ |F(B)| ≥ +�2n +3 +� +− 1 =⇒ n + 2 +3 +≥ 2n − 2 +3 +− 1 ⇐⇒ 7 ≥ n, +a contradiction. +• Suppose that |F(B)| ≤ ⌊2n/3⌋ − 2. Since |W| = |F(B)| + |F(B3)| = n − 3 and +|F(B3)| ≤ deg v3 − 1 by the connectedness of G, we see that +�n +3 +� +− 1 ≥ deg v3 − 1 ≥ |F(B3)| ≥ n − 3 − +�2n +3 +� ++ 2 = +�n +3 +� +− 1. +Hence, all the equalities of these inequalities are satisfied. In particular, we have +|F(B3)| = deg v3 −1 = ⌈n/3⌉−1. Since n−3 > ⌈n/3⌉ and G is connected, there is +w ∈ W \N(v3) adjacent to v1 or v2, say, v1. Then we can construct a new strongly +disjoint set of bouquets B′ consisting of bouquets N(v3) ∪ {v3} and {v1, w} such +that |F(B′)| = ⌈n/3⌉ + 1, a contradiction to (3.2). +7 + +□ +By using Propositions 3.1, 3.2 and 3.3, we can prove Corollary 1.4. +Proof of Corolalry 1.4. It is enough to show that Graphdepth,dim(n) ⊂ C∗(n) for n ≤ 12. +Take (a, b) ∈ Graphdepth,dim(n) arbitrarily. Then the inequalities 1 ≤ a ≤ b ≤ n − 1 +automatically hold. +• If a = b, then we see that (a, b) ∈ C∗(n) by Proposition 3.1. +• The case of b = n − 1 has been already discussed in Remark 2.3. +Hence, in what follows, we assume that 1 ≤ a < b ≤ n − 2. Here, we consider the set +C′(n) := {(a, b) ∈ Z2 : 1 ≤ a < b ≤ n − 2} \ C∗(n). +Then the direct computations show the following: +C′(n) = ∅ for 2 ≤ n ≤ 6, C′(7) = {(4, 5)}, C′(8) = {(5, 6)}, C′(9) = {(5, 7), (6, 7)}, +C′(10) = {(6, 7), (6, 8), (7, 8)}, C′(11) = {(6, 9), (7, 8), (7, 9), (8, 9)}, +C′(12) = {(7, 10), (8, 9), (8, 10), (9, 10)}. +Let n ≤ 12. Suppose that (a, b) ̸∈ C∗(n). Then (a, b) ∈ C′(n). By the above compu- +tations, we see that all elements of �12 +n=2 C′(n) satisfy b = n − 2 or b = n − 3. Hence, it +follows from Propositions 3.2 and 3.3 that (a, b) ̸∈ Graphdepth,dim(n), a contradiction. +Therefore, C∗(n) = Graphdepth,dim(n) holds if n ≤ 12. +□ +Remark 3.4. We see that +C′(13) = {(7, 11), (8, 9), (8, 10), (8, 11), (9, 10), (9, 11), (10, 11)}. +Since we cannot claim that (8, 9) ̸∈ Graphdepth,dim(13) only by Propositions 3.2 and 3.3, +the above proof of Corollary 1.4 is not available if n = 13. Instead, we can claim that +Graphdepth,dim(13) = C∗(13) or C∗(13) ∪ {(8, 9)}. +Example 3.5. For small n’s, the direct computations show the following: +C∗(n) = C−(n) for n = 2, 3, . . . , 8, 10, +C∗(9) = C−(9) ∪ {(5, 6)}, +C∗(11) = C−(11) ∪ {(6, 6), (6, 7), (6, 8)}, +C∗(12) = C−(12) ∪ {(7, 8), (7, 9), (8, 9)}. +Note that [6, Example 2.11] also mentions the similar equalities in the case 3 ≤ n ≤ 9. +4. In the case of chordal graphs +In this section, we discuss the behavior of pairs of dimensions and depths of edge ideals +of chordal graphs. +8 + +4.1. The inclusion C∗(n) ⊂ Graphchordal +depth,dim(n). First, we discuss the inclusion C∗(n) ⊂ +Graphchordal +depth,dim(n). Here, we remark that since we only know that Graphchordal +depth,dim(n) ⊂ +Graphdepth,dim(n) by definition, we cannot immediately claim C∗(n) ⊂ Graphchordal +depth,dim(n) +from Theorem 1.1. +Let (a, b) ∈ C∗(n). We divide the discussions into two cases: +(i) a + b > n; or +(ii) a + b ≤ n. +(i) In this case, we can apply the same proof as that of [6, Theorem 2.8]. In that proof, +the graphs G(m; s1, . . . , sm) given in [6, Construction 2.5] play the essential role and we see +that G(m; s1, . . . , sm) is chordal. Hence, we obtain the inclusion by the same discussion. +(ii) In this case, we cannot apply the same proof. In fact, Lemmas 2.3 and 2.4 are used +for the proof in the case a + b ≤ n. Especially, Lemma 2.3 uses the technique of S-cone +(also known as S-suspension). In general, the chordality is not preserved by taking S-cone. +Hence, Lemma 2.3 is not immediately available for chordal graphs. +Instead, we directly prove the following: +Proposition 4.1. Let a, b, n be integers satisfying 1 ≤ a ≤ b ≤ n−1 and a+b ≤ n. Then +there exists a chordal graph G ∈ Graph(n) such that (depth G, dim G) = (a, b). Namely, +we have (a, b) ∈ Graphchordal +depth,dim(n). +Note that the assumption a+b ≤ n immediately implies (a, b) ∈ C∗(n) (see Remark 2.4). +Proof. Let G be the graph defined as follows: +V (G) = U ⊔ W, where U = {u1, . . . , un−b} and W = {w1, . . . , wb}; +E(G) = {{u, u′} : u, u′ ∈ U} ∪ {{ui, wi} : 1 ≤ i ≤ a − 1} +∪ {{uj, wk} : a ≤ j ≤ n − b, a ≤ k ≤ b}. +It is straightforward to check that the ordering w1, . . . , wb, u1, . . . , un−b becomes a perfect +elimination ordering of G. Hence, G is chordal. Moreover, we can easily see that W is an +independent set with |W| = α(G). Hence, dim G = b. +In what follows, we prove that depth G = a, i.e., pdim G = n − a. +Regarding the minimal vertex covers of G, we observe that any minimal vertex cover +contains at least (n−b−1) vertices of U since U forms a clique. In particular, we see that +U itself becomes a minimal vertex cover. Fix uj ∈ U and let C be a minimal vertex cover +not containing uj. Then U \ {uj} ⊂ C. +• If 1 ≤ j ≤ a − 1, then wj ∈ C, so C = (U \ {uj}) ∪ {wj}. +• If a ≤ j ≤ n − b, then wa, . . . , wb ∈ C, so C = (U \ {uj}) ∪ {wa, . . . , wb}. +Therefore, by Theorem 2.2, we conclude that +pdim G = max{|C| : C is a minimal vertex cover of G} = max{|U|, |U| + b − a} = n − a. +□ +4.2. The inclusion Graphchordal +depth,dim(n) ⊂ C∗(n). For the proof of this inclusion, we pre- +pare the following lemma. +Lemma 4.2. (1) For any vertex v of a graph G, we have +dim G = max{dim(G \ N[v]) + 1, dim(G \ v)}. +9 + +(2) Let G be a chordal graph and let vn, . . . , v1 be a perfect elimination ordering of G. +Then we have +depth G = min{depth(G \ N[vn]) + 1, depth(G \ vn)}. +Proof. (1) Let W be an independent set of G with |W| = α(G). Then the required equality +directly follows from the following observations: +• If v ∈ W, then W \ {v} ⊂ V (G \ N[v]) becomes an independent set of G \ N[v] +with |W \ {v}| = α(G \ N[v]). +• If v ̸∈ W, then W ⊂ V (G \ v) becomes an independent set of G \ v with |W| = +α(G \ v). +(2) We prove the equality by induction on |V (G)|. +Let v = vn and deg v = m. Let C be a minimal vertex cover of G with |C| = pdim G +(cf. Theorem 2.2). +• If v ∈ C, then C′ = C \ {v} becomes a minimal vertex cover of G \ v with +pdim(G \ v) = |C′|. Hence, by the hypothesis of induction, we see that +depth G = |V (G)| − pdimG = |V (G)| − (|C′| + 1) = |V (G \ v)| − |C′| += |V (G \ v)| − pdim(G \ v) = depth(G \ v). +• If v ̸∈ C, since N[v] forms a clique by definition of perfect elimination ordering, +we see that N(v) ⊂ C. Thus, C′′ = C \ N[v] becomes a minimal vertex cover of +G \ N[v] with pdim(G \ N[v]) = |C′′|. Hence, we see that +depth G = |V (G)| − pdim G = |V (G)| − (|C′′| + m) = (|V (G \ N[v])| + 1) − |C′′| += |V (G \ N[v])| − pdim(G \ N[v]) + 1 = depth(G \ N[v]) + 1. +Here, Theorem 2.2 says that if C0 is a minimal vertex cover with |C0| ̸= pdim G, then +|C0| < pdim G. Thus, the required equality follows. +□ +Now, we prove the inclusion Graphchordal +depth,dim(n) ⊂ C∗(n). Take (a, b) ∈ Graphchordal +depth,dim(n) +arbitrarily. Then there is a chordal graph G such that (depth G, dim G) = (a, b). To prove +(a, b) ∈ C∗(n), it is enough to show that +b ≤ (n − b)(b − a + 1). +(4.1) +In fact, (4.1) is equivalent to the following: +b +n − b ≤ b − a + 1 ⇐⇒ +� +b +n − b +� +≤ b − a + 1 ⇐⇒ a ≤ b + 1 − +� +b +n − b +� +. +This means that (a, b) ∈ C∗(n). +Let vn, . . . , v1 be a perfect elimination ordering of G. We prove the inequality (4.1) by +induction on n. +Let v = vn and let +a′ = depth(G \ v), b′ = dim(G \ v), a′′ = depth(G \ N[v]) and b′′ = dim(G \ N[v]). +By Lemma 4.2 (1), either b = b′ or b = b′′ + 1 holds. +• Assume that b = b′. +Since a ≤ a′ holds by Lemma 4.2 (2), we see from the +hypothesis of induction that +b = b′ ≤ ((n − 1) − b′)(b′ − a′ + 1) ≤ (n − b)(b − a + 1). +10 + +• Assume that b = b′′ + 1. Since a ≤ a′′ + 1, we see that +b = b′′ + 1 ≤ ((n − deg v − 1) − b′′)(b′′ − a′′ + 1) + 1 +≤ (n − deg v − b)(b − a + 1) + 1 = (n − b)(b − a + 1) − (deg v(b − a + 1) − 1) +≤ (n − b)(b − a + 1). +References +[1] W. Bruns and J. Herzog, “Cohen-Macaulay rings, revised edition”, Cambridge University Press, 1998. +[2] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg, 25 (1961), 71—76. +[3] I. Gitler and C. Valencia, Bounds for invariants of edge-rings, Commun. Algebra 33 (2005), 1603–1616. +[4] H. T. H`a and A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti +numbers, J. Algebraic Combin. 27 (2008), no. 2, 215–245. +[5] J. Herzog and T. Hibi, “Monomial Ideals”, GTM, Springer, 2010. +[6] T. Hibi, H. Kanno, K. Kimura, K. Matsuda and A. Van Tuyl, Homological invariants of Cameron– +Walker graphs, Trans. Amer. Math. Soc. 374 (2021), 6559–6582. +[7] T. Hibi, H. Kanno, K. Matsuda, Induced matching numbers of finite graphs and edge ideals, J. Algebra +532 (2019), 311–322. +[8] M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. +A 113 (2006), no. 3, 435–454. +[9] K. Kimura, Non-vanishingness of Betti numbers of edge ideals, Harmony of Gr¨obner bases and the +modern industrial society, 153–168, World Sci. Publ., Hackensack, NJ, 2012. +[10] X. Zheng, Resolutions of facet ideals, Comm. Algebra 32 (2004), 2301–2324. +(A. Higashitani) Department of Pure and Applied Mathematics, Graduate School of Infor- +mation Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan +Email address: higashitani@ist.osaka-u.ac.jp +(A. Kanno) Department of Pure and Applied Mathematics, Graduate School of Information +Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan +Email address: u825139b@ecs.osaka-u.ac.jp +(R. Ueji) Department of Pure and Applied Mathematics, Graduate School of Information +Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan +Email address: u136547i@alumni.osaka-u.ac.jp +11 + diff --git a/ftE3T4oBgHgl3EQf3Qv5/content/tmp_files/load_file.txt b/ftE3T4oBgHgl3EQf3Qv5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e7493486903864b333e566e809e15bc8016c0b7c --- /dev/null +++ b/ftE3T4oBgHgl3EQf3Qv5/content/tmp_files/load_file.txt @@ -0,0 +1,558 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf,len=557 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='04763v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='AC] 11 Jan 2023 BEHAVIORS OF PAIRS OF DIMENSIONS AND DEPTHS OF EDGE IDEALS AKIHIRO HIGASHITANI, AKANE KANNO, AND RYOTA UEJI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Edge ideals of finite simple graphs G on n vertices are the ideals I(G) of the polynomial ring S in n variables generated by the quadratic monomials associated with the edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In this paper, we consider the possible pairs of dimensions and depths of S/I(G) for connected graphs with a fixed number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We discuss such pairs in the case where dimension is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' As a corollary, we completely determine the pairs for connected graphs with small number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We also study the possible pairs for connected chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Introduction Monomial ideals are one of the most well-studied objects in the area of combinatorial commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Especially, edge ideals of graphs are of particular interest in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' On the other hand, dimensions and depths are fundamental invariants on graded (or local) rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The goal of this paper is the investigation of the behaviors of pairs of dimensions and depths for the quotient rings by edge ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Throughout the present paper, we only treat finite simple graphs, so we omit “finite simple”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let Graph(n) denote the set of all connected graphs on the vertex set [n] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let S = k[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , xn] be a polynomial ring in n variables over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The edge ideal I(G) of G ∈ Graph(n) is defined by I(G) = (xixj : {i, j} ∈ E(G)) ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Invariants on edge ideals have been well studied in combinatorial commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Especially, the graded Betti numbers of the edge ideals of graphs have been considered in various viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since the (Castelnuovo–Mumford) regularity and the projective dimension of edge ideals can be obtained from the information on Betti numbers, we can also obtain some characterizations or bounds on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For example, see [3, 4, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Note that the depths of edge ideals and the projective dimensions are equivalent invariants in some sense (see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For more information on edge ideals, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', [5, Section 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In this paper, we focus on the relationship between dimensions and depths of edge ideals for connected graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The study of their relationship is initiated in [6, Section 2] as far as the authors know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let us briefly recall what is obtained there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Given a positive integer n, let Graphdepth,dim(n) := {(depth S/I(G), dim S/I(G)) : G ∈ Graph(n)} , where dim G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' depth G) denotes the dimension (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' depth) of S/I(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' On the other hand, let C∗(n) := � (a, b) ∈ Z2 : 1 ≤ a ≤ b ≤ n − 1, a ≤ b + 1 − � b n − b �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Primary: 13D02, Secondary: 13F55, 05E40, 05C70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Edge ideals, dimension, depth, projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 1 Then the following is proved: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1 ([6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we have C∗(n) ⊂ Graphdepth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' By taking this theorem into account, the following question naturally arises: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Does the equality C∗(n) = Graphdepth,dim(n) hold?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In [6, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2], it is mentioned that Graphdepth,dim(n) is computed by using com- puter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' As a consequence of the computations, we can verify that C∗(n) = Graphdepth,dim(n) holds when n ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In this paper, we develop more theoretical considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The first main result of this paper is the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 (See Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let (a, b) ∈ Graphdepth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If b ≥ n − 3, then (a, b) ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' This theorem enables us to show the equality C∗(n) = Graphdepth,dim(n) when n is relatively small as follows: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The equality C∗(n) = Graphdepth,dim(n) holds if n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For a graph G, we say that G is chordal if any cycle in G of length at least 4 has a chord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let Graphchordal depth,dim(n) = {(depth S/I(G), dim S/I(G)) : G ∈ Graph(n), G is chordal} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Namely, we restrict Graphdepth,dim(n) into chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The second main result of this paper is the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we have C∗(n) = Graphchordal depth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Note that in [6, Section 3], the possible pairs of dimensions and depths of the edge ideals of Cameron–Walker graphs are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' See [6, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We can observe that such pairs for Cameron–Walker graphs are quite small compared with C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' A brief structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In Section 2, we recall some fundamental materials used for the proofs of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In Section 3, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In Section 4, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The first named author is partially supported by JSPS Grant- in-Aid for Scientists Research (C) JP20K03513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The second named author is partially supported by Grant-in-Aid for JSPS Fellows JP21J21603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Preliminaries In this section, we prepare some terminologies, notions and statements for the proofs of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Terminologies on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let G be a graph on the vertex set V (G) with the edge set E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We call a subset W of V (G) an independent set (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' clique) if {v, w} ̸∈ E(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' {v, w} ∈ E(G)) for any v, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let α(G) = max{|W| : W is an independent set of G}, called the independence number of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' It is well known that dim S/I(G) = α(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let M ⊂ E(G) be a subset of the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We say that M is an induced matching of G if M satisfies that – e ∩ e′ = ∅ for any e, e′ ∈ M with e ̸= e′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' and – there is no e′′ ∈ E(G) such that e ∩ e′′ ̸= ∅ and e′ ∩ e′′ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For a vertex v of G, let N(v) = {w ∈ V : {v, w} ∈ E(G)}, let N[v] = N(v) ∪ {v} and let deg v = |N(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For a vertex v of G, let G \\ v denote the induced subgraph of G by V (G) \\ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Similarly, for U ⊂ V (G), let G\\U denote the induced subgraph of G by V (G)\\U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We say that C ⊂ V (G) is a vertex cover if C ∩ e ̸= ∅ for any e ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We call a vertex cover C minimal if C \\ {v} is not a vertex cover of G for any v ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' A perfect elimination ordering of G is an ordering vn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , v1 of the vertices of G such that {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , vk−1} ∩ N(vk) forms a clique for each k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' It is known that G is chordal if and only if G has a perfect elimination ordering ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In this paper, given a graph G, we use the notation dim G, depth G and pdimG instead of dim S/I(G), depth S/I(G) and pdim S/I(G) for short, respectively, where pdim denotes the projective dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Bouquets and projective dimensions of S/I(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We recall the terminology and the notation used in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The graph B on the vertex set V (B) = {w, z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , zd} with the edge set {{w, zi} : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , d} is called a bouquet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Namely, a bouquet is nothing but a star graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' A vertex w is called a root, the vertices zi are called flowers and the edges {w, zi} are called stems of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' These words were originally used in [10, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Given a graph G, we call B a bouquet of G if B is a (not necessarily induced) subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let B = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , Bp} be a set of bouquets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We define the following: F(B) := p� i=1 {z ∈ V (Bi) : z is a flower of Bi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' R(B) := p� i=1 {z ∈ V (Bi) : z is a root of Bi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' S(B) := p� i=1 {s ∈ E(Bi) : s is a stem of Bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let G be a graph and let B = {B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , Bp} be a set of bouquets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Assume that V (Bi) ∩ V (Bj) = ∅ holds for any 1 ≤ i < j ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We say that B is strongly disjoint if there are stems si ∈ E(Bi) for each i such that {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , sp} forms an induced matching of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We say that B is semi-strongly disjoint if R(B) forms an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 3 Note that if B is strongly disjoint, since any stem in Bi is incident to the root of Bi, we see that any two roots should not be adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Namely, B becomes semi-strongly disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1 ([9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Given any graph G, we have pdim G ≥ max{|F(B)| : B is a strongly disjoint set of bouquets of G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1, we see that given a graph G, we have deg v ≤ pdim G for any u ∈ V (G) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1) since N(u) ∪ {u} forms a (set of a single) bouquet with a root u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 ([9, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let G be chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we have pdim G = max{|F(B)| : B is a semi-strongly disjoint set of bouquets of G} = max{|C| : C is a minimal vertex cover of G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Auslander–Buchsbaum formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let G be a graph on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then S/I(G) is an S-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since S is a polynomial ring, pdim G < ∞ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, by the Auslander– Buchsbaum formula (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3]), we have pdim G + depth G = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We will often use this in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' On the sets C∗(n) and C−(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let us recall the set C−(n) defined in [6, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For n ≥ 2, let C−(n) := {(1, n − 1)} ∪ � (a, b) ∈ Z2 : 1 ≤ a ≤ b, a ≤ �n 2 � , b ≤ n − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then, by the proof of [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='9], we see that C−(n) ⊂ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The difference of C−(n) and C∗(n) will be mentioned below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' See Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' [6, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let G ∈ Graph(n) and assume that α(G) = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then G itself must be a bouquet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, G is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Thus, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2, we have pdimG = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, depth G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' These imply that if (a, n − 1) ∈ Graphdepth,dim(n), then a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, (1, n − 1) ∈ C−(n) ⊂ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Given integers a, b with 1 ≤ a ≤ b ≤ n − 1, assume that a + b ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then (a, b) ∈ C− automatically holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In fact, if b = n − 1, then a = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' if b ≤ n − 2, since 2a ≤ a + b ≤ n holds, we have a ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In the case of graphs with large independence number In this section, we consider the inclusion Graphdepth,dim(n) ⊂ C∗(n) for some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, we discuss when (a, b) ∈ Graphdepth,dim(n) belongs to C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' First, we discuss the case where a = b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', the corresponding graph is Cohen–Macaulay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Assume that (b, b) ∈ Graphdepth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then (b, b) ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We see that (b, b) ∈ C∗(n) holds if and only if ⌈b/(n−b)⌉ ≤ 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Here, it follows from [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3] that if G is Cohen–Macaulay, then b ≤ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' This implies that when G is Cohen–Macaulay with b = dim G, we have b ≤ n/2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', (b, b) ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' □ 4 Next, we discuss the case where b is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Note that we always have 1 ≤ b ≤ n − 1 and the case b = n − 1 has been already discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If (a, n − 2) ̸∈ C∗(n), then (a, n − 2) ̸∈ Graphdepth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let f(a, b) := b + 1 − � b n − b � − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Note that for integers a, b with 1 ≤ a ≤ b ≤ n − 1, f(a, b) ≥ 0 holds if and only if (a, b) ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In what follows, we lead a contradiction if there exists a graph G ∈ Graph(n) with (depth G, dim G) = (a, n − 2) such that f(a, n − 2) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since f(a, n − 2) = ⌊n/2⌋ − a, we have ⌊n/2⌋ < depth G = n − pdimG, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', pdimG < ⌈n/2⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' On the other hand, since dim G = n − 2, there is an independent set W of G with |W| = n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let v1, v2 be the remaining vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', {v1, v2} = V (G) \\ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since G is connected and W is an independent set, we see that n − 2 ≤ � w∈W deg w = |N(v1) ∩ W| + |N(v2) ∩ W| = � deg v1 + deg v2 − 2, if {v1, v2} ∈ E(G), deg v1 + deg v2, if {v1, v2} ̸∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' On the other hand, since pdimG < ⌈n/2⌉, we see that deg vi < ⌈n/2⌉ for i = 1, 2 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If {v1, v2} ∈ E(G), then we see that n − 2 ≤ deg v1 + deg v2 − 2 ≤ n − 1 2 + n − 1 2 − 2 = n − 3, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, {v1, v2} ̸∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Suppose that n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we see that n − 2 ≤ deg v1 + deg v2 ≤ n − 2 2 + n − 2 2 = n − 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', we have deg w = 1 for each w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since {v1, v2} ̸∈ E(G), we see that there is no path in G connecting v1 and v2, a contradiction to the connectedness of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, n should be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Moreover, there should be at least one vertex w0 in W with deg w0 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since � w∈W deg w ≥ n − 1 and deg vi ≤ (n − 1)/2 hold, we conclude that G should look like as follows: v1 v2 w0 Note that deg v1 = deg v2 = (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, G is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' By taking a set of bouquets consisting of two bouquets whose roots are v1 and v2, we see from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 that �n 2 � > pdim G ≥ n − 2, 5 a contradiction to n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' □ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let n ≥ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If (a, n − 3) ̸∈ C∗(n), then (a, n − 3) ̸∈ Graphdepth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let f(a, b) be the same thing as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then f(a, n − 3) = ⌊2n/3⌋ − a − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Suppose that there exists a graph G ∈ Graph(n) with (depth G, dim G) = (a, n − 3) such that f(a, n − 3) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since dim G = n − 3, there is an independent set W with |W| = n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let v1, v2, v3 be the remaining vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1), we have deg vi ≤ n − a < n − ⌊2n/3⌋ + 1 = ⌈n/3⌉ + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', deg vi ≤ �n 3 � for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1) Similarly, for any strongly disjoint set B of bouquets of G, we have |F(B)| ≤ �n 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2) Here, we have the following four possibilities on the adjacencies of v1, v2, v3: (i) v1 v2 v3 v1 (ii) v2 v3 (iii) v1 v2 v3 (iv) v1 v2 v3 We divide our discussions into these four cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (i) In this case, we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1) that n − 3 ≤ � w∈W deg w = 3 � i=1 (deg vi − 2) ≤ 3 ��n 3 � − 2 � ≤ 3 �n + 2 3 − 2 � = n − 4, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (ii) By the similar computation to (i), we have that n − 3 ≤ � w∈W deg w = 3 � i=1 deg vi − 4 ≤ 3 �n 3 � − 4 = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 n − 4 if n ≡ 0 (mod 3), n − 3 if n ≡ 2 (mod 3), n − 2 if n ≡ 1 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Thus, we get the following possibilities: n ≡ 2 (mod 3) and deg w = 1 for each w ∈ W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' n ≡ 1 (mod 3) and deg w = 1 for each w ∈ W;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' n ≡ 1 (mod 3), deg w0 = 2 for some w0 ∈ W and deg w = 1 for each w ∈ W \\{w0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In the first two cases (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' the third case), G should look like the left-most one (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' the central one or the right-most one) of the following figure: v1 v2 v3 v1 v2 v3 w0 v1 v2 v3 w0 6 Note that we have |N(vi) ∩ W| ≥ ⌈n/3⌉ − 2 > 1 for i = 1, 3 and |(N(v1) ∪ N(v3)) ∩ W| ≥ 2⌈n/3⌉ − 3 in any cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, we can always take a strongly disjoint set of bouquets B consisting of two bouquets whose roots are v1 and v3 satisfying |F(B)| ≥ 2⌈n/3⌉ − 3 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (Note that “+1” comes from a stem {v1, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=') By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2), we obtain that �n 3 � ≥ 2 �n 3 � − 2 ⇐⇒ 2 ≥ �n 3 � ⇐⇒ 6 ≥ n, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (iii) Take bouquets B1, B2, B3 of G whose roots are v1, v2, v3, respectively, such that V (Bi) ∩ V (Bj) = ∅ (i ̸= j) and F({B1, B2, B3}) = W hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Assume that N(v3) \\ N(v2) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let w ∈ N(v3) \\ N(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then, by taking a strongly disjoint set B′ of bouquets consisting of B′ 2 and a bouquet {v3, w}, where B′ 2 is constructed by attaching a new flower v1 to B2, we see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2) that �n 3 � ≥ |F(B′)| = |F(B2)| + 2 = n − 3 − |F({B1, B3})| + 2 ≥ n − 1 − �n 3 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, we obtain the inequality �n 3 � ≥ n − 1 − �n 3 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', n ≤ 7, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Assume that N(v3) \\ N(v2) = ∅, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', N(v3) ⊂ N(v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Take w ∈ N(v3) and let B′ be a new bouquet with its root w and flowers v2, v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then {B1, B′} becomes strongly disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2), we have �n 3 � ≥ |F({B1, B′})| = |F(B1)| + 2 ≥ n − 3 − |F({B2})| + 2 ≥ n − 1 − �n 3 � , a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (iv) Take bouquets B1, B2, B3 of G whose roots are v1, v2, v3, respectively, such that V (Bi) ∩ V (Bj) = ∅ (i ̸= j) and F({B1, B2, B3}) = W hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If the situation (iv) happens, then there are j, k such that {Bj, Bk} is strongly disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In fact, if not, then either F(Bj) ⊂ F(Bk) or F(Bk) ⊂ F(Bj) holds for each 1 ≤ j < k ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' This implies that we have F(B1) ⊂ F(B2) ⊂ F(B3) after properly changing the indices of B1, B2, B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, F(B3) = W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Thus, n − 3 = deg v3 ≤ ⌈n/3⌉ ≤ (n + 2)/3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', n ≤ 5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let, say, B = {B1, B2} be a strongly disjoint set of bouquets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Suppose that |F(B)| ≥ ⌊2n/3⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we see that �n 3 � ≥ |F(B)| ≥ �2n 3 � − 1 =⇒ n + 2 3 ≥ 2n − 2 3 − 1 ⇐⇒ 7 ≥ n, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Suppose that |F(B)| ≤ ⌊2n/3⌋ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since |W| = |F(B)| + |F(B3)| = n − 3 and |F(B3)| ≤ deg v3 − 1 by the connectedness of G, we see that �n 3 � − 1 ≥ deg v3 − 1 ≥ |F(B3)| ≥ n − 3 − �2n 3 � + 2 = �n 3 � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, all the equalities of these inequalities are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, we have |F(B3)| = deg v3 −1 = ⌈n/3⌉−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since n−3 > ⌈n/3⌉ and G is connected, there is w ∈ W \\N(v3) adjacent to v1 or v2, say, v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we can construct a new strongly disjoint set of bouquets B′ consisting of bouquets N(v3) ∪ {v3} and {v1, w} such that |F(B′)| = ⌈n/3⌉ + 1, a contradiction to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 7 □ By using Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3, we can prove Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proof of Corolalry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' It is enough to show that Graphdepth,dim(n) ⊂ C∗(n) for n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Take (a, b) ∈ Graphdepth,dim(n) arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then the inequalities 1 ≤ a ≤ b ≤ n − 1 automatically hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If a = b, then we see that (a, b) ∈ C∗(n) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The case of b = n − 1 has been already discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, in what follows, we assume that 1 ≤ a < b ≤ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Here, we consider the set C′(n) := {(a, b) ∈ Z2 : 1 ≤ a < b ≤ n − 2} \\ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then the direct computations show the following: C′(n) = ∅ for 2 ≤ n ≤ 6, C′(7) = {(4, 5)}, C′(8) = {(5, 6)}, C′(9) = {(5, 7), (6, 7)}, C′(10) = {(6, 7), (6, 8), (7, 8)}, C′(11) = {(6, 9), (7, 8), (7, 9), (8, 9)}, C′(12) = {(7, 10), (8, 9), (8, 10), (9, 10)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Suppose that (a, b) ̸∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then (a, b) ∈ C′(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' By the above compu- tations, we see that all elements of �12 n=2 C′(n) satisfy b = n − 2 or b = n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, it follows from Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 that (a, b) ̸∈ Graphdepth,dim(n), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Therefore, C∗(n) = Graphdepth,dim(n) holds if n ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We see that C′(13) = {(7, 11), (8, 9), (8, 10), (8, 11), (9, 10), (9, 11), (10, 11)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since we cannot claim that (8, 9) ̸∈ Graphdepth,dim(13) only by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3, the above proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4 is not available if n = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Instead, we can claim that Graphdepth,dim(13) = C∗(13) or C∗(13) ∪ {(8, 9)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For small n’s, the direct computations show the following: C∗(n) = C−(n) for n = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , 8, 10, C∗(9) = C−(9) ∪ {(5, 6)}, C∗(11) = C−(11) ∪ {(6, 6), (6, 7), (6, 8)}, C∗(12) = C−(12) ∪ {(7, 8), (7, 9), (8, 9)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Note that [6, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='11] also mentions the similar equalities in the case 3 ≤ n ≤ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In the case of chordal graphs In this section, we discuss the behavior of pairs of dimensions and depths of edge ideals of chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The inclusion C∗(n) ⊂ Graphchordal depth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' First, we discuss the inclusion C∗(n) ⊂ Graphchordal depth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Here, we remark that since we only know that Graphchordal depth,dim(n) ⊂ Graphdepth,dim(n) by definition, we cannot immediately claim C∗(n) ⊂ Graphchordal depth,dim(n) from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let (a, b) ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We divide the discussions into two cases: (i) a + b > n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' or (ii) a + b ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (i) In this case, we can apply the same proof as that of [6, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In that proof, the graphs G(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , sm) given in [6, Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='5] play the essential role and we see that G(m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , sm) is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, we obtain the inclusion by the same discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (ii) In this case, we cannot apply the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In fact, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4 are used for the proof in the case a + b ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Especially, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 uses the technique of S-cone (also known as S-suspension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In general, the chordality is not preserved by taking S-cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='3 is not immediately available for chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Instead, we directly prove the following: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let a, b, n be integers satisfying 1 ≤ a ≤ b ≤ n−1 and a+b ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then there exists a chordal graph G ∈ Graph(n) such that (depth G, dim G) = (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Namely, we have (a, b) ∈ Graphchordal depth,dim(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Note that the assumption a+b ≤ n immediately implies (a, b) ∈ C∗(n) (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let G be the graph defined as follows: V (G) = U ⊔ W, where U = {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , un−b} and W = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , wb};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' E(G) = {{u, u′} : u, u′ ∈ U} ∪ {{ui, wi} : 1 ≤ i ≤ a − 1} ∪ {{uj, wk} : a ≤ j ≤ n − b, a ≤ k ≤ b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' It is straightforward to check that the ordering w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , wb, u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , un−b becomes a perfect elimination ordering of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, G is chordal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Moreover, we can easily see that W is an independent set with |W| = α(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, dim G = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In what follows, we prove that depth G = a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', pdim G = n − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Regarding the minimal vertex covers of G, we observe that any minimal vertex cover contains at least (n−b−1) vertices of U since U forms a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' In particular, we see that U itself becomes a minimal vertex cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Fix uj ∈ U and let C be a minimal vertex cover not containing uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then U \\ {uj} ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If 1 ≤ j ≤ a − 1, then wj ∈ C, so C = (U \\ {uj}) ∪ {wj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If a ≤ j ≤ n − b, then wa, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , wb ∈ C, so C = (U \\ {uj}) ∪ {wa, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , wb}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Therefore, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2, we conclude that pdim G = max{|C| : C is a minimal vertex cover of G} = max{|U|, |U| + b − a} = n − a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' The inclusion Graphchordal depth,dim(n) ⊂ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' For the proof of this inclusion, we pre- pare the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (1) For any vertex v of a graph G, we have dim G = max{dim(G \\ N[v]) + 1, dim(G \\ v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 9 (2) Let G be a chordal graph and let vn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , v1 be a perfect elimination ordering of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then we have depth G = min{depth(G \\ N[vn]) + 1, depth(G \\ vn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (1) Let W be an independent set of G with |W| = α(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then the required equality directly follows from the following observations: If v ∈ W, then W \\ {v} ⊂ V (G \\ N[v]) becomes an independent set of G \\ N[v] with |W \\ {v}| = α(G \\ N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If v ̸∈ W, then W ⊂ V (G \\ v) becomes an independent set of G \\ v with |W| = α(G \\ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (2) We prove the equality by induction on |V (G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let v = vn and deg v = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let C be a minimal vertex cover of G with |C| = pdim G (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If v ∈ C, then C′ = C \\ {v} becomes a minimal vertex cover of G \\ v with pdim(G \\ v) = |C′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, by the hypothesis of induction, we see that depth G = |V (G)| − pdimG = |V (G)| − (|C′| + 1) = |V (G \\ v)| − |C′| = |V (G \\ v)| − pdim(G \\ v) = depth(G \\ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' If v ̸∈ C, since N[v] forms a clique by definition of perfect elimination ordering, we see that N(v) ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Thus, C′′ = C \\ N[v] becomes a minimal vertex cover of G \\ N[v] with pdim(G \\ N[v]) = |C′′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Hence, we see that depth G = |V (G)| − pdim G = |V (G)| − (|C′′| + m) = (|V (G \\ N[v])| + 1) − |C′′| = |V (G \\ N[v])| − pdim(G \\ N[v]) + 1 = depth(G \\ N[v]) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Here, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 says that if C0 is a minimal vertex cover with |C0| ̸= pdim G, then |C0| < pdim G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Thus, the required equality follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' □ Now, we prove the inclusion Graphchordal depth,dim(n) ⊂ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Take (a, b) ∈ Graphchordal depth,dim(n) arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Then there is a chordal graph G such that (depth G, dim G) = (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' To prove (a, b) ∈ C∗(n), it is enough to show that b ≤ (n − b)(b − a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1) In fact, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1) is equivalent to the following: b n − b ≤ b − a + 1 ⇐⇒ � b n − b � ≤ b − a + 1 ⇐⇒ a ≤ b + 1 − � b n − b � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' This means that (a, b) ∈ C∗(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let vn, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' , v1 be a perfect elimination ordering of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' We prove the inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='1) by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Let v = vn and let a′ = depth(G \\ v), b′ = dim(G \\ v), a′′ = depth(G \\ N[v]) and b′′ = dim(G \\ N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 (1), either b = b′ or b = b′′ + 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Assume that b = b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since a ≤ a′ holds by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='2 (2), we see from the hypothesis of induction that b = b′ ≤ ((n − 1) − b′)(b′ − a′ + 1) ≤ (n − b)(b − a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 10 Assume that b = b′′ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Since a ≤ a′′ + 1, we see that b = b′′ + 1 ≤ ((n − deg v − 1) − b′′)(b′′ − a′′ + 1) + 1 ≤ (n − deg v − b)(b − a + 1) + 1 = (n − b)(b − a + 1) − (deg v(b − a + 1) − 1) ≤ (n − b)(b − a + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' References [1] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Bruns and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Herzog, “Cohen-Macaulay rings, revised edition”, Cambridge University Press, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Matsuda, Induced matching numbers of finite graphs and edge ideals, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Algebra 532 (2019), 311–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Katzman, Characteristic-independence of Betti numbers of graph ideals, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' A 113 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' 3, 435–454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Kimura, Non-vanishingness of Betti numbers of edge ideals, Harmony of Gr¨obner bases and the modern industrial society, 153–168, World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=', Hackensack, NJ, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' [10] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Zheng, Resolutions of facet ideals, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Algebra 32 (2004), 2301–2324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Higashitani) Department of Pure and Applied Mathematics, Graduate School of Infor- mation Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan Email address: higashitani@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='jp (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Kanno) Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan Email address: u825139b@ecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='jp (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content=' Ueji) Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan Email address: u136547i@alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} +page_content='jp 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE3T4oBgHgl3EQf3Qv5/content/2301.04763v1.pdf'} diff --git a/ldAyT4oBgHgl3EQfyfmD/content/tmp_files/2301.00685v1.pdf.txt b/ldAyT4oBgHgl3EQfyfmD/content/tmp_files/2301.00685v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5c4cb801204f3bf24f244a08f9dd4c1b4632e49 --- /dev/null +++ b/ldAyT4oBgHgl3EQfyfmD/content/tmp_files/2301.00685v1.pdf.txt @@ -0,0 +1,536 @@ +arXiv:2301.00685v1 [math.SP] 2 Jan 2023 +ON THE CENTRAL LIMIT THEOREM FOR LINEAR +EIGENVALUE STATISTICS ON RANDOM SURFACES +OF LARGE GENUS +ZE´EV RUDNICK AND IGOR WIGMAN +Abstract. We study the fluctuations of smooth linear statistics +of Laplace eigenvalues of compact hyperbolic surfaces lying in short +energy windows, when averaged over the moduli space of surfaces +of a given genus. The average is taken with respect to the Weil- +Petersson measure. We show that first taking the large genus limit, +then a short window limit, the distribution tends to a Gaussian. +The variance was recently shown to be given by the corresponding +quantity for the Gaussian Orthogonal Ensemble (GOE), and the +Gaussian fluctuations are also consistent with those in Random +Matrix Theory, as conjectured in the physics literature for a fixed +surface. +1. Introduction +Let X be a compact hyperbolic surface of genus g ≥ 2, and λj = +1/4 + r2 +j be the eigenvalues of the Laplacian on X. We examine the +statistics of these eigenvalues in short(ish) intervals. For this purpose, +we use a smooth linear statistic as in [13]: for an even test function +f with compactly supported Fourier transform �f ∈ C∞ +c (R) and τ > 0 +define +Nf,L,τ(X) := +� +j≥0 +f (L (rj − τ)) + f (L (rj + τ)) . +(When τ = 0 the definition changes to � +j f(Lrj).) +In [13] it was shown that in the double limit, of first averaging over +the moduli space Mg of hyperbolic surfaces of genus g with respect to +the Weil-Petersson measure, taking the large genus limit g → ∞, and +then taking the limit L → ∞, the variance of Nf,L,τ is given by the +Date: January 3, 2023. +This research was supported by the European Research Council (ERC) under +the European Union’s Horizon 2020 research and innovation programme (grant +agreement No. 786758) and by the Israel Science Foundation (grant No. 1881/20). +1 + +2 +ZE´EV RUDNICK AND IGOR WIGMAN +GOE variance: denoting by EWP +g +the average over Mg, we have +(1.1) +lim +L→∞ +� +lim +g→∞ EWP +g +���NL,τ − EWP +g +(NL,τ) +��2�� += Σ2 +GOE(f) +where Σ2 +GOE(f) = 2 +� ∞ +−∞ |x| �f(x)2dx. This supports conjectures of Michael +Berry [3, 4] on number variance for chaotic systems. +In this note, we examine the distribution of NL,τ(X). The distri- +bution is conjectured to be Gaussian for any fixed hyperbolic surface +in the large energy limit [2], see also [12] for the modular surface, as +one can expect based on analogous behaviour in random matrix the- +ory [5], and for the zeros of the Riemann zeta function [15]. Here, we +take the surface X to be random with respect to the Weil Petersson +measure on the moduli space Mg, and show that in the double limit +the distribution tends to a normal distribution: +Theorem 1.1. For all bounded continuous functions h, we have +lim +L→∞ lim +g→∞ EWP +g +� +h +� +NL,τ − EWP +g +(NL,τ) +� +Σ2 +GOE(f) +�� += +1 +√ +2π +� ∞ +−∞ +h(t)e−t2/2dt. +The opening step in the proof of Theorem 1.1 is to express NL,τ +as a sum over closed geodesics via the Selberg Trace Formula. +We +then apply a recent result of Mirzakhani and Petri [10], that the set +of lengths of primitive closed geodesics, when thought of as a random +point process on the moduli space Mg, converge as g → ∞, to a Poisson +point process with a certain intensity. We thus obtain, for each L, a +certain random variable, the value of a functional HL,τ on the Poisson +point process, which we then show has a Gaussian limit distribution as +L → ∞. +A noteworthy feature of our proof of Theorem 1.1 is that we avoid +using the method of moments, and instead use various features of the +theory of point processes to pass directly to the limiting Poisson point +process, where the computation of the limit L → ∞ is greatly stream- +lined. +There are other random models of hyperbolic surfaces, and various +spectral statistics in these models have been explored recently [7, 14], +for instance in the random cover model, the analogue of [13] for the +smooth number variance has recently been obtained by Fr´ed´eric Naud +[11], who also obtained GUE statistics for the twisted Laplacian. The +CLT in the random cover model will be in the MSc thesis of Yotam +Maoz [9]. + +THE CLT ON RANDOM SURFACES OF LARGE GENUS +3 +2. Background on point processes +2.1. Generalities. We give brief background on point processes, using +the survey [6] as our basic reference. +Our background space will be the non-negative reals R≥0 = [0, ∞). +A point process on R≥0 is a random assignment of a set of points in +R≥0, each of which is assumed to be locally finite. We assume that +the origin is not one of these points. We denote by N the set of all +realizations of point processes on R≥0. These can be thought of as +atomic measures +(2.1) +µ = +∞ +� +j=1 +δ(xj) +with 0 < x1 ≤ x2 ≤ . . . a discrete set of points (so the only possible +accumulation point is at infinity) each occurring with finite multiplicity. +In particular, for such µ, we have µ(0) = 0 and µ{(0, t]} < ∞. +We have a topology on N , given by declaring that a sequence of +measures µn ∈ N converges vaguely to another measure µ∞ ∈ N , if +for each continuity point of µ∞{(0, t]}, we have limn→∞ µn{(0, t]} = +µ{(0, t]}. We denote this by µn +v→ µ. Equivalently [6, Theorem 1], for +all compactly supported continuous functions G on R≥0 we have +(2.2) +lim +n→∞ +� ∞ +0 +Gdµn = +� ∞ +0 +Gdµ∞. +With this topology, N becomes a separable, complete space which is +metrizable [6]. +For a point process N on R≥0 and a bounded Borel set B ⊂ R≥0, we +denote by N(B) the number of points in B; this is a random variable. +The measure λ(B) = E(N(B)) is called the intensity of the process. +As an important example, a Poisson point process with intensity λ +is a point process Pois(λ) so that for any Borel set B, the random +variables N(B) is a Poisson variable with intensity λ(B), and so that +for any choice of disjoint Borel sets B1, . . . , Bk, the random variables +N(B1), . . . , N(Bk) are independent. +Given point processes Nn, N∞ on R≥0, we say that Nn converges +in distribution to N∞ (written Nn +d→ N∞) if the sequence of random +vectors (Nn(B1), . . . , Nn(Bk)) converges in distribution to the random +vector (N∞(B1), . . . , N∞(Bk)) for all k ≥ 1 and all choices of bounded +Borel sets Bi with boundaries satisfying N∞(∂Bi) = 0 almost surely +for all i. This is equivalent to requiring that +E(h(Nn)) → E(h(N∞)) + +4 +ZE´EV RUDNICK AND IGOR WIGMAN +for all bounded continuous functions h : N → R (continuous means +that whenever we have a sequence νn ∈ N which converges vaguely to +ν∞, we have limn h(νn) = h(ν∞)). +We next recall the continuous mapping theorem (see e.g. [1, Theo- +rem 9.4.2]), which in our context, states that if we have a sequence of +random variables Xn : Ωn → N , each defined on its own probability +space Ωn, which converge in distribution to another random variable +X∞ : Ω∞ → N , and G : N → R is a continuous map1 then the ran- +dom variables G(Xn) : Ωn → R converge in distribution to the random +variable G(X∞) : Ω∞ → R. +2.2. A functional. It important for us to extend the functional form +(2.2) of vague convergence to allow taking G which is continuous on +the positive reals, but allowed not to extend continuously to all of R≥0, +e.g. to blow up at x = 0. We define a function G : N → R by +G : µ ∈ N �→ +� ∞ +0 +Gdµ := +� +x∈µ +G(x). +We claim that G : N → R is continuous: +Lemma 2.1. Assume G ∈ C(R>0) is continuous on the positive reals, +and supported in a bounded interval. Then G : N → R is a continuous +mapping. +Proof. What we need to show is that if µ∞, µn ∈ N with µn +v→ µ∞ +vaguely, then limn→∞ G(µn) = G(µ∞), that is (2.2) holds for G. +Denote the atoms of the measures µn, µ∞ in the representation (2.1) +by xn,j (n = 1, 2 . . . , ∞, j = 1, 2, . . . ): +µn = +� +j≥1 +δ(xn,j). +Note that vague convergence µn → µ∞ implies convergence of the point +sets, which are the discontinuity points of the distribution functions, cf +[6, Theorem 25]: Suppose that G is supported in (0, L). If x∞,1 . . . , x∞,J +are the atoms of µ∞ in (0, L), so that G(µ∞) = �J +j=1 G(x∞,j), then for +all n ≫ 1, there are exactly J atoms xn,1, . . . , xn,J of µn in (0, L) and +limn→∞ xn,j = x∞,j for all j = 1, . . . , J. Hence for n ≫ 1, +lim +n→∞ G(µn) = lim +n→∞ +J +� +j=1 +G(xn,j) = +J +� +j=1 +lim +n→∞ G(xn,j). +1We can also allow G to not be continuous in a set of measure zero w.r.t. the +distribution of X∞. + +THE CLT ON RANDOM SURFACES OF LARGE GENUS +5 +Since G(x) is continuous at xj, we have +lim +n→∞ G(xn,j) = G(lim +n xn,j) = G(x∞,j), +so that +lim +n→∞ G(µn) = +J +� +j=1 +G(x∞,j) = G(µ∞) +as claimed. +□ +3. Reduction to a Poisson approximation +3.1. An expansion. Using Selberg’s trace formula, we expand the +centered variable NL,τ − EWP(NL,τ) as a sum over closed geodesics [13] +NL,τ − EWP(NL,τ) = Nosc − EWP(Nosc) +with +(3.1) +Nosc(X) = +� +γ +HL,τ (ℓγ (X)) +the sum over all primitive, non-oriented, closed geodesics {γ} on X, +with lengths ℓγ(X) where +HL,τ(x) = 2x +L +∞ +� +k=1 +F(kx), +F(x) = +�f +� x +L +� +cos(xτ) +sinh(x/2) +. +For each x > 0, the sum defining HL,τ(x) is finite, ranging up to +k ≤ L/x, and so we get a continuous function on the positive reals. +But there need be no limit as x → 0; if �f(0) > 0 then for x small, +HL,τ(x) ≫ +� +k≪x−1/2 +x +sinh(kx) +�f(0) ≫ +� +k≪x−1/2 +�f(0) +k +≫ log 1 +x +which blows up as x → 0. +3.2. The Mirzakhani-Petri Theorem. For each genus g ≥ 2, the +moduli space Mg equipped with the Weil-Petersson probability mea- +sure gives a probability space, and the length spectrum gives a point +process Lg : Mg → N , assigning to a surface X ∈ Mg its (primitive, +unoriented) length spectrum Lg(X) = {0 < ℓ1 ≤ ℓ2 ≤ . . . }. For an +interval [a, b] ⊂ R≥0, we have the random variable +Ng([a, b]) : X �→ #Lg(X) ∩ [a, b]. +Further, let Pois(νMP) be the Poisson point process on R≥0 with +intensity +νMP(x) = 2 sinh2(x/2) +x +dx, + +6 +ZE´EV RUDNICK AND IGOR WIGMAN +that is for each interval I ⊂ R≥0, the counting function NMP(I) is a +Poisson random variable with intensity νMP(I) = +� +I +2 sinh2(x/2) +x +dx. +Mirzakhani and Petri [10] showed that for any set of disjoint intervals +Ii = [ai, bi], i = 1, . . . , k, the random variable (Ng(I1), . . . Ng(Ik)) con- +verge in distribution to the Poisson random variable (NMP(I1), . . . NMP(Ik)), +in other words that the point processes Lg converge in distribution to +the Poisson point process Pois(νMP). +3.3. A random approximation. Define a mapping N → R +HL,τ = +� +ℓ∈L +HL,τ(ℓ). +We note that HL,τ is a continuous function on the positive reals. There- +fore, by Lemma 2.1, the functional HL,τ : N → R is continuous with +respect to the vague topology on the space of point processes on the +positive reals. +By the Mirzakhani-Petri theorem combined with the continuous map- +ping theorem, we find that the random variables Nosc = HL,τ ◦ Lg : +Mg → R converge in distribution to the random variable +SL,τ := HL,τ ◦ Pois(νMP). +We recall that a sequence of real valued random variables Yi con- +verges in distribution to a random variable Y if the cumulative distri- +bution functions (CDF’s) Fi(t) = Prob(Yi ≤ t) converge pointwise to +FY (t) in every continuity point of FY . This is equivalent to requiring +that for every bounded continuous function h, we have +lim +i→∞ E (h (Yi)) = E (h (Y )) . +In turn this, by L´evy’s continuity theorem, is equivalent to pointwise +convergence of the characteristic functions: Recall that for a random +variable Y , the characteristic function is φY (t) = E(exp(itY )), t real. +Hence the characteristic functions of Nosc converge as g → ∞ to the +characteristic function of SL,τ: +Corollary 3.1. We have convergence of characteristic functions +lim +g→∞ EWP +g +(exp (it (Nosc))) = EPois (exp (it (SL,τ))) . +3.4. Working with the Poisson approximation. We will next show +that in the limit L → ∞, the characteristic function of SL,τ tends to +that of a Gaussian. By L´evy’s continuity theorem (see e.g. [1, Chapter +10]), this will prove Theorem 1.1. + +THE CLT ON RANDOM SURFACES OF LARGE GENUS +7 +Theorem 3.2. For all τ > 0, +lim +L→∞ EPois(itSL,τ − E(SL,τ) +� +Σ2 +GOE(f) +) = e−t2/2. +By its definition, the expected value is zero. That the variance is +correct was demonstrated in the course of the proof of [13, Lemma +5.2]. To show Gaussianity, we use a combinatorial shortcut, of using +cumulants. +Recall that the cumulants of a random variable S are the coefficients +in the Taylor expansion of the cumulant generating function log E(ezS) +about z = 0: +log E(ezS) = +∞ +� +m=1 +κm(S)zm +m!. +The first moment is the first cumulant, the second cumulant is just +the variance, and the third cumulant is the centered third moment. In +general, the n-th centered moment is an n-th-degree polynomial in the +first n cumulants, for instance, the fourth centered moment is κ4 +3κ2 +2. +Gaussianity is equivalent to vanishing of the higher cumulants κm(S) +for m ≥ 3. So we want to show: +Proposition 3.3. For all m ≥ 3, +lim +L→∞ κm(SL,τ) = 0. +We will use Campbell’s formula [8]: Let L be a Poisson point process +with intensity ν, let H : R → R be a measurable function, and define +the random sum +S = +� +ℓ∈L +H(ℓ). +Then (assuming everything converges) the expected value and variance +of S are given by +E(S) = +� +R +H(x)dν(x), +Var(S) = +� +R +H(x)2dν(x) +and the moment generating function is given by +(3.2) +E(ezS) = exp +�� +R +[ezH(x) − 1]dν(x) +� +. +Using the formula (3.2) gives +log E(ezS) = +� +m≥1 +zm +m! +� +R +H(x)mdν(x). +Therefore, Proposition 3.3 follows from: + +8 +ZE´EV RUDNICK AND IGOR WIGMAN +Proposition 3.4. For all m ≥ 3, uniformly in τ, +(3.3) +lim +L→∞ +� +R +HL,τ(x)mdνMP(x) = 0. +Proof. We use: +Lemma 3.5. [13, Lemma 6.3] Let L > 2. Then uniformly in τ, +i) For 0 < x < 1/2, we have |HL,τ(x)| ≪ 1 +L log(L/x). +ii) For x ≥ 1/2, we have |HL,τ(x)| ≪ 1 +Lx exp(−x/2). +Applying Lemma 3.5 we obtain +� ∞ +0 +HL,τ(x)mdνMP(x) ≪ +� 1/2 +0 +1 +Lm(log L +x )msinh(x/2)2 +x +dx ++ +� L +1/2 +1 +Lmxme−mx/2sinh(x/2)2 +x +dx. +For 0 < x < 1/2, use sinh(x/2) < 1.1 · x/2 ≪ x to bound +� 1/2 +0 +(log L +x )msinh(x/2)2 +x +dx ≪ +� 1/2 +0 +x(log L +x )mdx += L2 +� ∞ +2L +(log y)mdy +y3 +≪m L2 +� L2 +2L +(log L2)mdy +y3 + L2 +� ∞ +L2 y1/2dy +y3 +≪ (log L)m + L−3 ≪ (log L)m +so that the first integral contributes O((log L/L)m). For the second +integral, use sinh(x/2) < ex/2 for x > 0 to bound, for m ≥ 3, +� L +1/2 +xme−mx/2sinh(x/2)2 +x +dx < +� ∞ +0 +e−( m +2 −1)xxm−1dx = 2mΓ(m) +(m − 2)m +so that the second integral contributes Om(L−m). Thus we obtain for +m ≥ 3 +� ∞ +0 +HL,τ(x)mdνMP(x) ≪m L−m+o(1) +which proves (3.3). +□ +This concludes the proof of Theorem 3.2, hence of Theorem 1.1. + +THE CLT ON RANDOM SURFACES OF LARGE GENUS +9 +References +[1] Athreya, Krishna B.; Lahiri, Soumendra N. Measure theory and probability +theory. Springer Texts in Statistics. Springer, New York, 2006. +[2] Aurich, R., Bolte, and Steiner, F. Universal signatures of quantum chaos, Phys. +Rev. Lett. 73, no. 10, 1356–1359 (1994). +[3] Berry, M. V. Semiclassical theory of spectral rigidity. Proc. Roy. Soc. London +Ser. A 400 (1985), no. 1819, 229–251. +[4] Berry, M. V. Fluctuations in numbers of energy levels. Stochastic processes in +classical and quantum systems (Ascona, 1985), 47–53, Lecture Notes in Phys., +262, Springer, Berlin, 1986. +[5] Diaconis, P. and Evans, S. Linear functionals of eigenvalues of random matri- +ces. Trans. Amer. Math. Soc. 353 (2001), no. 7, 2615–2633. +[6] Grandell, J. Point Processes and Random Measures. Advances in Applied +Probability, Vol. 9, No. 3 (1977), pp. 502–526. +[7] Hide, W. and Magee, M. Near optimal spectral gaps for hyperbolic surfaces. +arXiv:2107.05292 [math.SP] +[8] Kingman, J. F. C. Poisson processes. Oxford Studies in Probability, 3. Ox- +ford Science Publications. The Clarendon Press, Oxford University Press, New +York, 1993. +[9] Maoz, Y. MSc thesis, Tel Aviv University 2023, in preparation. +[10] Mirzakhani, M. and Petri, B. Lengths of closed geodesics on random surfaces +of large genus. Comment. Math. Helv. 94 (2019), no. 4, 869–889. +[11] Naud, F. Random covers of compact surfaces and smooth linear spectral sta- +tistics. arXiv:2209.07941 [math.SP] +[12] Rudnick Z. A central limit theorem for the spectrum of the modular group, +Annales Henri Poincare 6 (2005), 863–883. +[13] Rudnick, Z. GOE statistics on the moduli space of surfaces of large genus . +arXiv:2202.06379 [math.SP] +[14] Shen, Y. and Wu, Y. The Cheeger constants of random Belyi surfaces. +arXiv:2204.09853 [math.DG]. International Mathematics Research Notices +(IMRN), to appear. +[15] Selberg, A. Contributions to the theory of the Riemann zeta-function. Arch. +Math. Naturvid., 48(5):89-155, 1946. +School of Mathematical Sciences, Tel Aviv University, Tel Aviv +69978, Israel +Email address: rudnick@tauex.tau.ac.il +Department of Mathematics, King’s College London, UK +Email address: igor.wigman@kcl.ac.uk + diff --git a/ldAyT4oBgHgl3EQfyfmD/content/tmp_files/load_file.txt b/ldAyT4oBgHgl3EQfyfmD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aeac47fe9f809e7fc679bd6da73c11298cabce37 --- /dev/null +++ b/ldAyT4oBgHgl3EQfyfmD/content/tmp_files/load_file.txt @@ -0,0 +1,302 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf,len=301 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='00685v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='SP] 2 Jan 2023 ON THE CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS ON RANDOM SURFACES OF LARGE GENUS ZE´EV RUDNICK AND IGOR WIGMAN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We study the fluctuations of smooth linear statistics of Laplace eigenvalues of compact hyperbolic surfaces lying in short energy windows, when averaged over the moduli space of surfaces of a given genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The average is taken with respect to the Weil- Petersson measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We show that first taking the large genus limit, then a short window limit, the distribution tends to a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The variance was recently shown to be given by the corresponding quantity for the Gaussian Orthogonal Ensemble (GOE), and the Gaussian fluctuations are also consistent with those in Random Matrix Theory, as conjectured in the physics literature for a fixed surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Introduction Let X be a compact hyperbolic surface of genus g ≥ 2, and λj = 1/4 + r2 j be the eigenvalues of the Laplacian on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We examine the statistics of these eigenvalues in short(ish) intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For this purpose, we use a smooth linear statistic as in [13]: for an even test function f with compactly supported Fourier transform �f ∈ C∞ c (R) and τ > 0 define Nf,L,τ(X) := � j≥0 f (L (rj − τ)) + f (L (rj + τ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' (When τ = 0 the definition changes to � j f(Lrj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=') In [13] it was shown that in the double limit, of first averaging over the moduli space Mg of hyperbolic surfaces of genus g with respect to the Weil-Petersson measure, taking the large genus limit g → ∞, and then taking the limit L → ∞, the variance of Nf,L,τ is given by the Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' This research was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 786758) and by the Israel Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 1881/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 1 2 ZE´EV RUDNICK AND IGOR WIGMAN GOE variance: denoting by EWP g the average over Mg, we have (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1) lim L→∞ � lim g→∞ EWP g ���NL,τ − EWP g (NL,τ) ��2�� = Σ2 GOE(f) where Σ2 GOE(f) = 2 � ∞ −∞ |x| �f(x)2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' This supports conjectures of Michael Berry [3, 4] on number variance for chaotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' In this note, we examine the distribution of NL,τ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The distri- bution is conjectured to be Gaussian for any fixed hyperbolic surface in the large energy limit [2], see also [12] for the modular surface, as one can expect based on analogous behaviour in random matrix the- ory [5], and for the zeros of the Riemann zeta function [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Here, we take the surface X to be random with respect to the Weil Petersson measure on the moduli space Mg, and show that in the double limit the distribution tends to a normal distribution: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For all bounded continuous functions h, we have lim L→∞ lim g→∞ EWP g � h � NL,τ − EWP g (NL,τ) � Σ2 GOE(f) �� = 1 √ 2π � ∞ −∞ h(t)e−t2/2dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The opening step in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1 is to express NL,τ as a sum over closed geodesics via the Selberg Trace Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We then apply a recent result of Mirzakhani and Petri [10], that the set of lengths of primitive closed geodesics, when thought of as a random point process on the moduli space Mg, converge as g → ∞, to a Poisson point process with a certain intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We thus obtain, for each L, a certain random variable, the value of a functional HL,τ on the Poisson point process, which we then show has a Gaussian limit distribution as L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' A noteworthy feature of our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1 is that we avoid using the method of moments, and instead use various features of the theory of point processes to pass directly to the limiting Poisson point process, where the computation of the limit L → ∞ is greatly stream- lined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' There are other random models of hyperbolic surfaces, and various spectral statistics in these models have been explored recently [7, 14], for instance in the random cover model, the analogue of [13] for the smooth number variance has recently been obtained by Fr´ed´eric Naud [11], who also obtained GUE statistics for the twisted Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The CLT in the random cover model will be in the MSc thesis of Yotam Maoz [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' THE CLT ON RANDOM SURFACES OF LARGE GENUS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Background on point processes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Generalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We give brief background on point processes, using the survey [6] as our basic reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Our background space will be the non-negative reals R≥0 = [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' A point process on R≥0 is a random assignment of a set of points in R≥0, each of which is assumed to be locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We assume that the origin is not one of these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We denote by N the set of all realizations of point processes on R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' These can be thought of as atomic measures (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1) µ = ∞ � j=1 δ(xj) with 0 < x1 ≤ x2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' a discrete set of points (so the only possible accumulation point is at infinity) each occurring with finite multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' In particular, for such µ, we have µ(0) = 0 and µ{(0, t]} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We have a topology on N , given by declaring that a sequence of measures µn ∈ N converges vaguely to another measure µ∞ ∈ N , if for each continuity point of µ∞{(0, t]}, we have limn→∞ µn{(0, t]} = µ{(0, t]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We denote this by µn v→ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Equivalently [6, Theorem 1], for all compactly supported continuous functions G on R≥0 we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2) lim n→∞ � ∞ 0 Gdµn = � ∞ 0 Gdµ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' With this topology, N becomes a separable, complete space which is metrizable [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For a point process N on R≥0 and a bounded Borel set B ⊂ R≥0, we denote by N(B) the number of points in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' this is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The measure λ(B) = E(N(B)) is called the intensity of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' As an important example, a Poisson point process with intensity λ is a point process Pois(λ) so that for any Borel set B, the random variables N(B) is a Poisson variable with intensity λ(B), and so that for any choice of disjoint Borel sets B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , Bk, the random variables N(B1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , N(Bk) are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Given point processes Nn, N∞ on R≥0, we say that Nn converges in distribution to N∞ (written Nn d→ N∞) if the sequence of random vectors (Nn(B1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , Nn(Bk)) converges in distribution to the random vector (N∞(B1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , N∞(Bk)) for all k ≥ 1 and all choices of bounded Borel sets Bi with boundaries satisfying N∞(∂Bi) = 0 almost surely for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' This is equivalent to requiring that E(h(Nn)) → E(h(N∞)) 4 ZE´EV RUDNICK AND IGOR WIGMAN for all bounded continuous functions h : N → R (continuous means that whenever we have a sequence νn ∈ N which converges vaguely to ν∞, we have limn h(νn) = h(ν∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We next recall the continuous mapping theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' [1, Theo- rem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2]), which in our context, states that if we have a sequence of random variables Xn : Ωn → N , each defined on its own probability space Ωn, which converge in distribution to another random variable X∞ : Ω∞ → N , and G : N → R is a continuous map1 then the ran- dom variables G(Xn) : Ωn → R converge in distribution to the random variable G(X∞) : Ω∞ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' A functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' It important for us to extend the functional form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2) of vague convergence to allow taking G which is continuous on the positive reals, but allowed not to extend continuously to all of R≥0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' to blow up at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We define a function G : N → R by G : µ ∈ N �→ � ∞ 0 Gdµ := � x∈µ G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We claim that G : N → R is continuous: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Assume G ∈ C(R>0) is continuous on the positive reals, and supported in a bounded interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Then G : N → R is a continuous mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' What we need to show is that if µ∞, µn ∈ N with µn v→ µ∞ vaguely, then limn→∞ G(µn) = G(µ∞), that is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2) holds for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Denote the atoms of the measures µn, µ∞ in the representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1) by xn,j (n = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , ∞, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' ): µn = � j≥1 δ(xn,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Note that vague convergence µn → µ∞ implies convergence of the point sets, which are the discontinuity points of the distribution functions, cf [6, Theorem 25]: Suppose that G is supported in (0, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' If x∞,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , x∞,J are the atoms of µ∞ in (0, L), so that G(µ∞) = �J j=1 G(x∞,j), then for all n ≫ 1, there are exactly J atoms xn,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , xn,J of µn in (0, L) and limn→∞ xn,j = x∞,j for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Hence for n ≫ 1, lim n→∞ G(µn) = lim n→∞ J � j=1 G(xn,j) = J � j=1 lim n→∞ G(xn,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 1We can also allow G to not be continuous in a set of measure zero w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' the distribution of X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' THE CLT ON RANDOM SURFACES OF LARGE GENUS 5 Since G(x) is continuous at xj, we have lim n→∞ G(xn,j) = G(lim n xn,j) = G(x∞,j), so that lim n→∞ G(µn) = J � j=1 G(x∞,j) = G(µ∞) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Reduction to a Poisson approximation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' An expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Using Selberg’s trace formula, we expand the centered variable NL,τ − EWP(NL,τ) as a sum over closed geodesics [13] NL,τ − EWP(NL,τ) = Nosc − EWP(Nosc) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1) Nosc(X) = � γ HL,τ (ℓγ (X)) the sum over all primitive, non-oriented, closed geodesics {γ} on X, with lengths ℓγ(X) where HL,τ(x) = 2x L ∞ � k=1 F(kx), F(x) = �f � x L � cos(xτ) sinh(x/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For each x > 0, the sum defining HL,τ(x) is finite, ranging up to k ≤ L/x, and so we get a continuous function on the positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' But there need be no limit as x → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' if �f(0) > 0 then for x small, HL,τ(x) ≫ � k≪x−1/2 x sinh(kx) �f(0) ≫ � k≪x−1/2 �f(0) k ≫ log 1 x which blows up as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The Mirzakhani-Petri Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For each genus g ≥ 2, the moduli space Mg equipped with the Weil-Petersson probability mea- sure gives a probability space, and the length spectrum gives a point process Lg : Mg → N , assigning to a surface X ∈ Mg its (primitive, unoriented) length spectrum Lg(X) = {0 < ℓ1 ≤ ℓ2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For an interval [a, b] ⊂ R≥0, we have the random variable Ng([a, b]) : X �→ #Lg(X) ∩ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Further, let Pois(νMP) be the Poisson point process on R≥0 with intensity νMP(x) = 2 sinh2(x/2) x dx, 6 ZE´EV RUDNICK AND IGOR WIGMAN that is for each interval I ⊂ R≥0, the counting function NMP(I) is a Poisson random variable with intensity νMP(I) = � I 2 sinh2(x/2) x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Mirzakhani and Petri [10] showed that for any set of disjoint intervals Ii = [ai, bi], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' , k, the random variable (Ng(I1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Ng(Ik)) con- verge in distribution to the Poisson random variable (NMP(I1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' NMP(Ik)), in other words that the point processes Lg converge in distribution to the Poisson point process Pois(νMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' A random approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Define a mapping N → R HL,τ = � ℓ∈L HL,τ(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We note that HL,τ is a continuous function on the positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' There- fore, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1, the functional HL,τ : N → R is continuous with respect to the vague topology on the space of point processes on the positive reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' By the Mirzakhani-Petri theorem combined with the continuous map- ping theorem, we find that the random variables Nosc = HL,τ ◦ Lg : Mg → R converge in distribution to the random variable SL,τ := HL,τ ◦ Pois(νMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We recall that a sequence of real valued random variables Yi con- verges in distribution to a random variable Y if the cumulative distri- bution functions (CDF’s) Fi(t) = Prob(Yi ≤ t) converge pointwise to FY (t) in every continuity point of FY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' This is equivalent to requiring that for every bounded continuous function h, we have lim i→∞ E (h (Yi)) = E (h (Y )) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' In turn this, by L´evy’s continuity theorem, is equivalent to pointwise convergence of the characteristic functions: Recall that for a random variable Y , the characteristic function is φY (t) = E(exp(itY )), t real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Hence the characteristic functions of Nosc converge as g → ∞ to the characteristic function of SL,τ: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We have convergence of characteristic functions lim g→∞ EWP g (exp (it (Nosc))) = EPois (exp (it (SL,τ))) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Working with the Poisson approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We will next show that in the limit L → ∞, the characteristic function of SL,τ tends to that of a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' By L´evy’s continuity theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' [1, Chapter 10]), this will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' THE CLT ON RANDOM SURFACES OF LARGE GENUS 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For all τ > 0, lim L→∞ EPois(itSL,τ − E(SL,τ) � Σ2 GOE(f) ) = e−t2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' By its definition, the expected value is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' That the variance is correct was demonstrated in the course of the proof of [13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' To show Gaussianity, we use a combinatorial shortcut, of using cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Recall that the cumulants of a random variable S are the coefficients in the Taylor expansion of the cumulant generating function log E(ezS) about z = 0: log E(ezS) = ∞ � m=1 κm(S)zm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='. The first moment is the first cumulant, the second cumulant is just the variance, and the third cumulant is the centered third moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' In general, the n-th centered moment is an n-th-degree polynomial in the first n cumulants, for instance, the fourth centered moment is κ4 +3κ2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Gaussianity is equivalent to vanishing of the higher cumulants κm(S) for m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' So we want to show: Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For all m ≥ 3, lim L→∞ κm(SL,τ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We will use Campbell’s formula [8]: Let L be a Poisson point process with intensity ν, let H : R → R be a measurable function, and define the random sum S = � ℓ∈L H(ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Then (assuming everything converges) the expected value and variance of S are given by E(S) = � R H(x)dν(x), Var(S) = � R H(x)2dν(x) and the moment generating function is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2) E(ezS) = exp �� R [ezH(x) − 1]dν(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Using the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2) gives log E(ezS) = � m≥1 zm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' � R H(x)mdν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Therefore, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='3 follows from: 8 ZE´EV RUDNICK AND IGOR WIGMAN Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For all m ≥ 3, uniformly in τ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='3) lim L→∞ � R HL,τ(x)mdνMP(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' We use: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' [13, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='3] Let L > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Then uniformly in τ, i) For 0 < x < 1/2, we have |HL,τ(x)| ≪ 1 L log(L/x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' ii) For x ≥ 1/2, we have |HL,τ(x)| ≪ 1 Lx exp(−x/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='5 we obtain � ∞ 0 HL,τ(x)mdνMP(x) ≪ � 1/2 0 1 Lm(log L x )msinh(x/2)2 x dx + � L 1/2 1 Lmxme−mx/2sinh(x/2)2 x dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For 0 < x < 1/2, use sinh(x/2) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1 · x/2 ≪ x to bound � 1/2 0 (log L x )msinh(x/2)2 x dx ≪ � 1/2 0 x(log L x )mdx = L2 � ∞ 2L (log y)mdy y3 ≪m L2 � L2 2L (log L2)mdy y3 + L2 � ∞ L2 y1/2dy y3 ≪ (log L)m + L−3 ≪ (log L)m so that the first integral contributes O((log L/L)m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' For the second integral, use sinh(x/2) < ex/2 for x > 0 to bound, for m ≥ 3, � L 1/2 xme−mx/2sinh(x/2)2 x dx < � ∞ 0 e−( m 2 −1)xxm−1dx = 2mΓ(m) (m − 2)m so that the second integral contributes Om(L−m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Thus we obtain for m ≥ 3 � ∞ 0 HL,τ(x)mdνMP(x) ≪m L−m+o(1) which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' □ This concludes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='2, hence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' THE CLT ON RANDOM SURFACES OF LARGE GENUS 9 References [1] Athreya, Krishna B.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' The Cheeger constants of random Belyi surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='09853 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='DG].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' International Mathematics Research Notices (IMRN), to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' [15] Selberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Contributions to the theory of the Riemann zeta-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' Naturvid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=', 48(5):89-155, 1946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content=' School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel Email address: rudnick@tauex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='tau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='il Department of Mathematics, King’s College London, UK Email address: igor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='wigman@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} +page_content='uk' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldAyT4oBgHgl3EQfyfmD/content/2301.00685v1.pdf'} diff --git a/ndE3T4oBgHgl3EQfiwq0/content/tmp_files/load_file.txt b/ndE3T4oBgHgl3EQfiwq0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbdf754d44e6eda8b763839804b11be7d43b9915 --- /dev/null +++ b/ndE3T4oBgHgl3EQfiwq0/content/tmp_files/load_file.txt @@ -0,0 +1,903 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf,len=902 +page_content='Magneto optics of a doubly charged quantum dot - Observation of a negative diamagnetic shift Giora Peniakov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ∗ Ayal Beck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ∗ Eilon Poem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='3 Zu-En Su,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 Boaz Taitler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 and David Gershoni1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' † 1The Physics Department and the Solid State Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Technion–Israel Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 3200003 Haifa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Israel 2Technische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Physikalisches Institï¿œt and Wilhelm Conrad Rï¿œntgen-Center for Complex Material Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Universitat Wï¿œrzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Am Hubland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' D-97074 Wï¿œzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Germany 3Department of Physics of Complex Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Weizmann Institute of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 7610001 Rehovot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Israel (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 2023) We present magneto-optical studies of a self-assembled semiconductor quantum dot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' concentrating specifically on the case in which the dot is doubly positively charged,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' studying this way the confined hole - hole exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' A simple harmonic potential model, which we extend to capture the influence of an externally applied magnetic field in Faraday configuration fully describe the observed polarization sensitive magneto-photoluminscence spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We deduce the effective compo- sition of the quantum dot from its measured electronic g-factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Using this value we determine the dot effective permittivity and quantitatively describe various measured excitonic transitions, their measured Zeeman splittings and their diamagnetic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In particular, the model quantitatively accounts for an observed pronounced negative diamagnetic shift, which provides a direct measure for the hole-hole exchange interaction and its dependence on the externally applied magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' INTRODUCTION Self-assembled Quantum Dots (QDs) in semiconduc- tors form a well-known platform for quantum technolo- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' They have proven to be the best contemporary single-photon sources [1–4], while providing an excellent interface between anchored spin qubits and “flying” pho- ton qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Much progress has been made in controlling confined-spin qubits in QDs [5–8] and entangling them with photons Sen[9–14] , enabling deterministic genera- tion of long strings of entangled photons [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In addition, QDs still provide a convenient platform for studying the many-body states of confined many carri- ers complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Interesting properties of such complexes include the relative interactions between the consisting particles, the form of their spatial wavefunctions, and their response to externally applied fields, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In particular, an externally applied magnetic field causes the associated optical transitions to energetically shift - an effect known as the diamagnetic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Modeling those shifts in confined systems is still an ongoing effort [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Here we present a magneto-optical study of semicon- ductor QDs in which we focus our attention on the optical properties of a doubly positively charged QD and its fun- damental excitonic transition denoted by X+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The QD confined X+2 exciton contains three heavy holes and an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' After radiative recombination of an electron- hole pair, the QD remains with two heavy-holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The pairs of holes may form either three spin triplet states or one spin singlet state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Our work was spurred by notic- ing an anomaly in the diamagnetic shifts of the opti- cal transitions into the singlet state, the X+2 S0 , which we ∗ These authors contributed to this work equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' † giora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='peniakov@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='uni-wuerzburg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='de found to be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In the effort to understand this phenomenon, we found that the X+2 excitonic transi- tions form an excellent platform for studying the hole- hole Coulomb exchange interaction and its dependence on externally applied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We show, using a simple harmonic model for the QD spatial potential, that the measured diamagnetic shifts of the X+2 transitions and the measured electron and hole g-factors can be ob- tained using one free fitting parameter which describes the effective composition of the QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' First, we present our measurements of the electron and hole g-factors by measuring the spectral Zeeman splittings of the bright and dark neutral excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' From the values of the mea- sured g-factors, we extract the composition of the QD as captured by the parameter x, defining the ratio of In- dium and Gallium in the QD, InxGa1−xAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Next, we present full polarization-sensitive magneto-PL measure- ments displaying the diamagnetic shifts of various opti- cal transitions and their optical transition selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We then concentrate on the anomalous negative shift of the doubly positively charged exciton, the X+2 S0 , and fit its optical transition field dependence using no additional free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Finally, we use a Hartree-Fock approx- imation to calculate the absolute values of the X+2 dia- magnetic shifts in terms of the diamagnetic shift of the bright exciton, X0 BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' EXPERIMENTAL SYSTEM We studied a single InxGa1−xAs self-assembled QD embedded in a planar microcavity grown along the [001] direction [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We used an Attocube closed-cycle cryo- stat to cool the sample down to 4 Kelvin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' A built-in vec- tor magnet enabled us to apply a magnetic field in any arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='04583v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='mes-hall] 11 Jan 2023 2 desired direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The emitted photoluminescence (PL) was collected by a ×60 objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Its polarization was analyzed by pairs of liquid crystal variable retarders and polarizing beam splitters, enabling PL polarization pro- jecting on any direction in the Poincarï¿œ sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The PL was then spectrally analyzed using an 80 cm double monochromator, providing a spectral resolution of ∼ 20 µeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The QD was optically excited using an above band-gap CW red HeNe or a blue diode laser, emitting at 633 or 445 nm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The excitation colors induce the average charge state of the QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' While HeNe illumina- tion results in positive charging blue excitation leads to negative charging [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We defined the symmetry axis of the QD and the op- tical beam direction as the z-axis of the experimental system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The x and y axes were defined along the polar- ization eigenstates of the QD’s bright exciton, X0 BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The X0 BE is an electron-hole pair which can be expressed in the spin basis {|+z⟩ = |⇑↓⟩ , |−z⟩ = |⇓↑⟩} with ⇑\\⇓ and ↑\\↓ denoting the spin projections of the heavy-hole and electron onto the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Since a heavy-hole and an elec- tron have total angular momenta of 3/2 and 1/2, respec- tively, the angular momentum projection of a |⇑↓⟩ (|⇓↑⟩) pair along this axis is +1 (-1) [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Consequently, op- tical recombination of the |⇑↓⟩ and |⇓↑⟩ pairs results in a right-handed (R) and left-handed (L) circularly po- larized photon emission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The anisotropic electron-hole exchange interaction in this QD lifts the degeneracy of the above basis by δ1 ≈ 30µeV [26] thus forming new eigenstates, √ 2 |±x⟩ = |⇑↓⟩ ± |⇓↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Recom- bination of those excitonic eigenstates results in either horizontal, √ 2H = R + L, or vertical √ 2V = R − L rectilinear photon emission, enabling a one-to-one corre- spondence between the X0 BE’s two-level system and the two-dimensional space of light polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The dark exciton (X0 DE) is another electron-hole state, but with parallel spins √ 2 |ψDE⟩1,2 = |⇑↑⟩ ± |⇓↓⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In general, this state is optically inactive because the an- gular momentum is not conserved upon recombination [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' However, small optical activity of the X0 DE was mea- sured [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Zielinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' explained it by small mix- ing of the X0 DE and X0 BE eigenstates [28], which can be enhanced by applying an in-plane magnetic field perpen- dicular to the QD optical axis (Voigt configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For a magnetic field parallel to the symmetry axis (Faraday configuration) no additional mixing occurs, and the X0 DE barely emits [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Measuring single-carrier g-factors The Zeeman interaction between an externally applied magnetic field and QD confined carriers’ spin removes the Kramers’ degeneracy between the confined carriers spin state which is parallel and anti-parallel to the field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The Zeeman interaction linearly depends on the magnetic field magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This dependence is most generally expressed in terms of a 3 × 3 g-factor tensor [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For simplicity we assume here that this tensor is diagonal and have only two different components: along the symmetry axis (gz e and gz h) and perpendicular to it (g⊥ e and g⊥ h ) [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In the first part of the experiment, we measured the confined electron and hole g-factors tensor components along the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This was done by measuring the Zeeman splitting of the X0 BE and X0 DE under B-field in the ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Assuming that the absolute magnitude of the g-factors of those transitions are given by the sum and difference of the absolute magnitudes of the single-carrier g-factors |gz BE(DE)| = |gz e| ± |gz h| (1) [33, 34], we were able to extract gz e and gz h from the mea- sured gz BE and gz DE [30, 32, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 1 is derived from the parallel and anti-parallel spin nature of the dark and bright excitons, using the sign convention given by the Zeeman Hamiltonian H = −µBgz eBzSz + 1 3µBgz hBzJz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' (2) Here, µB is the Bohr magneton, Sz and Jz are the angu- lar momentum z-projections ± 1 2 and ± 3 2, and Bz is the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We readily measured the Zeeman split- ting of the X0 BE since its spectral doublet appears bright in the PL, as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In contrary, the low op- tical activity of the X0 DE made its Zeeman splitting mea- surement more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' To overcome this problem, we added a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5T Voigt component to our measurement (in-plane B-field), enhancing the X0 DE emission to a mea- surable amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Although the total B-field direction was no longer in the ˆz direction, we found that the in-plane field effect on the measured ˆz direction Zeeman splitting could be safely neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We veified it by measuring the influence of the in-plane field on the bright exciton split- ting (X0 BE), and found that it stayed unaffected within our experimental precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' One can also notice in Figure 1 that the X0 DE cross- polarized doublet is not equally intense: at 0 Tesla, its horizontally (H) polarized component is much stronger than the vertically (V) polarized one, a phenomenon ob- served and explained in previous publications [36–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Adding magnetic field in Faraday configuration enhances the weaker component and gradually adds cross-circular polarization terms to the X0 DE doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' However, up to the maximal field strength of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5 Tesla, the two X0 DE’s components remain unequal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Nonetheless, we extracted the g-factors of the X0 BE and X0 DE by fitting their mea- sured Zeeman splittings to the following expression: ∆EBE(DE) = � δ2 1,2 + (µBgBE(DE)B)2 (3) , where δ1,2 are the fine-structure splittings of the X0 BE and X0 DE at 0 Tesla, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We summarize the val- 3 Normalized Intensity Split [meV] 𝐵� = 0𝑇 𝐵� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4𝑇 𝐵� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8𝑇 𝐵� = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2𝑇 𝑋�� � 𝑋�� � Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Polarization-sensitive magneto-PL of the bright ex- citon (X0 BE) and dark exciton (X0 DE), for various magnetic field strengths in the ˆz direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' A constant, ˆx-directional magnetic field of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5T was applied during all the measure- ments to allow the X0 DE optical transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Inset: Zeeman splitting of the X0 BE and X0 DE versus Bz-field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The g-factors of the two transitions are extracted by fitting the measured splitting with ∆E = � δ2 + (µBgB)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Spectral line δ[µeV ] gz-factor α[ µeV T 2 ] X0 BE 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='14 X0 DE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4∗ ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4 gz e −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='02 gz h −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='02 ∆0 [µeV ] 270 ± 10 Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Summary of the measured excitonic fine structure and Zeeman parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' δ is the natural splitting at B = 0, and α is the diamagnetic shift coefficient capturing the quadratic dependence of the energy in B (αB2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' gz (e/h) is the measured g-factor of the electron and hole in Faraday configuration, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ∗The dark exciton (X0 DE) splitting is too small to be directly observed in PL measurement, but can be mea- sured using time-resolved spectroscopy [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For reference, we also added the X0 DE − X0 BE splitting denoted by ∆0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ues of the measured excitonic and single-carrier g-factors in table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Estimating the QD effective composition from the measured electronic g-factor The isotropic electronic g-factor in bulk semiconduc- tors can be analytically calculated by the Roth’s formula [39]: ge = 2 − 2 3 Ep∆ Eg(Eg + ∆) (4) where Eg is the band gap energy between the valence and conduction bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ∆ is the split-off gap (between the valence band and the spin-orbit band) at k = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' and Ep is the Kane energy defined as Ep ≡ 2ℏ2 m |⟨s| ∂x |x⟩|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' where |s⟩ and |x⟩ are the crystal Bloch functions of the electron in the conduction band and in one of the three p−like degenerate valence band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' QDs are not bulk semiconductors, and applying Roth’s formula to them requires adjustment [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In principle, the confinement effect of the QD breaks the periodicity of the electronic wavefunctions, and the derivation of Roth’s formula collapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Nevertheless, as long as the confine- ment energy is much smaller than the parameters ∆, Ep and Eg, we expect Roth’s formula to be a good approx- imation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Indeed, a typical separation between confine- ment energy levels in our QD is of order 10 − 30 meV s [41], much smaller than ∆, Ep and Eg (see table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Since our QD comprises two semiconductors, GaAs and InAs, we averaged the values of ∆ and Ep over the two bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We introduced a weight parameter x to quantify the composition of the QD, InxGa1−xAs, and define: ∆x = x∆In+(1−x)∆Ga, Ex p = xEp(In)+(1−x)Ep(Ga) For the band gap, Eg, we preferred to use the directly measured value of the X0 BE emission, as it takes into account the confinement and lattice mismatch strain ef- fects, omitted in the simple average [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We defined a corrected band gap � Eg by adding ∼ 50meV to the X0 BE emission energy, thus accounting for the binding energy of the exciton [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Combining all three parameters, we obtained an x-dependent Roth formula ge(x) = 2 − 2 3 Ex p∆x � Eg(� Eg + ∆x) (5) that we can fit to the electronic g-factor as measured in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Fitting ge(x) to the measured value of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55 yields x ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We summarize the parameter values of that calculation in table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Measuring a negative diamagnetic shift for X+2 In Figure 2, we present a full Magneto-PL measure- ment in Faraday configuration of the various spectral lines of our QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We present it for two average charge states of the QD: negative and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The charge state is apparent in each case by considering the emission ratio between the positive and negative trions, X+ and X−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Many identified lines are marked in the PL following pre- vious studies [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' BZEnergy mey4 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Polarization-sensitive magneto-PL spectra in Faraday configuration for various magnetic field strengths, for negatively (a) and positively (b) charged QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The upper panel shows polarization-sensitive magneto-PL spectra (except at B = 0T, where the rectilinear polarization is shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The panels below show the degree of circular and rectilinear polarizations (given by the color bars to the right) as a function of the photon energy and the externally applied magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The identified spectral lines are marked: X0 - the exciton, XX0 - the biexciton, X+ - (X−-) positively (negatively) charged trion, XX0 T0(T3) metastable biexcitons with the two holes in T0(T3) spin Triplet configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' X+ T0(T3), and XX+ T0(T3) are similar positively charged excitons and biexcitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The X+2 lines result from the recombination of the doubly positively charged exciton, leaving behind two holes which can form either a singlet S0 or one of the triplets, T±3 or T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Note the negative diamagnetic shift of the X+2 S0 (marked with an oval dash line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The energy scale is relative to the X0 BE spectral line at zero magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' GaAs InAs In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='75Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25As Ep[eV ] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='85 Eg(4K)[eV ] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='519 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='418 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='334* ∆[eV ] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='363 ge Calculated -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='317 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55 ge Measured -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='484 -14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55 Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Comparison between measured electronic g-factors to calculated values using Roth’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The measured val- ues for bulk GaAs and InAs semiconductors are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The parameters Ep and ∆ for In0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='75Ga0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25As are weighted averages of their values in GaAs and InAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ∗The corrected band gap � Eg (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' On top of the Zeeman splitting of the spectral lines, they undergo a quadratic-in-B diamagnetic shift, which we characterize by the coefficient α in the term αB2 added to the Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For each spectral line in Figure 2, the shift is attributed to its spectral “center of mass” (the spectral center of the doublet), which in most cases shifts towards higher energy (hence the ter- minology of “diamagnetic” versus “paramagnetic” shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We explain this tendency by considering the areas of the initial and final states of each optical transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' It is a well-known result for quantum wells that the diamagnetic shift of a neutral exciton is proportional to its wavefunc- tion area [43] in a plane which is normal to the direction of the magnetic field: α = e2 8µ∥c2 ⟨f|ˆρ2|f⟩ = π 4 e2 µ∥c2 � f 2 (ρ) ρ3dρ (6) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ρ is the relative in-plane coordinate between the electron and hole,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' f (ρ) is the excitonic envelope wave- function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' µ∥ = memh/ (me + mh) is the in-plane reduced mass of the electron and hole,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' and e and c are the elec- tron charge and the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The final state of the QD after the excitonic recombination is just the vacuum, possessing no magnetic dependence, so the overall diamagnetic shift is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Extending the area interpretation to other optical transitions, it seems that in most cases the radiative recombination results in a fi- nal configuration with a reduced area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' As a result, most lines follow positive diamagnetic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Quantitatively, plugging in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 6 mh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25m0, me = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='065m0 (the effective masses of the hole and electron in the quantum dot [44] with m0 the free electron mass), and the mea- sured diamagnetic coefficient α = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='14 µeV/T 2, one calculates the exciton Bohr radius to be ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='7 nm – a compatible result with the ∼ 30 nm estimated diameter of our QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Figure 3 summarizes the diamagnetic shifts of several selected lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' One can see that many lines, including the X0 BE, XX0 and the trions, X− and X+, exhibit very similar diamagnetic shifts of ∼ 8µeV/T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We explain this (a) (b) nsity 5T 4T 3T 2T 1T Xx XX% XX3 X+2* Bz =:OT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2 V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5 3 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6 5 4 3 2 0 6 3 2 0 Energy [meV] Energy [meV]5 0 5 10 15 20 25 2 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='3 X0 XX0 X+ X- XX0 T3 XX0 T0 XX+ T0 X+ T0 XX+ T3 X+ T3 XX- T1 X- T1 XX- T0 X- T0 X+2 T0 X+2 T3 X+2 S0 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Measured energy shifts of various optical transitions as a function of B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' One spectral line is a prominent exception the X+2 S0 similarity by arguing that in all those transitions both the initial and final states contain only charge carriers occupying the QD ground-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We observe that when additional charge carriers occupy higher confined levels, the diamagnetic shift coefficients change (see for example X− T ±1, X+ T ±3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Interestingly enough, one prominent line that we at- tribute to the doubly charged exciton transition X+2 S0 ex- hibits a distinctive negative shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In what follows, we will try to explain this observation in terms of the ex- change interaction between the two heavy holes of the X+2 transitions’ final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Let us start by describing those transitions in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The doubly charged exciton X+2 Figure 4 schematically describes the energy levels and the optical transitions associated with the doubly charged exciton, X+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This exciton comprises one elec- tron in the ground-level 1e1, and three holes: two of them forming a singlet in the s-orbital ground-level, 1h2, and the third one occupies the first excited p-level 2h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Here, npm means: n - the energy level order, p - the particle type (e or h), and m - the number of particles occupying this level (either 1 or 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The exchange interaction be- tween the unpaired electron (in the 1st level) and hole (in the 2nd level) removes the degeneracy between the four possible two-carriers’ spin configurations, forming four distinct eigenstates similar to the case of the neutral ex- citon (X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' As such, we borrow the exciton “bright” and “dark” terminology to describe the eigenstates of the X+2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' States with anti-parallel e-h spins would be called “bright-like”, while states with parallel spins - “dark-like” (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We emphasize that the dark and bright 𝛿� ���� 𝛿� ���� 𝛿� ���� 𝑋"������"� �� 𝑋"����"± �� 𝑆� 𝑇� 𝑇±� V H H V R L 𝑋"������"� �� 𝑬 𝑋�� �� 𝑋�±� �� 𝑋�� �� 𝚫𝑬 𝛿� ���� Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Schematic description of the energy levels and opti- cal transitions associated with the doubly positively charged exciton X+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The configuration of each state is presented on the left, where thin blue arrows represent electrons with spin 1 2, and thick arrows represent heavy-holes with spin 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The polarization selection rules are marked by colored down- ward arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' H (V ) marks the horizontal (vertical) rectilinear polarization, while R (L) marks right (left) circular polariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' A schematic description of the emitted PL is drawn at the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The X+2 T±3 spectral line is drawn in green with a pink edge, symbolizing that the H and V polarizations over- lap such that the emission is unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' states are both optically active since the optical recombi- nation occurs between the unpaired s−electron and one of the s−level singlet holes, rather than the unpaired p−hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The final states of the X+2 recombination contain two holes - one in the ground level and one in the first ex- cited level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' As identical particles, they form either one singlet spin state denoted by S1h2h 0 or three triplet states denoted by � T 1h2h 0 , T 1h2h ±3 � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The two initial bright-like exciton states can only recombine to the sin- glet S1h2h 0 or triplet T 1h2h 0 final states (but not to the T 1h2h ±3 ), resulting in two pairs of cross-rectilinearly polar- ized doublets [45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' the dark-like states can only recom- bine to the T 1h2h ±3 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Since in the absence of external magnetic field (B = 0) both the dark-like and the T 1h2h ±3 states are almost degenerate, the recombination results in a single, unpolarized, strong spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We label the X+2 optical transitions by their final states, specified by the subscripts: X+2 T0 , X+2 T±3 and X+2 S0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The latter transi- tion, X+2 S0 , is the one exhibiting a negative diamagnetic shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We note that in the absence of external field, the unpolarized X+2 T±3 spectral line is positioned exactly in be- tween the two cross linearly polarized components of the X+2 T0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This indicates that δ1e2h 0 , denoting the split- X土 4土 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 1← 土6 Spectral line δ1[µeV ] g-factor α[ µeV T 2 ] Model X0 BE 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='14 gz 1e + gz 1h X+2 T3 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25 X+2 T0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='13 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5 gz 1e + gz 2h X+2 S0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='08 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='7 X0 DE 0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='02 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4 gz 1e − gz 1h Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Summary of the measured g-factors for the X+2 transitions, compared to those of the bright and dark excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The lines are classified by a simple model which assumes that the g-factor of a given transition can be decomposed to the sum of the comprising charge carrier g-factors of the initial and final states of that transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' gz n(e/h) denotes the g-factor of the electron (hole) in the n energy level of the QD, where n = 1 is the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ting between the dark-like and bright-like X+2 states, is equal to δ1h2h T , the splitting between the holes’ triplet states T 1h2h 0 and T 1h2h ±3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The reason why these two terms, one due to isotropic e-h exchange and the other due to h-h anisotropic exchange, are almost equal, remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The measured diamagnetic shifts and g-factors of the X+2 transitions are summarized in table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' To qualita- tively explain the measured g-factors, we assume that a g-factor of a state can be deduced by summing up its indi- vidual single carrier component’s g-factors, and that the total g-factor of a transition results from the difference between its initial and final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' By further assuming that charge carriers in the well-defined symmetry config- urations S0 and T0 do not exhibit Zeeman splitting, we conclude that X+2 T3 ’s g-factor behaves as the bright ex- citation’s (X0 BE) factor, while the g-factors of X+2 T0 and X+2 S0 depend on the excited hole’s g-factor, gz 2h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' More measurements justifying the above classification as a gen- eral result can be found in the authors’ theses [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' It is interesting to note that plugging into the gz 1e + gz 2h sum the measured gz 1e-factor (∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55), and using for the sum an averaged value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 (see table III), one finds that gz 2h = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This value is opposite in sign compared to the ground state g-factors of the hole and the electron (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='26 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='55, respectively, according to Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' A detailed polarization-sensitive magneto-PL spectra of the X+2 spectral lines are presented in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' One can see that while the triplet lines shift towards higher energy with increasing B-field, the singlet lines shift to- wards lower energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Since the initial states of the X+2 S0 and X+2 T0 transitions are the same (the bright-like exci- ton states), we conclude that the difference in the sign of the diamagnetic shift between the two transitions stems Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Rectilinear polarization-sensitive PL spectra of the X+2 spectral lines relative to the neutral exciton state a) at zero magnetic field, b) as function of the externally applied field in Faraday configuration, and c) in magnetic field of 5T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The transitions are marked by their final spin configurations (S0,T0, T±3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The energy difference between the X+2 T0 and the X+2 S0 doublets (marked) equals twice the hole-hole exchange interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' from the different influence that the external magnetic field has on the final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The h-h singlet final state rises in energy faster than the initial state such that the overall spectral shift is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' On the other hand, the h-h triplet state rises in energy slower than the initial state, and thus the total spectral shift is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Magnetic field dependence of the exchange integral We now use a simple harmonic oscillator model to quantitatively describe how the hole-hole exchange in- teraction is affected by the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The exchange integrals between various states confined by a 2D har- monic potential with circular symmetry were calculated in Ref [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For two identical particles, one in an s-shell and one in a p-shell the exchange energy is: Ksp,0 = 1 4 �π 2 e2 4πϵ0ϵr 1 l0 (7) where e is the electron charge, ϵ0 is the vacuum per- mittivity and ϵr is the relative permittivity of the QD material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The effective length l0 characterizes the extent of the harmonic potential and is equal to: l0 = � ℏ mω0 (8) where m is the in-plane effective mass of the charge carri- ers, in our case holes, and ω0 is the harmonic frequency of 2000 +2 400 +2 ±3 1500 300 +2 Bz = 5T (c) 1000 8 200 100 500 0 0 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 2(Ksp,0 + βB2) H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4 (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5 0 Bz V 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4 2Ksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 500 4000 X+2 400 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ±3 H 3000 V V 300 (a 2000 B, = OT 200 1000 100 0 0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='2 0 Eenrgy [meV] Eenrgy [meV]7 the confining potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' To include the effect of the mag- netic field, we replace ω0 with ω ≡ � ω2 0 + e2B2z 4m2 , obtained by adding a magnetic field hamiltonian to the harmonic oscillator one and solving for the eigenenergies (harmonic spectrum + Landau levels spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The expression for the effective length then becomes: l = l0 � 1 + � eB 2mω0 �2�1/4 (9) Inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 9 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 7 yields an expression for the field dependence of the exchange energy: Ksp(B) = Ksp,0 � 1 + � eB 2mω0 �2�1/4 (10) Furthermore, mω0 can be expressed in terms of Ksp,0 by using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 7 and 8: mω0 = 512 e4 πℏϵ2 0ϵ2 r(Ksp,0)2 (11) Inserting this expression into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' we obtain an ex- pression for the field dependence of the hole-hole ex- change energy Ksp(B) = Ksp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 � 1 + � e5B 1024πℏϵ2 0ϵ2r(Ksp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0)2 �2�1/4 (12) When the magnetic energy is much smaller than the zero- field exchange energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' we can expand this expression to the first non-vanishing order in B: Ksp(B) ≈ Ksp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 + βB2 (13) where: β(theory) = e10 222π2ℏ2ϵ4 0ϵ4r(Ksp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0)3 (14) To compare this result to the measured diamagnetic shift coefficients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' we further extract Ksp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 and ϵr from our measurements: Ksp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='0 equals half of the energy separation between the X+2 S0 and X+2 T0 optical transitions at B = 0 (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' From the magneto-PL in Figure 5, we obtain Ksp,0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='79(1)meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' To estimate ϵr, we average its value over the InAs and GaAs constituents of the QD, ϵx r = xϵx r(In) + (1 − x)ϵx r(Ga), like we did for the ∆ and Ep parameters in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Using x ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='75, as we found by fitting Roth formula to the measured g-factor, we obtain ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='75 r ≈ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Combining those values, we conclude βtheory ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6 µeV T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In the experiment, β is directly measured as half the difference between the diamagnetic shifts of the X+2 T0 and X+2 S0 spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' To see this, note that the initial states of X+2 T0 and X+2 S0 transitions are the same and thus cancel Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The calculated electronic g-factor ge(x) and the calculated relative diamagnetic shift β(x) between the X+2 singlet and triplet transitions as function of the composition ratio x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Note that the calculation of the g-factor uses constant band gap energy as measured in the experiment, and thus at x = 0, 1 it does not reproduce the values for pure InAs and GaAs bulk materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' out upon subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The only contribution, then, is the final hole-hole state which is either a singlet or a triplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We observe: βmeasured = αX+2 T0 − αX+2 S0 2 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6 µeV T 2 − (−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8 µeV T 2 ) 2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='7(4)µeV T 2 (15) The calculated and the measured values of β agree up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='1 µeV T 2 , well within our experimental error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We see this compliance as a strong validation of the hole-hole exchange interaction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For further conviction, we test the sensitivity of our result to the value of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' One can see that the dependencies g (x) and β (x) are close to linear, and that a deviation of x by more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='02 would cause those parameters to miss the measured values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Diamagnetic shifts of the singlet and triplet lines After explaining the relative diamagnetic shift between the X+2 S0 and X+2 T0 transitions, we proceed by calculating the absolute values of those shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We find that we can figure those values using the measured emission energy of the X+2 spectral lines relative to the neutral exciton X0, and its diamagnetic shift coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In those calcu- lations, we use the Hartree-Fock approximation to evalu- ate the energies of the optical transitions’ initial and final many-body states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The energy of a many-body state is calculated, within this approximation, by summing over its individual-particle confining energies and the interac- tions between all the particle pairs involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Under this approximation, the energy of the ground state neutral exciton is: EX0 = Ee s + Eh s − Jeh ss (16) (X) calculatecg mesaurec0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='25X) ( calculatecmesatrec178 where Ee(h) s = 1 2ℏωe(h) is the energy of the electron (hole) in the s-level and Jeh ss is the direct Coulomb in- teraction between the electron and the hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' This ex- pression describes the exciton center-of-splitting (includ- ing at B ̸= 0) as it does not include the e-h exchange Coulomb interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In the following derivation, we use Jp1p2 n1n2 (Kp1p2 n1n2) to describe the direct (exchange) Coulomb interaction between orbitals n1 and n2 of the charge car- riers p1 and p2, the latter representing either holes (h) or electrons (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The emission energies of the X+2 S0 and X+2 T0 optical transitions are given by the difference between the en- ergies of the final and initial states (see figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Einital X+2 T0(S0) = Ee s + 2Eh s + Eh p + Jhh ss + 2Jhh sp − 2Jeh ss − Jeh sp Efinal X+2 T0(S0) = Eh s + Eh p + Jhh sp ∓ Khh sp (17) Note that the hole singlet Efinal X+2 S0 is higher in energy than the triplet Efinal X+2 T0 (Khh sp > 0) due to the different symme- tries of the associated spatial wavefunctions upon particle exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' for the optical transition we have: EX+2 T0(S0) = Efinal X+2 T0(S0) − Einital X+2 T0(S0) (18) =Ee s + Eh s + Jhh ss + Jhh sp − 2Jeh ss − Jeh sp ± Khh sp We can cast this in terms of the Hartree-Fock neutral exciton transition energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' EX0 = Ee s + Eh s − Jeh ss ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' as: EX+2 T0(S0) = EX0 + Jhh ss + Jhh sp − Jeh ss − Jeh sp ± Khh sp (19) All the elements in this equation can be expressed by the effective length of the electron (le = � ℏ meωe ) and hole (lh = � ℏ mhωh ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' using Ref [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Summing them up gives the following expression: EX+2 T0(S0) = EX0 + (a ± 1) Khh sp (20) where we defined the constant a as a ≡ 7 − 4 � 2 1 + γ2 �1/2 − (1 + 2γ2) � 2 1 + γ2 �3/2 (21) and γ ≡ le lh is the ratio between the effective lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' As Khh sp is the exchange energy between two holes, we have already found in the previous section that it can be expressed as Khh sp (B) ≈ Ksp,0 + βB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Using this, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 20 becomes: EX+2 T0(S0)(B) = EX0(B)+(a±1)Ksp,0 +β(a±1)B2 (22) The ratio γ can not be determined directly, as we lack the knowledge about the ratio between the effective in- plane masses of the electron and hole, or the ratio be- tween their related harmonic confinement frequencies ωe and ωh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' However, we can extract γ from a, since it is a function of directly measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' At zero mag- netic field: EX+2 T0 = EX0 + (a + 1)Ksp,0 ⇒ a = EX+2 T0 − EX0 Ksp,0 − 1 EX+2 S0 = EX0 + (a − 1)Ksp,0 ⇒ a = EX+2 S0 − EX0 Ksp,0 + 1 (23) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='EX+2 T0 − EX0 and EX+2 S0 − EX0 are the energies of the X+2 transitions relative to the neutral exciton, which at zero magnetic field, according to figure 5, equal respec- tively −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='38(1)meV and −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='95(1)meV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Using these val- ues and the previously obtained Ksp,0, both expressions in 23 yield a ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='13, which in turn implies γ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Finally, we are ready to calculate the absolute diamag- netic shifts of the hole-hole triplet and singlet lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Ac- cording to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 22, the diamagnetic coefficients of the X+2 transitions are: αX+2 T0(S0) = αX0 + β(a ± 1) (24) where αX0 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='4(2) µeV T 2 is the diamagnetic shift coef- ficient of the exciton, and β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='7(4) µeV T 2 is the relative X+2 diamagnetic shift calculated in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We find: αX+2 T0 = αX0 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='13β = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='5(2)µeV T 2 αX+2 S0 = αX0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='13β = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='9(8)µeV T 2 (25) which agree with our measured values 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='6(4) µeV T 2 and −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content='8(7) µeV T 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' SUMMARY We performed magneto - PL spectroscopy on a well- characterized InxGa1−xAs/GaAs QD in Faraday config- uration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' From the measurements we extracted the g- factors and the diamagnetic shifts of many excitonic tran- sitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' In particular, we observed an anomalous negative diamagnetic shift of spectral lines resulting from the ra- diative recombination of a doubly charged exciton (X+2 S0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Our results are explained using simple models for the Zee- man interaction and for the measured diamagnetic shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' For both interactions we use one free parameter: x, the effective relative Indium content of the ternary QD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' We use this parameter to linearly interpolate the QD elec- tronic g-factor and permittivity, from those of its binary components GaAs and InAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' By analysis of the measured g-factors of various opti- cal transitions we show that while the g-factors of the electron in the first and second level have the same sign, the g-factors of the hole in these levels are opposite in sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' 9 We explain the difference between the diamagnetic shifts of the optical transitions of the doubly positively charged exciton which result in the remaining holes in a singlet (X+2 S0 ) and that in which they form a triplet (X+2 T0 ) using a simple circular harmonic potential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The model describes in analytical form the hole-hole exchange interaction including the influence of the externally ap- plied magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Finally, using the Hartree-Fock ap- proximation we calculate the absolute diamagnetic shifts of these spectral lines using the measured diamagnetic shift of the neutral exciton (X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Tilchin, Professor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Lifshitz, and Professor E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Ivchenko for their help and valuable dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' The support of the Israeli Science Foundation (ISF), and that of the German Israeli Research Coop- eration—DIP (DFG-FI947-6-1) are gratefully acknowl- edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Dekel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Gershoni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Ehrenfreund, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Garcia, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Petroff, Physical 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} +page_content=' Huant, Physical Review B 58, 16221 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ndE3T4oBgHgl3EQfiwq0/content/2301.04583v1.pdf'} diff --git a/otE0T4oBgHgl3EQfZwCz/content/tmp_files/2301.02326v1.pdf.txt b/otE0T4oBgHgl3EQfZwCz/content/tmp_files/2301.02326v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..811b09088a2fec07ec2bf271f0587e6725bd59f1 --- /dev/null +++ b/otE0T4oBgHgl3EQfZwCz/content/tmp_files/2301.02326v1.pdf.txt @@ -0,0 +1,3896 @@ +SciPost Physics +Submission +Entanglement Rényi Entropies from Ballistic Fluctuation Theory: the +free fermionic case +Giuseppe Del Vecchio Del Vecchio1, Benjamin Doyon2, Paola Ruggiero3 +Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK +January 9, 2023 +Abstract +The large-scale behaviour of entanglement entropy in finite-density states, in and out of +equilibrium, can be understood using the physical picture of particle pairs. However, the +full theoretical origin of this picture is not fully established yet. In this work, we clarify this +picture by investigating entanglement entropy using its connection with the large-deviation +theory for thermodynamic and hydrodynamic fluctuations. We apply the universal frame- +work of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, +to correlation functions of branch-point twist fields, the starting point for computing Rényi +entanglement entropies within the replica approach. Focusing on free fermionic systems +in order to illustrate the ideas, we show that both the equilibrium behavior and the dy- +namics of Rényi entanglement entropies can be fully derived from the BFT. In particular, +we emphasise that long-range correlations develop after quantum quenches, and account- +ing for these explain the structure of the entanglement growth. We further show that this +growth is related to fluctuations of charge transport, generalising to quantum quenches the +relation between charge fluctuations and entanglement observed earlier. The general ideas +we introduce suggest that the large-scale behaviour of entanglement has its origin within +hydrodynamic fluctuations. +Contents +1 +Introduction +2 +2 +Ballistic Fluctuation Theory and twist fields +5 +2.1 +General setting +5 +2.2 +Large deviation theory of currents +6 +2.3 +F(λ) in MES: biased measure, flow equation +8 +2.4 +Application of the BFT to twist fields +9 +2.5 +Explicit expression of F(λ) in free fermionic theories +12 +3 +Entanglement and branch-point twist fields +14 +3.1 +Replicas and branch-point twist fields +14 +1giuseppe.del_vecchio_del_vecchio@kcl.ac.uk +2benjamin.doyon@kcl.ac.uk +3paola.ruggiero@kcl.ac.uk +1 +arXiv:2301.02326v1 [quant-ph] 5 Jan 2023 + +SciPost Physics +Submission +3.2 +Enhanced symmetry in free fermions: from Zα to U(α) +15 +4 +Entanglement entropies from BFT +17 +4.1 +Rényi entropies of a finite interval in a GGE (charge fluctuations in space) +17 +4.2 +Long-range correlations due to correlated particle pairs in homogeneous global +quenches +19 +4.3 +Rényi entropies of half system after a quench (current fluctuations in time) +20 +4.4 +Single-mode and pair-mode twist fields +23 +4.5 +Rényi entropies of an interval after a quench (fluctuations of single-mode densities +and currents) +25 +5 +Discussion and conclusion +29 +A +Remarks on notions of locality and twist fields +32 +A.1 +Unbounded observables and topological charges +32 +A.2 +Descendant twist fields and semilocality sectors +33 +A.3 +Non-abelian semilocality +33 +A.4 +Twist fields in the literature +33 +A.5 +Concepts of locality in the literature +34 +B +Correlations after a quench from an initial state with pair structure +34 +B.1 +Global U(1) densities and currents and decay of correlations in GGEs +35 +B.2 +Quench protocol and initial state +38 +B.3 +Approach to the GGE +40 +B.4 +Single-mode density and currents and decay of correlations in GGEs +42 +B.5 +Approach to the GGE +45 +C +S matrix in the α−copy theory +47 +References +48 +1 +Introduction +The understanding of entanglement in quantum many-body systems received a considerable boost +in the last decades, with the introduction and characterization of many different quantities which +“measure” the amount of entanglement in a given quantum state [1–4]. An important set of such +measures are the so-called entanglement Rényi entropies. Given a quantum system described by a +density matrix ρ and a subsystem A of the total system, with ¯A denoting its complement, consider +the associated reduced density matrix ρA = tr¯Aρ. Then, for any α ∈ �+, the α-Rényi entropy is +defined as +Sα = +1 +1 − α logtrρα +A. +(1) +They are good entanglement measures for all pure quantum states, i.e. states of the form ρ = |Ψ〉〈Ψ|. +They fully characterise the entanglement spectrum, and an important property is that in the limit +2 + +SciPost Physics +Submission +α → 1 they reduce to the famous entanglement Von Neumann entropy +S = −tr(ρA logρA). +(2) +In the context of one-dimensional systems, which is the focus of this paper, several exact results +are available for such quantities. For example, at equilibrium, Rényi and entanglement entropies +or their asymptotic behaviours can be obtained in the ground state states of critical [5], gapped [6] +and more general integrable [7] field theories, as well as beyond integrability [8] (note that for +free theories results were first obtained in [9]). In the case of critical systems described by a +conformal field theory (CFT), such results are easily generalized to finite temperature states (i.e., +Gibbs ensembles) [5], and also results for generic thermodynamic macrostates (i.e., generalized +Gibbs ensembles [10]) have been obtained [11–13] in the context of integrable models relying on +(thermodynamic) Bethe ansatz [14] methods. +When moving to out-of-equilibrium scenarios, the situation is more complicated and available +results are mainly qualitative or in the form of conjecture (an exception, however, is the exact +result in [15]). For example, an imaginary time path-integral formulation, together with confor- +mal invariance, has been used for a qualitative understanding of the ubiquitous linear growth of +entanglement [16] observed after quantum quenches [17,18]. Moreover, the dynamics of the en- +tanglement entropy (2) for a generic integrable system was understood in terms of a semiclassical +“quasiparticle picture” (whose original version was proposed in [16]), complemented with the +Bethe ansatz knowledge of the stationary state attained at late times, as conjectured in [19] (see +also [20]). These results have been extensively verified numerically (see, e.g., [20]). An impor- +tant point to stress is that the quasi-particle picture does not admit a generalization for describing, +for generic α, the growth of Rényi entropies [21,22] (with the exception of free systems [11]). +A common starting point for (most of) these results is the so-called “replica approach”, whose +main idea is that trρα +A (cf. (1)) can be computed by considering α copies (with α an integer) of +the original model, ending up with a “replicated” theory. Appropriate analytic continuation to +α ∈ �+, gives the Rényi and the Von Neumann entropy (see, e.g., [6]). +In particular, within this approach, powerful tools are the so-called branch point twist fields, +T α and its hermitian conjugate ¯T α. Twist fields, in general, are special fields associated to a given +symmetry of the theory; they exist, in a many-body system, for every symmetry transformation. +The branch point twist fields are special kind of those: as the replicated theory is invariant un- +der permutations of the copies, T α, ¯T α are the twist fields associated to the generator of cyclic +permutations i �→ i + 1 mod α and its inverse, respectively. The quantities trρα +A can be related +to correlation functions of such twist fields, as first pointed out in quantum field theory in [7] +clarifying ideas from [6], and as shown in quantum chains in [23]. +In this work, we make use of the large-deviation theory for ballistic transport, dubbed ballis- +tic fluctuations theory (BFT), introduced in [24, 25], in order to study the Rényi entanglement +entropy. The BFT, which is based on hydrodynamic projection principles, gives access to the large- +deviation theory for fluctuations of total 2-currents on arbitrary rays in space-time, in homoge- +neous and stationary states. It generalises in a natural fashion the specific free energy from ther- +modynamics. By the relation between currents and twist fields, the BFT, as pointed out in [24], +also gives access to two-point functions of twist fields. +Concentrating on (generic) free fermionic systems, we show that both the equilibrium and +the dynamics of Rényi entropies at large scales of space and time can be obtained from large- +deviation principles and the BFT as applied to branch-point twist fields. The resulting form of the +Rényi entanglement entropy growth and saturation agree with previous results based on counting +3 + +SciPost Physics +Submission +particle pairs, but the method is new, and brings out, we believe, important new physics underlying +the entanglement entropies. The main two observations are: +(1) We obtain an exact relation between the growth of the Rényi entanglement entropies af- +ter a so-called integrable [26–28], pair-production quench, and static and dynamic “full counting +statistics" in the final GGE. Consider N<,> := +� +|v(θ)|<,>ξ/2 dθ ψ† +θψθ the conserved quantity giv- +ing the total number of “slow" and “fast" fermionic modes ψθ, with speeds |v(θ)| < ξ/2 and +|v(θ)| > ξ/2, respectively, where ξ = x/t is a spacetime ray. Let us denote +F<,ξ +dyn (λ) = lim +t→∞ t−1 log〈eλJN<(t)〉 +the scaled cumulant generating function for the total current JN<(t) of slow modes passing through +a point in the time interval [0, t] in the final GGE; and +F>,ξ +stat (λ) = lim +x→∞ x−1 log〈eλN>(x)〉 +the scaled cumulant generating function for the total number N>(x) of fast modes lying on the +spatial interval [0, x] in the final GGE. Consider for simplicity α to be even. Then, as x, t → ∞ +with x/t = ξ fixed, the Rényi entanglement entropy on the interval [0, x], at time t after the +quench, has asymptotic form: +Sα(x, t) ∼ +1 +1 − α +� +2t +α/2 +� +q=−α/2+1 +F<,ξ +dyn (ih2q−1) + x +α/2 +� +q=−α/2+1 +F>,ξ +stat (ih2q−1) +� +, +hp = πp +α . +(3) +This extends earlier observations of the connection between entanglement entropy and full count- +ing statistics [29–31] to non-equilibrium quenches. Our calculations also provide a fundamental +explanation of such relations in terms of twist fields and the large-deviation theory for their asymp- +totic behaviours, which, as far as we know, has not been noticed before. +(2) We give a new exact derivation of the so-called quasi-particle picture in the case of free +fermions with generic dispersion relation. Our derivation is completely independent from the +other exact result for the Ising model in [15], which was based instead on Toeplitz matrix rep- +resentation and multidimensional phase methods. In particular, our method makes transparent +how it is the simple structure of long-range correlations induced by particle pairs in integrable +quenches that allows one to describe both the growth and saturation of entanglement in a simple +and universal way in terms of the long-time GGE, as this structure allows the separation of the +contributions of fast and slow modes as per (3). The emphasis on the structure of long-range cor- +relations also gives a clear understanding as to why for quenches starting from more complicated +states, for instance producing correlated groups of more than two particles, more information +about the initial state is needed to describe the entanglement growth; in these case no simple +formula exists (as showed for example in [32,33]). +We concentrate on free fermion models for simplicity and in order to most clearly illustrate +the method and physics. However, as the method is based on general large-deviation and hydro- +dynamic principles, it is expected to be much more widely applicable, which we leave for future +works. In particular, it suggests that hydrodynamic modes and hydrodynamic projections are the +more accurate notions at the root of the large-scale behaviour of the entanglement dynamics, +rather than particles and their productions. +The paper is organized as follows. Sec. 2 is an introduction to BFT and its relation to twist +fields. In Sec. 3 we review the replica approach and the associated branch-point twist fields, +4 + +SciPost Physics +Submission +and we discuss the simplifications occurring in the free fermionic case. Sec. 4 is the core of the +paper, where we derive an expression for correlation function of twist fields from BFT, both in- +and out-of- equilibrium, and use them to obtain (3) and recover the known formulas for Rényi and +entanglement entropies. A discussion of our method and results is given in Sec.5. The appendices +complement the main text with observations and details of the calculations. In particular, App. A +contains remarks on notions of locality and twist fields. App. B contains all the details about the +applicability of BFT in the different situations we consider, by explicitly computing correlations, +and their long-range behaviour, after a quench from a state with pair structure. Finally, App. C is +about the structure of the S-matrix in the α-copy theory. +2 +Ballistic Fluctuation Theory and twist fields +The BFT [24, 25], detailed below, is a theory describing the large-scale, ballistic fluctuations. It +is expected to apply to a large class of quantum and classical many-body, extensive systems. It +applies to generic systems with space-translation invariant dynamics and interaction range that +is short enough. It has been developed originally for states that are spacetime stationary and +clustering in space, but many of the ideas have been extended to more general situations [34,35]. +In this paper, we use the BFT as originally developed in [24, 25],and show how, and under +which assumptions, it can be applied to states that emerge after quantum quenches as well. Quan- +tum quenches give rise to states that are locally spacetime stationary, but present time-varying +long-range correlations, as we will explain below. We will explain how simple ideas based on the +principles explained in [24] allow us to nevertheless use the BFT. We mention that long-range cor- +relations can also, in principle, be accounted for directly by using the more sophisticated ballistic +macroscopic fluctuation theory (BMFT) [35], which we leave for further studies. +2.1 +General setting +The main strength of the BFT is that it stipulates that only some emergent properties of the system +are required in order to describe the large-scale, ballistic fluctuations: the data of its Euler hydro- +dynamics. We assume the system of interest to have a certain number N (which in our application +to free fermions will be infinite, as the system is integrable) of conserved quantities +Qi = +� +d x qi(x, t) +(4) +such that dQi/dt = 0. They are assumed to be hermitian (in quantum systems), or real (in +classical systems). They include the Hamiltonian H = +� +d x h(x, t), which generates time trans- +lations. For simplicity of the discussion we assume these to be in involution, [Qi,Q j] = 0 for all +i, j ∈ {1,··· , N}, however this is not necessary in general. They have associated conservation laws +∂tqi + ∂x ji = 0. The observables qi, ji are the corresponding charge density and current, assumed +to be “local". In the present paper, locality of qi and ji simply means that Qi has appropriate ex- +tensivity properties; we keep the general discussion formal in order to avoid technicalities, but see +the remark about locality concepts in the literature in App. A. +Within such systems, we focus on states belonging to the manifold of maximal entropy states +(MES). Each state is characterized in terms of a vector β = {β1,··· ,βN} of “Lagrange multipli- +ers", with N components (there are as many number of components as the number of conserved +5 + +SciPost Physics +Submission +quantities Qi). Given such a vector, the density matrix defining the system reads4 +ρβ ∝ e− +� +i βiQi. +(5) +These include the GGEs studied in integrable systems. We consider the system to be in infinite +volume. Below, when no ambiguity occurs, expectation values on such states will be denoted +simply as 〈·〉. Importantly, we assume such states to be clustering strongly enough: connected +correlations tend to zero quickly enough at large spatial separations, +〈a(x,0)b(y,0)〉c := 〈a(x,0)b(y,0)〉 − 〈a(x,0)〉〈b(y,0)〉 → 0 +(|x − y| → ∞). +(6) +Here and below, a(x, t) is a local observable at the position x and evolved to time t. +As per basic principles of statistical mechanics, the averages of densities are generated by the +free energy +〈qi〉 = +∂ +∂ βi +f (β) +(7) +and the mapping +〈q〉 ↔ β +(8) +is bijective (in appropriate regions of values of 〈qi〉 and βi). +As mentioned, states that are spacetime stationary for local observables, but with spacetime +varying long-range correlations, arise naturally in quantum quenches even after long times. This +is because for thermalisation to happen on large regions of the system takes a long time. In such +cases, the state is not described by the MES (5). A precise description of states with long-range +correlations is more involved, see e.g. [35, 38]. But the concept of MES is still useful in these +situations, as, nevertheless, averages of all local observables, or observables supported on regions +smaller than the correlation range, still are described by (5). We will explain how to use this fact +in order to “avoid” long-range correlations and apply the results of the BFT. +2.2 +Large deviation theory of currents +Consider some conserved quantity Q = Qi∗ (for a given i∗ ∈ {1,··· , N}), with associated density +q = qi∗ and current j = ji∗. +It is instructive to start with a description of the large-deviation theory for extensive charges +at equilibrium, before discussing currents. In a given state ρβ, a natural question is to charac- +terize the restriction ∆J(x) = − +� x +0 d x′ q(x′,0) of the charge Q to a spatial interval [0, x], and +its fluctuations within this interval. Here x denotes the horizontal path from (0,0) to (x,0) (and +the notation ∆J(x) is adapted to generalising to currents, as done below). The fluctuations are +fully characterized by the cumulants of ∆J(x). It is a simple result that the cumulant generating +function at large x is given by a difference of specific free energies f (β) of the system, +〈eλ∆J(x)〉 ≍ e−x∆f (λ), +∆f (λ) = f ({βi + δii∗λ}i) − f (β). +(9) +Here and below, we use the notation A(s) ≍ B(s) with the meaning that lims→∞ +logA(s) +log B(s) = 1. +Therefore +∆f (λ) = − lim +x→∞ x−1 log〈eλ∆J(x)〉, +(10) +4In (5), an infinite-volume limit needs to be taken. In quantum spin chains, it is shown that if the weight determining +the density matrix, +� +i βiQi, is short-range, then this defines a state that is unique and exponentially clustering in +space [36]. More generally, one can construct states using the Hilbert space of extensive charges, see [37]. +6 + +SciPost Physics +Submission +and this generates the scaled cumulants cm, +∆f (λ) = +∞ +� +m=1 +λm +m! cm, +cm = − lim +x→∞ x−1〈 +� +∆J(x) +�m 〉c. +(11) +When studying transport, similarly, we are interested in characterizing the total current pass- +ing by a given spatial point (e.g., the origin) in a given interval of time [0, t]. One is therefore +interested in the total transfer of Q in time t, i.e., ∆J(t) = +� t +0 dt′ j(0, t′), where now t denotes the +vertical path from (0,0) to (0, t). As argued in [24] using hydrodynamic principles, the structure +parallels closely the equilibrium case in ballistic systems, such as those admitting many conserved +charges. At large times t → ∞, a large-deviation principle holds generically for linear scaling +with t. +In fact, one can go further and consider the 2-current j = (j,q), and the integral along a more +general path ℓ, starting in (0,0) and ending in (x, t), over its perpendicular component to the +path. This defines the following object +∆J(ℓ) = +� (x,t) +(0,0) +j(x′, t′) ∧ dℓ +(12) +where dℓ = (d x′, dt′) and j ∧ dℓ = jdt′ − qd x′. By current conservation, the integral (12) is +in fact independent of the path chosen, and therefore the result only depends on the end-points +(0,0) and (x, t) (for lightness of notation, we keep implicit the dependence on (0,0)) +∆J(ℓ) = ∆J(x, t). +(13) +For example, we may choose to connect the initial and final points of the path via a segment of +ray +x +t = tanγ, +|γ| < π +2 , +(14) +as will be done below. +Let us consider the Euclidean distance between the initial and final points, +ℓ = +� +x2 + t2 +(15) +(we do not assume Euclidean spacetime symmetry, this is simply a convenient way of control- +ling the scale of x and t). As mentioned, in ballistic systems a large-deviation principle holds +generically for ∆J(x, t) ∝ ℓ. Then, as in the equilibrium case, we can define the scaled cumulant +generating function (SCGF) F(λ) of ∆J(x, t) as +〈eλ ∆J(x,t)〉 ≍ eℓF(λ), +F(λ) = lim +ℓ→∞ℓ−1 log〈eλ ∆J(x,t)〉 = +∞ +� +m=1 +λm +m! cm(γ) . +(16) +The function F(λ) also depends on the angle γ, which we keep implicit. The coefficients cm(γ) +are the scaled cumulants of ∆J(x, t), +cm(γ) = lim +ℓ→∞ℓ−1〈 +� +∆J(x, t) +�m〉c, +(17) +which are m-point correlation functions of currents and densities integrated over the path ℓ. The +finiteness of scaled cumulants depends on the asymptotic behavior of density and current corre- +lation functions at large spacetime separations. See App. B for more details. +7 + +SciPost Physics +Submission +2.3 +F(λ) in MES: biased measure, flow equation +From the explanations above, at γ = π/2 (in the spatial direction) we have that F(λ) = −∆f (λ), +explicitly given in terms of thermodynamic quantities (cf. (9)). What is the corresponding quantity +for finite values of γ (i.e., involving the time direction)? It turns out that the answer is completely +given using the data of the Euler hydrodynamics of the model. +The Euler hydrodynamics controls the motion and correlations of the many-body system at +large scales of space and time. For our purposes, it is sufficient to recall that it is completely fixed +by the flux jacobian matrix, defined as (using the bijection (8)) +Ai j = ∂ 〈ji〉 +∂ 〈qj〉. +(18) +In integrable systems, the flux Jacobian is in fact an infinite dimensional matrix, or more precisely +an integral operator [39]. +From the definition it is clear that Ai j is basis-dependent. Its fundamental information is con- +tained in its spectrum +� +veff +k +� +k=1,···,N, which is composed of eigenvalues if N is finite, and which +admits a continuum in integrable systems (where N = ∞). The spectrum is interpreted as the +set of effective velocities, or “generalized sound velocities,” associated to normal modes of the +hydrodynamics [39]. +In order to compute F(λ) in (16), the idea is to bias the measure ρβ defining the MES (where +expectation values are considered), by multiplying it by the exponential of the integrated 2-current +∆J(x, t), as appears in (16). The state thus becomes λ−dependent, and it is in fact a MES, which +we write as ρλ;β. Associated to the segment of path from (0,0) to (x, t), with tanγ = x/t, one +can derive a flow equation for ρλ;β, within the space of MES, starting from ρ0;β = ρβ. Conve- +niently, this can be written as a flow equation for the Lagrange multipliers β(λ;γ) themselves as +follows [24] +∂ +∂ λβj(λ;γ) = −sgn[A(λ;γ) − tanγ �N]i∗ j , +βj(0;γ) = βj +(19) +where we explicitly introduced the dependence on λ as well as on the path (through γ) in all +quantities. +The main result of BFT is that, solving the flow (19), one can get an expression of the SCGF +directly in terms of the 2-current j evaluated along the flow, namely +F(λ) = +� λ +0 +dλ′ (cosγ〈j〉λ′ − sinγ〈q〉λ′) +(20) +where 〈·〉λ denotes expectation values on ρλ;β (recall that 〈·〉0 ≡ 〈·〉) and 〈j〉λ′ = 〈j(0,0)〉λ′, +〈q〉λ′ = 〈q(0,0)〉λ′. Crucially, from Eq. (20), one has that F(λ) is given in terms of thermodynamic +and Euler hydrodynamic objects only. +The result (20) with (19) follows from a large-deviation principle. The full derivation of Eqs. +(19-20) is not reported here, but can be found in [24]. +The main assumption underlying the validity of (20) is that of “strong enough” clustering along +the space-time ray of velocity x/t = tanγ. Specifically, spatial clustering (6) is not enough: one +needs vanishing of correlation functions of perpendicular currents j⊥(x, t) := j(x, t) ∧ dℓ, the +integrand in (12), when the distance along the ray (0,0) → (x, t) goes to infinity, +〈j⊥(y,s)j⊥(y + ℓsinγ,s + ℓcosγ)〉c → 0 +(ℓ → ∞, y/s = tanγ) +(21) +8 + +SciPost Physics +Submission +and similar requirements for all multipoint functions of perpendicular currents. The vanishing +must be fast enough to make integrals defining the cumulants rapidly converging. +A crucial remark for our work, concerning the requirement of clustering, is Remark 3.3 of [24]. +Recall that ∆J(x, t) = ∆J(ℓ) in (12) is independent of the path ℓ, and only depends on the end- +points (0,0) (which we have kept implicit) and (x, t). One may therefore hope to apply the general +result of the BFT for each path element and obtain, in place of (20), the expression +F(λ) = lim +ℓ→∞ℓ−1 +� λ +0 +dλ′ +� (x,t) +(0,0) +〈j(x′, t′)〉λ′ ∧ dℓ, +(22) +for any path ℓ (with a well-defined large-scale limit ℓ → ∞). As explained in [24, Rem 3.3], this +is expected to be correct if and only if there are no strong correlations between the perpendicular +currents amongst different points on the path. +Eq. (22) is the main result from BFT which will be applied below to expectation values of ob- +servables (specifically, of twist fields) both at equilibrium in states described by GGEs (in Sec. 4.1), +and out-of-equilibrium in states emerging after quenches (in Sec. 4.2 and the following ones). In +the latter case, this can be done by choosing a path that “avoids" the long-range correlations that +such states present, so that expectation values in the pre-quench state can be approximated with +the ones in the corresponding long-time GGE (where correlations can be shown to decay fast +enough). +Remark (clustering and the ballistic large-deviation principle). If there is no path that satisfies strong +clustering, then typically the ballistic large deviation principle is broken, and the SCGF resulting +from the BFT may be infinite or zero. Exponential clustering, which is sufficiently strong, is ex- +pected to hold on rays away from the fluid velocities, tanγ ̸= veff +i +∀i, in generic equilibrium states. +In GGEs of integrable systems at nonzero entropy density, the spectrum of the flux Jacobian con- +tains a continuum, and clustering is in fact a power-law, with power 1/ℓ2 for the perpendicular +currents, Eq. (21), for all rays within this continuum, see App. B.1 for the case of free fermions. +This is still strong enough for the BFT results to hold, as confirmed numerically for the hard +rods [25]. By contrast, in non-integrable systems, where the flux Jacobian has a discrete spec- +trum, clustering is too weak on rays along any of the eigenvalues of the spectrum of the flux +Jacobian (fluid velocties), tanγ = veff +i +for some i. In these directions, the ballistic large-deviation +principle is broken. See the discussion in [24]. In general, at zero temperatures when there is +no gap, weak power-law clustering is seen and the ballistic large-deviation principle is also bro- +ken. When the breaking of the large-deviation principle happens as we change a parameter (a +(generalised) temperature, a coupling), this can be seen as a “dynamical phase transition". +2.4 +Application of the BFT to twist fields +One of the most important application of the BFT is to two-point correlation functions of so-called +“twist fields”. This is useful, because, as explained in the introduction, twist fields are probably +the most efficient way of studying entanglement entropy, the main object of this work. +There are a number of ways of defining twist fields, and we will discuss two natural approaches. +The first is natural in the context of the large-deviation theory as recalled above and based on the +explicit knowledge of extensive conserved quantities; it applies to classical and quantum systems +alike. The second is a more abstract formulation that does not require the explicit knowledge of +extensive conserved quantities, but that is better adapted to quantum systems. +9 + +SciPost Physics +Submission +Consider as above an extensive conserved quantity Q. Recall that Q has associated density and +current q(x, t) and j(x, t). It is convenient to define the height field ϕ(x, t) via the relations +q(x, t) = ∂xϕ(x, t), +j(x, t) = −∂tϕ(x, t). +(23) +This ensures that the continuity equation ∂tq +∂x j = 0 is automatically satisfied. The height field +is unbounded (because the charge is extensive), and we note in particular that differences grow +linearly, +ϕ(0,0) − ϕ(x, t) = ∆J(x, t) ∝ ℓ. +(24) +The height field may be written as an integral over half space5, +ϕ(x, t) = − +� ∞ +x +d y q(y, t) + ϕ(∞, t). +(25) +One finds that the boundary term at infinity is constant in time, ϕ(∞, t) = ϕ(∞) (which can +be chosen to vanish). Further, it is clear that the result is independent from the choice of path in +spacetime thanks to the conservation laws, +ϕ(x, t) = +� (∞,t) +(x,t) +j ∧ dℓ + ϕ(∞), +(26) +and thus the height field ϕ(x, t) only depends on the point (x, t). This justifies the notation. One +may also choose a different direction for the half-space integral, the difference being encoded +within +ϕ(∞) − ϕ(−∞) = Q. +(27) +Exponentials of ϕ(x, t), that is +Tλ = eλϕ, +(28) +are known in general as “twist fields”. Because of the expression (25), they are not “local" in the +naïve sense, but are usually referred to as “semilocal" in the literature (see App. A for an overview); +this is made clearer below using exchange relations. As an immediate result of the large-deviation +analysis, using +Tλ(0,0)T−λ(x, t) = eλ(ϕ(0,0)−ϕ(x,t)) +(29) +with (24), we get the leading exponential behavior of the two-point correlations of twist fields as +〈Tλ(0,0)T−λ(x, t)〉 ≍ eℓF(λ) . +(30) +That is, if the ballistic large-derivation principle holds for the charge Q, then the associated twist +field shows an exponential behaviour at large space-time separations. Because eλϕ is not bounded +for λ ∈ �, Tλ should be referred to as an “unbounded twist field". +In quantum models, because of the lack of commutativity of observables, and a fortiori in +quantum field theory (QFT), because, additionally, of the necessary renormalisation procedure +applied to the twist fields, the relation (29) is not strictly valid. However, corrections do not affect +the leading exponential behaviour of the two-point function, as argued for the XX quantum chain +in [41]. +5This expression is somewhat formal. In quantum spin chains, by applying a time derivative to (25), one can make +the result mathematically rigorous using an appropriate Hilbert space of the Gelfand-Naimark-Segal type, as proved +in [40]. +10 + +SciPost Physics +Submission +A second viewpoint on twist fields is as follows. A twist field T(x, t) is in general an opera- +tor associated to a symmetry transformation that “acts locally enough". This is a transformation +a(x, t) �→ ˜a(x, t) of the model, that maps local observables to local observables, and that pre- +serves the operator algebra; one usually also requires that it preserves the Hamiltonian density, +˜h(x, t) = h(x, t). The property of an operator T(x, t) that makes it a twist field associated to such +a symmetry, is the following equal-time exchange relation: +T(x, t)a(y, t) = +� +˜a(y, t)T(x, t) +y ≫ x +a(y, t)T(x, t) +y ≪ x +(31) +for every local observable a(y, t). Corrections at larges distances |y − x| should be small enough, +for instance exponentially decaying. This equal-time exchange relation is known in the literature +as expressing the “semilocality" of the twist field6 T(x, t). +In general, symmetry transformations act via unitary operators U as ˜a(x, t) = Ua(x, t)U−1. +For instance, in quantum spin chains, one often can write U = +� +x∈� Ux where Ux acts non-trivially +on a neighbourhood of x (possibly up to exponentially decaying corrections); this is indeed local +enough. In such instances, one can simple set +T(x,0) = +� +y≥x +Uy. +(32) +How does the exchange-relation formulation (31) connect with the height-field formulation +(28) of twist fields? This is from the general principle that every extensive conserved quantity +gives rise to a continuous, one-parameter unitary group of symmetry transformations that act locally +enough. That is, given an extensive Q (recall that it is assumed to be hermitian), we may consider +the symmetry transformation +˜a(x, t) = eiηQa(x, t)e−iηQ +(33) +for a real parameter η ∈ �. It is simple to see that this acts locally enough, as described above. +Indeed, by conservation we can replace Q by Q(t) in (33), and by locality of the density q(x, t), +we have that +� L +−L d x [q(x, t), a(0, t)] approaches its limit as L → ∞ quickly enough, and gives in +the limit a local observable supported at x = 0 at time t. Therefore, by using the Baker-Campbell- +Hausdorff formula, at least for all η small enough, eiηQa(x, t)e−iηQ gives rise to a local observable +at (x, t). Using the same principles, it is then a simple matter to show that (31) holds with the +choice (25) for the height field, and the identification in (28) +T = T−iη. +(34) +Because e−iηϕ is bounded, we will referred to these as “bounded twist fields”. Bounded twist fields +are the ones usually considered in the literature. In particular, because they are bounded, their +two-point functions should decay in spacetime, +ℜF(−iη) ≤ 0. +(35) +One may in fact argue that, viceversa, to every symmetry transformation that acts locally enough, +we can associate an extensive conserved quantity. In quantum field theory, Noether’s theorem shows +6If the twist field indeed preserves the hamiltonian density, it commutes with it at large distances, up to small +(e.g. exponential) corrections. This has important implications, which justifies considering twist fields, for many pur- +poses, on the same footing as local fields; for instance, the time-evolved twist field is analytic in the time variable at +small enough times. +11 + +SciPost Physics +Submission +that, for continuous symmetry groups, we indeed have U = eiηQ for some conserved quantity Q +associated to a conserved 2-current; again this is local enough. In general, for any local enough +transformations, one can identify, formally, an extensive charge Q with the operator −ilog U (thus +taking (33) with η = 1), a formal construction that, we expect, could be used fruitfully within +the BFT. In the present paper, we will consider a twist field associated to a discrete symmetry +transformation, but this will be embedded within a continuous symmetry group thanks to the +free-fermion structure, thus the charge Q will be explicit. +Finally, we observe that applying the BFT for the bounded twist fields, using the identification +(34), requires an analytic continuation of λ in the BFT formulae, to purely imaginary values −iη. +This is a subtle aspect, as for purely imaginary values of λ, the modified state by the flow equation +is not strictly a MES (because the resulting linear functional on the algebra of observables is not +necessarily positive). We believe that if fluid velocities are well separated, as is typically the case in +non-integrable systems, then the analytic continuation can be obtained meaningfully by keeping +the sign of the eigenvalues constant in the flow equation (19) (as the analytic continuation will not +“see" the jumps in eigenvalues), and integrating the flow in the complex λ-plane. In free fermion +models, the analytic continuation can be performed directly on the explicit result for F(λ), as +done in [41] and in the next section. We will show below that the BFT indeed predicts decay of +correlation functions in this case. We leave the discussion for interacting integrable systems to +future works. +See App. A for a brief discussion of notions of locality and twist fields. +2.5 +Explicit expression of F(λ) in free fermionic theories +Up to now, the theory was general, and all equations correctly give the ballistic part of large de- +viations in general systems with the properties detailed in Secs. 2.1 and 2.3. When considering +integrable systems, Eq. (20) gets further simplified. In fact, using the theory of generalized hydro- +dynamics (GHD) [42,43], an explicit expression of the hydrodynamic quantities 〈q〉λ,〈j〉λ along +the flow can be worked out. +We now overview the simplified expression for F(λ) in the special case of free fermions in +the continuum, arguably the easieast among 1D integrable systems, which is our focus in this +paper (the corresponding results in the case of generic interacting integrable models can be found +in [25]). +In order to keep the structure general, we simply assume that a fermionic, complex field ψ(x, t) +exists with interactions that are quadratic and short-range. Its Fourier modes are denoted ψθ, with +anti-commutation relation {ψ† +θ,ψθ ′} = δ(θ −θ ′). Here θ represents the momentum, which we as- +sume takes values in � for simplicity (for quantum chains, this would be a bounded subset instead, +but the general ideas are not affected). We also denote the dispersion relation as E(θ), which we +assume is strictly convex and symmetric E(θ) = E(−θ). Thus, under canonical normalisation, +ψ(x, t) = +1 +� +2π +� +dθ eiθ x−iE(θ)tψθ. +(36) +As it is integrable, the model possesses an infinite number of conserved quantities. A “scattering +basis" for these is given by Qθ = ψ† +θψθ, θ ∈ �. Strictly speaking, the Qθ’s are not linearly ex- +tensive, but (for generic dispersion relation) any extensive conserved quantity can be obtained +by a suitable “linear combination", or basis decomposition, Qi = +� +dθ hi(θ)Qθ. Here, hi(θ) is +the one-particle eigenvalue of the extensive charge Qi. Examples are the number of fermions +N = +� +dθ Qθ, the total momentum P = +� +dθ θQθ, and the total energy H = +� +dθ E(θ)Qθ. +12 + +SciPost Physics +Submission +A typical GGE (5) takes the form ρw := ρβ ∝ e− +� +dθ w(θ)Qθ , where w(θ) = +� +i βihi(θ) is the +generalised Boltzmann weight in the particle basis . For example, for a thermal state, we have +ρw ∝ e−β(H−µN) = e− +� +dθ β(E(θ)−µ)Qθ , so w(θ) = β(E(θ) − µ). +In general, the physical meaning of the Lagrange multipliers βi depends on the choice of +the set of charges Qi, i.e. on the choice of the set of one-particle eigenvalues hi(θ). But there +is no need to choose any particular infinite set of charges Qi, or to write explcitly w(θ) in a +basis decomposition w(θ) = +� +i βihi(θ). The function w(θ) fixes the GGE, and only few basic +requirements constrain w(θ) for ρw to be a valid GGE (we will ask that it be positive and grow +sufficiently fast as |θ| → ∞). We note that the conserved charge densities take the standard form +〈qi〉 = +� +dθ/(2π) n(θ)hi(θ) in terms of the occupation function +n(θ) = +1 +1 + ew(θ) . +(37) +and that, in a system of length L with periodic boundary conditions, we have 〈Qθ〉 = +L +2πn(θ). +In our calculations, we will assume that n(θ) has an analytic extension in a neighbourhood of +�, and that n(θ) → 0 as |θ| → ∞. +Consider the large-deviation problem for the charge Q = Qi∗, with one-particle eigenvalue +hi∗(θ) = h(θ). For free fermions, F(λ) simplifies to +F(λ) = − +� +dθ +2π |v(θ) cosγ − sinγ|[f (ελ(θ;γ)) − f (w(θ))] +(38) +where v(θ) = dE(θ)/dθ is the group velocity7. The function f (ε) is the fermionic free energy +function (the free energy density per distance and per unit rapidity θ), +f (ε) = −log(1 + e−ε), +(39) +and the function ελ(θ;γ) is the Boltzmann weight along the flow (19) in the particle basis. Its +initial condition is ε0(θ;γ) = w(θ), and the corresponding flow equation (which simply follows +from (19) in terms of βi) simplifies to +∂λελ(θ;γ) = sgn(tanγ − v(θ))h(θ) , +(40) +which is explicitly solved as +ελ(θ;γ) = w(θ) + λ sgn(tanγ − v(θ))h(θ) . +(41) +As mentioned above, in order to apply the BFT to bounded twist fields associated to symmetry +transformations, one needs to perform an analytic continuation in λ to the purely imaginary di- +rection λ = −iη, η ∈ �. In free Fermion systems this is simple to do, as the above formulae can be +directly analytically continued. The resulting F(λ) possesses, in general, both a real and an imag- +inary part. The real part describes the exponential decay of the two-point correlation functions of +twist fields, while the imaginary part describes oscillations. In the following, we will not discuss +oscillations, as their full description would require a more in-depth analysis; we will concentrate +on the exponential decay, hence the real part of F(λ). +7The effective velocity of GHD is just the group velocity in free particle models, veff(θ) = v(θ). +13 + +SciPost Physics +Submission +Correlation functions of twist fields are expected to be decaying at large spacetime distances. +It is simple to show from (38) that indeed8, ℜF(−iη) ≤ 0. +One important remark is that the only information required about the current ∆J(x, t) whose +SCGF is taken, is the one-particle eigenvalue h(θ) of the corresponding total charge Q. Thus, the +BFT predicts that only a limited amount of information about the twist field is required in order +to evaluate the exponential asymptotic of its two-point function. Note that this is true also in the +interacting case. +3 +Entanglement and branch-point twist fields +In this section, we recall how entanglement entropies can be computed using a certain type of +twist fields, called branch-point twist fields, associated to permutation symmetries. We then recall +that, in free fermionic theories, these can be re-written in terms of U(1) twist fields. This will be +used in the next section in order to apply the BFT to the calculation of entanglement entropies. +3.1 +Replicas and branch-point twist fields +Within the replica method, in order to compute entanglement entropies (cf. Eqs. (1)-(2)) in a +given theory, one re-writes the quantity trρα +A in terms of an appropriate expectation value in the +replica model. This is the model composed of α independent, commuting copies of the original +model (α ∈ �). For a one-dimensional system in a state with density matrix ρ, and with the +subsystem A being a single interval, e.g., A = [x1, x2], it is a simple matter to show [7, 23] that +trρα +A is exactly identified with the two-point function of branch-point twist fields, +trρα +A = 〈T α(x1,0)¯T α(x2,0)〉ρ⊗α. +(42) +The expectation value on the r.h.s. is computed in the density matrix ρ⊗α = ⊗α +i=1ρi, where ρi +is the original density matrix, on copy i. Branch-point twist fields in the replica theory are twist +fields associated to the symmetry under replica cyclic permutations of order α (which generate the +group Zα). They take the product form (32), involving on-site copy-permutation operators9 [23]: +T α(x,0) = +� +y≥x +Py +(43) +and ¯T α(x,0) = +� +T α(x,0) +�†. Here, denoting by ai(x) observables lying on (that is, acting non- +trivially only on) copy i ∈ {1,2,...,α} and position x, and identifying aα+1(x) ≡ a1(x), the +permutation unitary is defined by +Pxai(y)P−1 +x += +� +ai+1(y) +y = x +ai(y) +y ̸= x. +(44) +This implies the equal-time exchange relations (see (31)) +T α(x, t)ai(y, t) = +� +ai+1(y, t)T α(x, t) +y ≥ x +T α(x, t)ai(y, t) +y < x +(45) +8This is because in (38) one has e−ε−iη(θ;γ) = ru where r = e−w(θ) > 0 and u is a pure phase, |u| = 1, and +|1 + r| ≥ |1 + ru| for any r > 0 and any pure phase u. +9Here we omit any regularisation issue that may arise in models on a continuous space, which, as mention, do not +affect exponential asymptotic behaviours. +14 + +SciPost Physics +Submission +and +¯T α(x, t)ai(y, t) = +� +ai−1(y, t)¯T α(x, t) +y ≥ x +¯T α(x, t)ai(y, t) +y < x . +(46) +From Eq. (42), Rényi entanglement entropies can be simply obtained via Eqs. (1)-(2). +In the context of (1+1)-dimensional QFT, exchange relations of the form (45), (46) give the +most appropriate formulation for working definitions of the branch-point twist field. It is in this +context that they were first introduced [7], as a way of evaluating partition functions on branched +surfaces, taking inspiration from [6]. +We note that the action of branch-point twist fields can be diagonalized by going to the Fourier +basis in the replica index, +ap(x, t) = Fi→p[ai(x, t)] := +1 +�α +α +� +i=1 +eiπpi/αai(x, t) +(47) +for p ∈ {0,1,...,α − 1}. This gives +T α(x, t)ap(y, t) = +� +e−iπp/αap(y, t)T α(x, t) +y ≥ x +ap(y, t)T α(x, t) +y < x +(48) +and similarly for ¯T α(x, t). In the next subsection we will use a similar construction, albeit in a +different basis of the replica model. +As will be explained in Sec. 4, for our purposes, the most general object we need to consider +is the two-point correlation functions +〈T α(x1, t1)¯T α(x2, t2)〉ρ⊗α +(49) +at different spacetime points. +3.2 +Enhanced symmetry in free fermions: from Zα to U(α) +Consider the special case of free fermions, see Sec. 2.5. In this case, because of the quadratic nature +of free fermion Hamiltonians, the Zα symmetry of the replicated theory turns out to be embedded +into the larger symmetry group U(α), which accounts for not only permutation of replicas, but also +rotations amongst them. Thus, the branch-point twist field is a twist field associated to a particular +symmetry transformation, part of a continuous symmetry group. As explained in Sec. 2.4, using +Noether’s theorem, this then allows one to write an explicit extensive charge associated to the +twist field [7]. +The U(α) symmetry is most clearly expressed in a different basis of the replica theory than that +used above, obtained by “fermionising" the replica theory. By the basic construction of the replica +theory, different replicas commute with each other. However, in order to extract the symmetry +U(α), one needs fermions in different copies to anti-commute. One simply defines the replica +theory by asserting that fermion fields anti-commute. This is of course natural, but changes the +action of the branch-point twist field (the exchange relations (45), (46)) by introducing an extra +minus sign, as worked out in [7]. From now on, we denote +ψi(x, t) = +1 +� +2π +� +dθ eiθ x−iE(θ)tψθ,i +(50) +15 + +SciPost Physics +Submission +the Dirac fermion on the i-the copy in this new basis; thus ψi(x, t)ψj(x′, t′) = −ψj(x′, t′)ψi(x, t) +if i ̸= j, and the canonical anti-commutation relations hold, {ψi(x, t),ψ† +j(x′, t)} = δi jδ(x − x′). +We now recall the main arguments of [7] in order to obtain a useful form of the branch-point +twist field. In the new basis, the U(α) symmetry is explicitly a linear action on the fermions, in its +fundamental representation. Most importantly, in this basis, the cyclic permutation ψi → ψi+1 is a +particular element of U(α), which is in fact an element of a U(1) subgroup. The Fourier transform +(47) can be performed in this basis, +ψp(x, t) = +1 +�α +α +� +i=1 +eiπpi/αψi(x, t). +(51) +This diagonalises that U(1) subgroup; the action of the twist field is then diagonalised. In fact, it +turns out that the anti-commuting basis is also the one that guarantees that the Fourier transform +operation keeps the S-matrix diagonal, see Appendix C. Thus both the charge associated to the +twist field, and the S-matrix, are diagonal in terms of the particles corresponding to ψp – this is +at the root of the simplification. +The fermion fields ψp(x, t) after Fourier transform are still independent free fermions with +canonical anti-commutation relations. Each Fourier sector admits an independent U(1) symmetry, +and, as shown in [7], the branch-point twist field can be written as a product of U(1) twist fields +acting nontrivially on each Fourier sector. Because of the extra minus sign in the twist field action, +it is simpler to concentrate on the case of α even (the full dependence on α is obtained by analytic +continuation), in which case the product goes over the following values of momenta: +p ∈ Iα := {−α + 1,−α + 3,··· ,α − 1}. +(52) +Specifically, it is found that [7] +T α = +� +p∈Iα +τα +p = +α/2 +� +q=−α/2+1 +τα +2q−1 +(53) +with τα +p being a U(1) twist field acting non-trivially only on ψp (as a phase), +τα +p(x, t)ψq(y, t) = +� +e−iπp/αψq(y, t)τα +p(x, t) +y ≥ x and p = q +ψq(y, t)τα +p(x, t) +y < x or p ̸= q +(54) +(cf. (48)). +The decomposition (53) allows us to factorise the branch-point twist field two-point functions +into products of U(1) twist field two-point functions. This however only holds if the state can be +likewise factorised. This is nontrivial: the state ρ⊗α is naturally factorised in copy space, but not +necessarily in the Fourier-copy space. It is a simple matter to verify that if ρ satisfies Wick theorem, +then ρ⊗α also factorises as a tensor product of states ρ in Fourier-copy space; this is because such +states are completely determined by fermion two-point functions, which stay diagonal in Fourier- +copy space. Therefore, we have, in Wick-theorem states ρ, +〈T α(0,0)¯T α(x, t)〉ρ⊗α = +α/2 +� +q=−α/2+1 +〈τα +2q−1(0,0)¯τα +2q−1(x, t)〉ρ . +(55) +Note how on the right-hand side, each factor is evaluated in the state ρ for the fermion ψ2q−1. +16 + +SciPost Physics +Submission +In the following we are going to apply the BFT machinery to each correlation function of the +U(1) twist fields. The crucial fact that makes it simple is that, for any given p, τα +p(x, t) is the +(bounded) twist field associated to the U(1)-charge +Qp = πp +α +� +d x ψ† +p(x)ψp(x) = πp +α +� +dθ ψ† +θ,pψθ,p , +(56) +with explicit expressions as exponential of half-space integrals of charge densities, as per Eq. (28): +τα +p(x, t) = exp +� +i +� ∞ +x +d x′ qp(x′, t) +� +, +qp(x, t) = πp +α ψ† +p(x, t)ψp(x, t) . +(57) +Qp acts on the single-particle basis as +Qp|θ,q〉 = hpδp,q|θ,q〉, +with hp = πp +α +(58) +(note that ψp(x) has Qp-charge −hp, in agreement with (54)). With Q = Qp, the twist field τα +p +is identified with τα +p = T−i in the notation of (34) (that is, with η = 1), acting on the sector p. +Recall that the action of the charge on the single-particle basis is all we need to know in order to +apply the BFT (cf. (38) and (40)). +4 +Entanglement entropies from BFT +We arrived to a rewriting of the two-point function of the branch-point twist fields as product of +two-point functions of U(1) twist fields, Eq. (55). From there, using BFT, all such components +can be accessed via Eq. (30) specified to the twist fields τα +p, which in the notation of Eq. (30) is +identified with T−i, with the r.h.s. evaluated via Eq. (38), thus exploiting the free fermionic nature +of the problem. Considering different choices of the points in spacetime where the global fields +T α, ¯T α are located, we are able to access Rényi entropies both at equilibrium and after a quench, +as we are now going to discuss. +4.1 +Rényi entropies of a finite interval in a GGE (charge fluctuations in space) +We start by considering the α−Rényi entropy of a finite interval A = [0, x] within a generic GGE +ρw uniquely defined by the function w(θ) (see Sec. 2.5). This means that we are interested in the +following two-point function +〈T α(0,0)¯T α(x,0)〉ρ⊗α +w . +(59) +From the BFT perspective, this is obtained by focusing on the purely spatial direction, namely, we +consider an “horizontal path” by setting γ = π/2 in (38) (and we take h(θ) = hp). Each two-point +function of U(1) twist fields in (55) reads +〈τα +p(0,0)¯τα +p(x,0)〉ρw ≍ exp +� +x Fp(−i) +� +, +Fp(−i) = +� +dθ +2π log +� +1 + eihp−w(θ) +1 + e−w(θ) +� +. +(60) +Then we consider the product in Eq. (55), which turns into a sum in the exponent, i.e., +〈T α(0,0)¯T α(x,0)〉ρ⊗α +w ≍ exp{x Fα(−i)}, +with +Fα(−i) = +α/2 +� +q=−α/2+1 +F2q−1(−i) . +(61) +17 + +SciPost Physics +Submission +We may further evaluate those sums, by considering separately the part which depends and the +part which does not depend on p (equivalently q,q′, Eq. (53)). The latter is trivial and simply +gives a contribution to Fα(−i) which is − +� +dθ/(2π) of +2 +α/2 +� +q=1 +log +� +1 + e−w(θ)� += αlog +� +1 + e−w(θ)� +. +(62) +For the remaining part, let us start by focusing on half of the sum, the terms from q = 1 to α/2, +in (61). By defining z = 2πi +α , s = w + πi +α , we get +α/2 +� +q=1 +log(1 + ezq−s) += +∞ +� +r=1 +(−1)r+1 +r +e−r(s−z) +� +1 − erzα/2 +1 − erz +� +(63) +where we used the Taylor expansion log(1+ x) = +�∞ +r=1(−1)r+1x r/r (which converges for w > 0), +and we performed the sum over q. Next, we want to perform the sum in r in the r.h.s. of (63). To +do that, we substitute the values of z and w first: +∞ +� +r=1 +(−1)r+1 +r +e−r(w− πi +α ) +� +1 − erπi +1 − er 2πi +α +� +(64) +where now we should consider separately three cases: +1. r = αm for integer m: in this case r is even (as α is even), and we have +∞ +� +m=1 +(−1)αm+1 +αm +e−αmw+mπi +� +rπi +2πir/α +� += +∞ +� +m=1 +(−1)m+1 +2m +e−αmw +(65) += 1 +2 log +� +1 + e−αw� +. +(66) +2. r even but r ̸= αm for any integer m: in this case each term of the sum (64) is zero due to +the vanishing of the numerator, i.e., (1 − erπi) = 0. +3. r odd: this gives +� +r odd +2 +r e−rw +� +er πi +α +1 − er 2πi +α +� += +� +r odd +i +r +e−rw +sin πr +α +. +(67) +The sum of the terms for q = −α/2 + 1 to 0 in (61) give exactly the complex conjugate of this +result. Thus we get +α/2 +� +q=−α/2+1 +log(1 + ezq−s) = log(1 + e−αw) . +(68) +Putting everything together, Fα(−i) in (61) can be written as +Fα(−i) = +� +dθ +2π +� +log +� +1 + e−αw(θ)� +− αlog +� +1 + e−w(θ)�� +. +(69) +Finally, it is a matter of simple algebra to show that, in terms of the occupation function n(θ) (37), +we get +Fα(−i) = (1 − α) +� +dθ +2π Hα(θ) +(70) +18 + +SciPost Physics +Submission +where we defined +Hα(θ) += +1 +1 − α log[n(θ)α + (1 − n(θ))α] . +(71) +The α−Rényi entropy is finally given by +Sα(x) = +1 +1 − α log〈T α(x,0)¯T α(0,0)〉ρ⊗α +w ∼ x +� +dθ +2π Hα(θ) , +(72) +which coincides with the results obtained in [11,13] (there in the more general context of inter- +acting integrable models). +4.2 +Long-range correlations due to correlated particle pairs in homogeneous global +quenches +We now review the main concepts underlying quantum quenches, restricting to “integrable" pair- +producing initial states, and we explain how long-range correlations develop after such quenches. +A quantum quench is an initial value problem for the many-body system where the initial state +is the ground state of a different Hamiltonian than that used for the time evolution. Typically, one +imagines a sudden change of parameter, for instance of the mass parameter. In integrable models, +certain quenches are known to be of “integrable" type [26–28]. In these cases, the initial state can +be written explicitly in terms of the scattering states (or Bethe ansatz states) of the post-quench, +evolution Hamiltonian, as a so-called “squeezed state": +|Ψ〉 = 1 +N exp +� +1 +2 +� +dθ Kθ,−θψ† +θψ† +−θ +� +|0〉 +(73) +for some (θ-dependent) factor Kθ,−θ, with Nθ denoting a normalization constant, and |0〉 being +the ground state of the post-quench Hamiltonian. The squeezed state is generically a finite-density +state, where the energy (of the post-quench Hamiltonian) is extensive with the system size. See +App. B.2 for a discussion of such integrable initial states in free fermion models. We will use later +the fact there is a Bogolioubov transformation of the fermionic mode operators (a transformation +between the post-quench and pre-quench fermions), +ψ(x, t) ↔ ˜ +ψ(x, t) +(Bogolioubov) +(74) +such that the squeezed state satisfies (is defined by) +˜ +ψ(x, t)|Ψ〉 = 0. +(75) +After a long time in a quench problem, the state locally approaches a GGE. In integrable +quenches, there is a well-known relation between the squeezed-state representation of the ini- +tial state, and the long-time GGE (see e.g. [44]). The statement of convergence to a GGE pertains +only to local operators, or operators supported on finite intervals (that do not grow with time): +〈Ψ|a(x, t)|Ψ〉 → 〈a(x)〉ρw, +t → ∞ . +(76) +The limit in (76) is expected to be valid everywhere in space. The relation between initial state +and long-time GGE in free fermions can be worked out explicitly (see (150)) +e−w(θ) = |Kθ,−θ|2 . +(77) +19 + +SciPost Physics +Submission +Namely, we see that the map from squeezed states to GGEs is in fact one-to-one. +Because of this one-to-one correspondence, it is clear that it is sufficient to know the long- +time GGE in order to know the full behaviour of correlation functions in spacetime, as the GGE +fixes the initial state uniquely (naturally under the condition that it be a pair-producing squeezed +state). However, this relation can be relatively complex. Indeed, the statement of generalised +thermalisation – that a GGE is reached – is true, generically, only on finite regions of space (see +App. B.3). As is typically the case out of equilibrium, on large regions, say regions that grow +linearly with the time after the quench, the state might not correctly be described by a GGE; and +this may even be true for all times! Instead, the state may admit long-range spatial correlations, +for instance correlations that have a large weight on distances that grow linearly with the time. +These are not present in GGEs; recall that, as discussed above, GGEs typically have correlations +that decay quickly enough in space (see App. B.1). Thus, even at long times, there may remain +effects of the initial state that are not described by a GGE. +In the case of a squeezed state, such long-range correlations indeed exist, as we show in +App. B.3 (and also in App. B.5 for “single-mode densities and currents", introduced below in +Sec. 4.4). Their interpretation is that they are due to production of correlated pairs of opposite- +momentum particles by the quench protocol. These particle pairs carry correlations to large dis- +tances as they separate. We evaluate explicitly these long-range correlations for conserved den- +sities and currents in App. B.3 (and App. B.5). For instance, we find, for the charge density +q(x, t) = ψ†(x, t)ψ(x, t), that 〈Ψ|q(x, t)q(x′, t)|Ψ〉 − 〈Ψ|q(x, t)|Ψ〉〈Ψ|q(x′, t)|Ψ〉 exhibits strong, +ballistic-scale correlations, in accordance with the picture according to which particle pairs are +emitted at all velocities admitted by the dispersion relation. +In order to evaluate the Rényi entanglement entropy, as is clear from the calculation for GGEs +in Sec. 4.1, we must evaluate the large-deviation theory for fluctuations of charges on large regions +of space, and / or, as we will see below, fluctuations of current on large intervals of time. The BFT +allows us to do that. However, as mentioned, the BFT requires no long-range correlations along +the path ℓ in (12), as the flow equation (cf. (19)) assumes that the state along the path is a GGE. +Long-range correlations may break the assumption that scaled cumulants are evaluated within a +GGE – the effects of long-range correlations on the 2nd cumulants is illustrated in App. B.3. Below, +we take them into account by choosing appropriately the path ℓ in order to avoid such correlations! +Thus, the knowledge of where such correlations exist, and the knowledge of the long-time GGE, +is sufficient. +The fact that there exists a path ℓ that avoids long-range correlations explains why, in pair- +production states, the full behaviour of the Rényi entropies can be written in a simple and uni- +versal way in terms of the long-time GGE. The same is not true when considering non-integrable +quenches, namely quenches from more complicated states where groups of more than two cor- +related particles are emitted. Without the constraint of pair-production, the one-to-one corre- +spondence between the state and the GGE is lost, and long-range correlations carry additional +information not present in the GGE. Then, from our approach, we see that a universal description +in terms of the long-time GGE is lost because such an “avoiding path” does not exist in general +anymore. +4.3 +Rényi entropies of half system after a quench (current fluctuations in time) +We now turn to the calculation of the α−Rényi entropy of a semi-infinite interval A = [0,∞) +after a global homogeneous quantum quench, at long times t → ∞. This is obtained from the +20 + +SciPost Physics +Submission +KING’S COLLEGE LONDON +P AO L A R U G G I E RO +• + : + (for time-dependence) +• Correlations and breaking of LDT : further info + +• BFT (“vertical” path) : + +• Rényi entropy of semi-infinite system after a quench: + +A = [0, ∞] ⟨Tα(0,t)⟩ ≃ ⟨Tα(0,t) ¯Tα(0,0)⟩ +Ψin⟩ = ∏ +θ>0 +1 +Zθ +eWθψ† +θψ† +−θ ∅⟩ +⟨Tα(0,t) ¯Tα(0,0)⟩ ≍ exp {t Fα(1)} +Sα(t) = t∫ +dθ +2π |v(θ)|Hα(θ) + : HALF SYSTEM AFTER A QUANTUM QUENCH +Sα(t) +t +x +Tα(0,t) +0 +t +x +Tα(0,t) +0 +¯Tα(0,0) +−vθ +vθ +−vθ +vθ +[Alba,Calabrese,2017] +Figure 1: Evolution of Rényi entropies of half system A = [0,∞] within BFT. Left: Ini- +tial integration path. Because of initially entangled pairs, points along this path at time +t will be correlated, which prevents us from applying BFT directly. Right: Deformed +integration path. Along this new path one can show that point are not correlated any- +more. Moreover the only term contributing to the growth in time of entanglement is the +vertical path from (0, t) to (0,0). +branch-point twist field one-point function +〈Ψα|T α(0, t)|Ψα〉 +(78) +in the state |Ψα〉 = |Ψ〉⊗α = +�α +i=1 |Ψi〉, the α-copy replica of (73), +|Ψα〉 = +α +� +i=1 +1 +N exp +� +1 +2 +� +dθ Kθ,−θψ† +θ,iψ† +−θ,i +� +|0, i〉 . +(79) +As expressed in (76), one-point functions of local observables converge to averages within +GGEs. However, as we discussed, twist-fields are “semi-local" observables; from the point (0, t) +emanates a branch cut, which is sensitive to the state where it passes. The branch cut can be taken +on the horizontal half-line {(x, t) : x ∈ [0,∞)} going from (0, t) to (∞, t), as done in the explicit +construction of the field in Sec. 3. As explained in Sec. 4.2, and analysed in App. B.3, along this +half-line, there exist long-range correlations due to coherent particle pairs emitted by the initial +state. This prevents us from applying the BFT along this path (see Fig.1 (left)). +Using path independence of twist fields correlation functions, we can deform the path, between +its initial and final points, in a way to avoid such correlations. Specifically, we choose the piece- +wise linear path joining the points (0, t) → (0,0) → (∞,0). This is shown in Fig. 1 (right). +We note that as the final point is at spatial infinity, it can be displaced to time 0 – this in fact +implements the correct physics of the entanglement entropy due to the single boundary at x = 0. +Then, we may represent the one-point function as +〈Ψα|T α(0, t)|Ψα〉 ≍ 〈Ψα|T α(0, t)¯T α(0,0)T α(0+,0)|Ψα〉 +(80) +where the factors T α(0, t)¯T α(0,0) represent the segment of path (0, t) → (0,0), and the factor +T α(0+,0), the segment (0,0) → (∞,0). This is valid as an asymptotic relation for large t, where +the UV singularity due to the proximity of the fields ¯T α(0,0) and T α(0+,0) (which occurs because +of renormalisation effects) is neglected. +We simplify the expression (80) in two steps. +First, we note that the segment of path (0,0) → (∞,0) does not provide any contribution to +the result. This is because we may re-write the branch-point twist field T α(0+,0) as is done in +21 + +SciPost Physics +Submission +Sec. 3, but in the basis of the before-quench canonical free fermions of the replica theory, ˜ +ψi(x,0), +Eq. (74). The exchange relations (45) (here at t = 0) hold for any field ai(x,0), and in particular +hold for ai(x,0) = ˜ +ψi(x,0). Therefore, by the same arguments, we obtain a decomposition as in +(53), +T α = +� +p∈Iα +˜τα +p +(81) +but for different U(1) twist fields +˜τα +p(x,0) = exp +� +i +� ∞ +x +d x′ ˜qp(x′,0) +� +, +˜qp(x,0) = πp +α +˜ +ψ† +p(x,0) ˜ +ψp(x,0) +(82) +instead of (57). By (75), we have ˜ +ψi(x,0)|Ψj〉 = 0 for all i, j, so it is clear that +˜qp(x,0)|Ψα〉 = 0 , +(83) +therefore +˜τα +p(x,0)|Ψα〉 = |Ψα〉 . +(84) +Hence +〈Ψα|T α(0, t)|Ψα〉 ≍ 〈Ψα|T α(0, t)¯T α(0,0)|Ψα〉 . +(85) +Second, we note that along the path (0, t) → (0,0), generic observables do not have long- +range correlations coming from pair productions: correlations of generic observables approach +those within the final GGE fast enough, in such a way that corrections due to the quench give +only sublinear corrections to cumulants of time-integrated fermion bilinears (such as conserved den- +sities and currents). This is because particle pairs always create correlations between points at +separate spatial coordinates: it is not possible to create two co-propagating fermions, with the +same momentum (here they would be with vanishing momentum, as the total momentum has +to be zero). This fact is discussed in App. B.3; the discussion there is for a single copy, but it +extends immediately to the α-copy state |Ψα〉. Therefore, rewriting the branch-point twist fields +in terms of U(1) τα +p with branches in the time direction, using (53), and expanding in cumulants +of U(1) currents, we see that on the segment (0, t) → (0,0), for the purpose of the BFT, the state +is correctly described by a GGE10. +An important consequence of these arguments is that we may evaluate the twist field one-point +function after a quench, as an equal-space, different-time two-point function within the GGE repre- +senting the final state, Eq. (77): +〈Ψα|T α(0, t)|Ψα〉 ≍ 〈T α(0, t)¯T α(0,0)〉ρ⊗α +w . +(86) +This is valid at long times, and omits small-time effects that occur before generalised thermalisa- +tion (which do not affect the asymptotic regime we look at). +In order to apply BFT to the r.h.s. of (86) a last observation is needed. As the GGE is a Wick- +theorem state, we can use (55), thus we are interested in the separate two-point functions of U(1) +twist fields τα +p with branches in the time direction. It turns out that, as emphasised in Sec. 2.3, the +currents jp(0, t′), t′ ∈ [0, t] in GGEs have time-correlations that decay fast enough so as to give +only linearly growing cumulants: this is what allows the application of the BFT (see App. B.1 for +10We remark that for bosonic system, this argument would break, as pairs of particles with equal, zero momenta +are emitted with a finite density. However, it turns out that this correction due to the quench would not affect the +cumulants of time-integrated currents, as such pairs, being of zero momentum, do not carry any current. +22 + +SciPost Physics +Submission +a full discussion). We note that this is not true in general of other observables: in GGEs, generic +fermion bilinears have cumulants that grow faster than linearly with time. But we are intersted +in the currents only. +We are now in position to apply standard BFT. This amounts to repeating the same calculation +as above in Sec. 4.1, but now in the purely temporal direction. We use the general formula (38) +for the “vertical” path connecting initial and final point by choosing γ = 0, and similarly get (note +that the path is in opposite direction as that of formula (38), and thus we must take h(θ) = −hp) +〈τα +p(0, t)¯τα +p(0,0)〉ρw ≍ exp +� +t Fp(−i) +� +, +Fp(−i) = +� +dθ +2π|v(θ)|log +� +1 + eihp sgn(θ)−w(θ) +1 + e−w(θ) +� +(87) +where we used sgn(v(θ)) = sgn(θ). Again, after performing the product all two-points functions +of U(1) twist fields, we get +〈T α(0, t)¯T α(0,0)〉ρ⊗α +w ≍ exp{t Fα(−i)}, +Fα(−i) = +� +dθ +2π|v(θ)|log +� +1 + e−αw(θ) +� +1 + e−w(θ)�α +� +. +(88) +Using Hα(θ) as defined in (71), the α−Rényi entropy reads +Sα(t) = +1 +1 − α log〈T α(0, t)¯T α(0,0)〉ρ⊗α +w ∼ t +� +dθ +2π|v(θ)|Hα(θ) . +(89) +This is the result obtained both from exact calculation in [15] and within the quasi-particle picture +in [19,20]. +4.4 +Single-mode and pair-mode twist fields +We have discussed in Sec. 2.5 the conserved quantities Qθ = ψ† +θψθ, forming a “scattering" or +continuous basis for the extensive conserved quantities of the free fermion model. In Sec. 3.2, we +discussed the replica model with α copies, and the U(1) charges Qp, which are just the integration +Qp = hp +� +dθ Qθ,p (with hp = πp +α ) over all momenta θ of the continuous basis Qθ,p = ψ† +θ,pψθ,p in +the Fourier-copy p. There, we have also discussed the twist fields τα +p associated to these charges, +which turned out to be useful in the computation of the Rényi entanglement entropies in Subsec- +tions 4.1 and 4.3. A natural extension of these constructions is to the twist fields associated to +each conserved quantity Qθ,p. As we will see, these are indeed useful in evaluating the behaviour +of Rényi entanglement entropies for intervals that grow linearly with time. +In order to simplify the notation, we consider a single copy of the fermion, and the scattering +basis Qθ; the discussion immediately adapts to the Fourier-copy p. +In the study of the ballistic behaviours of many-body systems, and in particular in the BFT, it +is essential that the conserved charge Q considered be extensive – scale linearly with the volume +(typically one requires 〈Q2〉 +c ∝ L [37,45]). The charges Qθ are not extensive. However, as they +form a continuous basis, integrals on small θ-intervals are extensive; thus it is better to define, for +ε > 0 as small as desired, +Qθ = +� θ+ε/2 +θ−ε/2 +dθ ′ ψ† +θ ′ψθ ′ . +(90) +These act as +Qθ |θ ′〉 = Θ(ε/2 − |θ ′ − θ|)|θ ′〉 +(91) +23 + +SciPost Physics +Submission +hence have one-particle eigenvalues +hθ(θ ′) = Θ(ε/2 − |θ ′ − θ|). +(92) +We show in App. B.4 that such Qθ are indeed extensive in GGEs, and we evaluate explicitly their +associated densities and currents qθ(x, t) and jθ(x, t), +Qθ = +� +d x qθ(x, t), +∂tqθ(x, t) + ∂x jθ(x, t) = 0 . +(93) +From this, one can immediately construct the associated twist field +τθ(x, t) = exp +� +i +� ∞ +x +d x′ qθ(x′, t) +� +(94) +and, for its correlation functions, apply the corresponding BFT based on the one-particle eigen- +value (92). +In fact, we are interested in studying the squeezed state (73). It is clear that this state factorises +into momentum intervals as follows: +|Ψ〉 = +� +θ∈(�+ 1 +2 )ε +|Ψ|θ|〉 +(95) +where +|Ψ|θ|〉 = +1 +N|θ| +exp +�� θ+ε/2 +θ−ε/2 +dθ ′ Kθ,−θψ† +θψ† +−θ +� +|0|θ|〉 +(96) +and we write the ground state in a naturally factorised way as |0〉 = +� +θ∈(�+ 1 +2 )ε |0|θ|〉. Likewise, +we will consider the pair-mode charges Q|θ| = Qθ + Q−θ and the associated densities +q|θ|(x, t) = qθ(x, t) + q−θ(x, t) . +(97) +Both act trivially (as zero) on |Ψ|θ ′|〉 if θ ′ ̸= θ (θ,θ ′ ∈ (�+ 1 +2)ε). From these, we get the pair-mode +twist fields +τ|θ|(x, t) = exp +� +i +� ∞ +x +d x′ q|θ|(x′, t) +� +, +(98) +which acts trivially (as the identity) on |Ψ|θ ′|〉 if θ ′ ̸= θ. +These are still U(1) twist fields, for the sub-U(1) symmetry acting on the tensor factor of modes +within [θ − ε/2,θ + ε/2]. Note in particular that the global U(1) twist field τ(x, t) associated to +the total charge Q = +� +dθ ψ† +θψθ = +� +d x ψ†(x)ψ(x) can be factorised as +τ(x, t) = +� +θ∈(�+ 1 +2 )ε +τ|θ|(x, t) +(99) +and that, by factorisation of its action on the state, we have +〈Ψ|τ(x, t)τ(x′, t′)|Ψ〉 = +� +θ∈(�+ 1 +2 )ε +〈Ψ|θ||τ|θ|(x, t)τ|θ|(x′, t′)|Ψ|θ|〉. +(100) +Clearly, as the pair-mode twist fields act trivially on other tensor factors in the state, we may also +write, more simply, +〈Ψ|τ(x, t)τ(x′, t′)|Ψ〉 = +� +θ∈(�+ 1 +2 )ε +〈Ψ|τ|θ|(x, t)τ|θ|(x′, t′)|Ψ〉. +(101) +24 + +SciPost Physics +Submission +4.5 +Rényi entropies of an interval after a quench (fluctuations of single-mode den- +sities and currents) +We finally extend the result for the entanglement growth after a quench to a finite, but ballistically +growing interval A = [0, x], with x = ξt. To this aim, we should consider the following two-point +correlation function in a squeezed state Eq. (73) (or Eq. (79) in the replicated theory): +〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉, +x = ξt, +t → ∞. +(102) +The idea is the same as that used in Sec. 4.3, that we need to deform the integration path in +such a way that, everywhere along the path, all points remain uncorrelated (on large scales), thus +enabling us to apply BFT. The choice of the path will now depend on the values of ξ, and in fact, +we will need re-write the two-point function as a product of two-point functions of pair-mode +twists fields, and to choose different paths for each such two-point function. +It will simplify the discussion to already re-write the two-point function in terms of U(1) twist +field. As the squeezed state is a Wick-theorem state, we can directly use (55): +〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉 = +α/2 +� +q=−α/2+1 +〈Ψ2q−1|τα +2q−1(0, t)¯τα +2q−1(x, t)|Ψ2q−1〉 +(103) +where |Ψp〉 is the squeezed state |Ψ〉 for the fermions ψp(x),ψ† +p(x) on Fourier-copy space p. +We start by considering the two asymptotic regimes: +• At short times (more precisely in the limit ξ → ∞ of the scaled, long-time asymptotic be- +haviour), entangled particle pairs coming out from the initial state will correlate points +within the original integration path. +To apply BFT, then, we deform the straight path +(0, t) → (x, t) to the piece-wise straight path (0, t) → (0,0) → (x,0) → (x, t), made of +three segments (see Fig. 2 (left)). By the same arguments as in Sec. 4.3, the space-like +segment will not contribute to the entanglement growth, and the time-like segments will +give separated, identical contributions given by the long-time GGE. We are thus left with +the contribution of the two, separate time-like segments. The fact that the segments do +not correlate with each other is thanks to the assumption that the GGE satisfies n(θ) → 0 +as |θ| → ∞ (that is, the density of pairs produced at large momenta tends to zero), as is +discussed in App. B.3. +• At long enough times (either the limit ξ → 0 of the scaled, long-time asymptotic behaviour, +or the long-time limit followed by the long-distance scaling), the particles generated from +the initial state do not correlate points within the path (0, t) → (x, t): cumulants scaled +by the distance x do not receive contributions from such particle pairs. Corrections terms +to the GGE values of cumulants can only come from pairs of particles at infinitesimally +small momenta, and, it turns out, such corrections become zero when the total number of +correlated pairs on the interval [0, x] tend to zero. As there is at most a finite density of +pairs produced per unit momenta, there remain no pairs on infinitesimally small momentum +intervals11. Thus the asymptotic behaviour is that obtained from the long-time GGE. This +is discussed in App. B.3. Particle pairs of finite momenta would, of course, correlate points +between the path segments (0, t) → (0,0) and (x,0) → (x, t) (also discussed in App. B.3), +11In fact, as we are looking at fermionic models, the density tends to zero at zero momenta, but this is not required +in this argument. +25 + +SciPost Physics +Submission +KING’S COLLEGE LONDON +P AO L A R U G G I E RO +• + : + +• Asymptotic regimes : “small” and “large” time ( + ) + + +• Rényi entropies of finite interval after a quench : +A = [0, x] +⟨Tα(0,t) ¯Tα(x, t)⟩ +x/t → ∞, 0 +⇒ +⟨Tα(0,t) ¯Tα(x, t)⟩ = { +|⟨Tα(0,t) ¯Tα(0,0)⟩|2 +t ≪ x +⟨Tα(0,t) ¯Tα(x, t)⟩ +t ≫ x +Sα(x, t) = +2t∫ dθ +2π |v(θ)|Hα(θ) +t ≪ x +x∫ dθ +2π Hα(θ) +t ≫ x + : FINITE INTERVAL AFTER A QUANTUM QUENCH +Sα(x, t) +t +x +Tα(0,t) +0 +¯Tα(x, t) +−vθ +vθ +t +x +Tα(0,t) +0 +¯Tα(x, t) +vθ +−vθ +Figure 2: Evolution of Rényi entropies of finite subsystem A = [0, x] within BFT. The +integration path that we need to chose (continuous dark-gray line) in order for BFT to +apply is different at short (left) and long (right) times. The choice depends on which +points in spacetime get correlated because of initially entangled pairs produced by the +initial state. +thus we must avoid the piece-wise straight path. Therefore the correct way to use BFT is by +using the original path (see Fig. 2 (right). +So we arrive to the following asymptotic results for x, t → ∞: +〈Ψα|T α(0, t)¯T α(x, t|Ψα〉 +≍ +� +〈T α(0, t)¯T α(0,0)〉ρw〈T α(x,0)¯T α(x, t)〉ρw +t ≪ x +〈T α(0, t)¯T α(x, t)〉ρw +t ≫ x += +���〈T α(0, t)¯T α(0,0)〉ρw +��2 +t ≪ x +〈T α(0, t)¯T α(x, t)〉ρw +t ≫ x +(104) +where we used 〈T α(0, t)¯T α(0,0)〉∗ +ρw = 〈T α(0,0)¯T α(0, t)〉ρw = 〈T α(x,0)¯T α(x, t)〉ρw (by the fact +that T(x, t)† = ¯T(x, t)). This leads to +Sα(x, t) = +1 +1 − α log〈T α(0, t)¯T α(x, t)〉 +∼ +� +� +� +� +� +� +� +� +� +2t +� +dθ +2π|v(θ)|Hα(θ) +t ≪ x +x +� +dθ +2π Hα(θ) +t ≫ x. +(105) +Namely, at short times (but much larger than microscopic times), the growth is described by the +path in the purely temporal direction (89), and at long times, the system goes, uniformly as a +function of the velocity x/t → 0, to the equilibrium GGE and there the result is the one of the +purely spatial path (72). +These are, however, only asymptotic results in ξ, within the scaled regime x, t ∝ ℓ. It turns +out that we can access all values of ξ = x/t within this regime, by using similar arguments, but +now for the single-mode twist fields introduced in Sec. 4.4 (in fact, we need the pair-mode twist +fields). Effectively, using these, we will be able to take into account that the meaning of “short” +and “long” time depend directly on the speed of the travelling particles v(θ) = E′(θ). +We start with the decomposition of global U(1) twist fields into pair-mode twist fields (99), +which we write in the replica model for each Fourier-copy p, and with single-particle charge eigen- +26 + +SciPost Physics +Submission +value hp = πp +α instead of 1 in (99) (as done in (57)) +τα +p(x, t) = +� +θ∈(�+ 1 +2 )ε +τα +|θ|,p(x, t) +(106) +where +τα +|θ|,p(x, t) = exp +� +i +� ∞ +x +d x′ q|θ|,p(x′, t) +� +(107) +and q|θ|,p(x, t) has the form (173) times hp. Thus, by factorisation of two-point functions (101), +we re-write (103) as +〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉 = +� +θ∈(�+ 1 +2 )ε +α/2 +� +q=−α/2+1 +〈Ψ2q−1|τα +|θ|,2q−1(0, t)¯τα +|θ|,2q−1(x, t)|Ψ2q−1〉 . (108) +Having made this re-writing, the analysis now follows that of the ξ → ∞ and ξ → 0 limits +made above: there is an exact parallel for each individual two-point function +〈Ψp|τα +|θ|,p(0,0)¯τα +|θ|,p(x, t)|Ψp〉 +(p = 2q − 1), +with the only difference that it is not necessary to take the asymptotic limit in ξ. For each θ (and +each p), the factor |Ψ|θ|,p〉 of the full state |Ψp〉, on which τα +|θ|,q act non-trivially, correlates points +(x, t), (x′, t′) only for +|x − x′| +|t + t′| ∈ [v(θ − ε/2), v(θ + ε/2)] +(recall that θ ∈ (� + 1 +2)ε). The analysis of single-mode correlations is made in App. B.4. +Therefore, for ξ > 2v(θ + ε/2) correlations occur on the horizontal path (0, t) → (x, t), but +no correlations occur on (0, t) → (0,0) → (x,0) → (x, t) (note that, again, the segment of path +(0,0) → (x,0) does not contribute). Thus we must choose the latter path (Fig. 2 (left)). On the +contrary, ξ < 2v(θ − ε/2), correlation occur between the segment of paths (0, t) → (0,0) and +(x,0) → (x, t), but not on the horizontal path (0, t) → (x, t). Thus we must choose the latter +(Fig. 2 (right)). In making these right choices, the correlation functions of pair-mode twist fields +tend to their values in the long-time GGE, +〈Ψp|τα +|θ|,p(0,0)¯τα +|θ|,p(x, t)|Ψp〉 +(109) +≍ +� +〈τα +|θ|,p(0, t)¯τα +|θ|,p(0,0)τα +|θ|,p(x,0)¯τα +|θ|,p(x, t)〉ρw +ξ > 2v(θ + ε/2) +〈τα +|θ|,p(0,0)¯τα +|θ|,p(x,0)〉ρw +ξ < 2v(θ − ε/2) . +Note how in the first line, it is a four-point function that appears. +Now again, in order to apply the BFT, we need to consider the correlations between twist +fields within GGE. We already argued in Sec. 4.3 (with supporting calculations in App. B.3) that +no strong correlation occurs between local operators at equal times and different points in space, +thus on the second line of (109) we may apply the BFT. We also argued that no strong correlation +occurs between current operators at equal space and different times, and in fact this also holds +for single mode currents. However, in order to simplify the first line of (109), we need to address +correlations between currents on the path segments (0, t) → (0,0) and (x,0) → (x, t). In general, +local observables have strong correlations at different space-time points, due to hydrodynamic +27 + +SciPost Physics +Submission +modes propagating in space-time. As we do not make any strong assumption about the disper- +sion relation, all hydrodynamic velocities occur, hence correlations occur between generic local +observables on these separate path segments. However, single-mode currents only produce hy- +drodynamic modes at the corresponding velocities; supporting calculations are found in App. B.5. +Thus, as long as ξ > v(θ + ε/2), no correlation occurs between these paths for the single mode +currents j±θ,p. As v(θ) > 0, then ξ > 2v(θ + ε/2) ⇒ ξ > v(θ + ε/2), hence on the first line +simplifies and we have +〈Ψp|τα +|θ|,p(0,0)¯τα +|θ|,p(x, t)|Ψp〉 ≍ +� +� +� +���〈τα +|θ|,p(0, t)¯τα +|θ|,p(0,0)〉ρw +��� +2 +ξ > 2v(θ + ε/2) +〈τα +|θ|,p(0,0)¯τα +|θ|,p(x,0)〉ρw +ξ < 2v(θ − ε/2) +(110) +where the BFT can be applied for all two-point functions. +For the two-point functions with spatial separation 〈τα +|θ ′|,p(0,0)¯τα +|θ ′|,p(x,0)〉ρw, we can use the +analysis of (60) made in Sec. 4.1, where we only have to replace, on the right-hand side of (60) +inside the θ-integral, the constant one-particle eigenvalue hp by the piece-wise constant function +hp +� +Θ(ε/2 − |θ − θ ′|) + Θ(ε/2 − |θ + θ ′|) +� +. +Thus the same analysis goes through, but with the integral restricted to +θ ∈ Iθ ′,ε := [θ ′ − ε/2,θ ′ + ε/2] ∪ [−θ ′ − ε/2,−θ ′ + ε/2]. +Likewise for the two-point functions with temporal separation 〈τα +|θ ′|,p(0, t)¯τα +|θ ′|,p(0,0)〉ρw, with the +analysis of Sec. 4.3. Putting the results together, we obtain +1 +1 − α log +� +α/2 +� +q=−α/2+1 +〈Ψ2q−1|τα +|θ|,2q−1(0, t)¯τα +|θ|,2q−1(x, t)|Ψ2q−1〉 +� +∼ +� +Iθ,ε +dθ ′ +2π min(x,2t|v(θ ′)|) Hα(θ ′) +(111) +which is valid for x/t < 2v(θ − ε/2) or x/t > 2v(θ + ε/2). +For the case of x/t within this excluded region, we do not have explicit results, but the scaled +cumulants still are finite (as one can see by doing a calculation similar to App. B.3, for instance). +Thus, the result may be deemed valid as well within this excluded region, up to an error of order +ε. +Taking the product over θ’s as per (108), +1 +1 − α log〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉 ≍ +� +dθ +2π min(x,2t|v(θ)|) Hα(θ) + O(ε) +(112) +and as this holds for all ε > 0, we can take the limit ε → 0 and we obtain +Sα(x, t) = +1 +1 − α log〈T α(0, t)¯T α(x, t)〉 ∼ +� +dθ +2π min(x,2t|v(θ)|) Hα(θ) . +(113) +This is in full agreement with the quasiparticle picture [19,20]. +28 + +SciPost Physics +Submission +Finally, we note that the relation (3) between this formula for Rényi entanglement entropy +growth, and the static and dynamic fluctuations, is directly obtained from the above discussion, +by identifying +JN<(t) = +� t +0 +dt′ +� +θ∈(�+ 1 +2 )ε +2v(θ)(x) = +� x +0 +d x′ +� +θ∈(�+ 1 +2 )ε +2v(θ)>x/t +q|θ|(x′,0) +(115) +using the explicit expressions of pair-mode densities and currents (176) in App. B.4, and taking +the limit ε → 0. +5 +Discussion and conclusion +In this paper we have studied the Rényi entanglement entropy in GGEs and after quenches from +integrable (pair-production) states in free fermion theories. Although this has been relatively well +studied in the literature, most results were based on specific ways of writing the Rényi entangle- +ment entropy using the free fermion structure (e.g. in terms of determinants), and on the idea of +entanglement due to engangled pairs produced by the quench [19, 20]. A first-principle deriva- +tion that generalises beyond free fermions was still largely missing, while it is known that the +simple quasi-particle picture fails for α-Rényi entanglement entropies (with α ̸= 1) in interacting +models [22]. +We have proposed a new approach based on twist-field correlation functions and hydrody- +namic fluctuations. This uses the hydrodynamic theory for free fermions, which is a special case +of generalised hydrodynamics (GHD), and the ballistic fluctuation theory (BFT), which relates +the exponential decay of twist-field correlation functions to hydrodynamic large-deviation theory. +Crucially, in order to have a full understanding of the quench dynamics, we have introduced a +new concept: that of single-mode twist fields. These are twist fields associated to the quasi-local +charge counting the number of fermions within a small interval of momentum; or more generally +twist fields “acting" on the quasi-locality sector of observables supported on a small momentum +interval. The approach is potentially more general and more fundamental, as hydrodynamics, the +BFT and single-mode twist fields – twist fields associated to individual hydrodynamic modes – are +applicable much beyond free fermions. Perhaps most interestingly, it reveals the new physics of +thermodynamic and hydrodynamic fluctuations behind the behaviour of the Rényi entanglement +entropies. +Three important concepts are brought forward: +• Entanglement is deeply connected to fluctuations, and the large-scale behaviour of entan- +glement, both static and dynamic, is controlled by large-deviation and hydrodynamic prin- +ciples. +• Hydrodynamic modes and projections onto such modes are more accurate and general no- +tions which replace the idea of particle-pair productions used to understand entanglement +dynamics in integrable systems. +29 + +SciPost Physics +Submission +• The fact that the entanglement growth in quenches from “integrable” states can be written +as a simple and universal function of the generalised Gibbs ensemble (GGE) reached at long +times comes from the particularly simple structure of the long-range correlations that such +states present. +To elaborate on these concepts: first we have confirmed that the Rényi entanglement entropy +in GGEs is controlled by thermodynamic fluctuations, and related to (a simple analytic continua- +tion of) a difference of thermodynamic free energies. This is in agreement with the observations, +made earlier [29–31], that the large-deviation theory for charge fluctuations is closely related to +the entanglement entropy. Here, this relation appears naturally from completely general concepts: +branch-point twist fields and the BFT. Using these, in fact, the conclusion is pushed further: we +show that the growth of Rényi entanglement entropy after a quench is controlled by hydrodynamic +current fluctuations, and related to a dynamical free energy associated to the large-deviation the- +ory for charge transport, as fully encoded in Eq. (3) in the introduction (we mention that some +qualitative arguments in this direction were already present in [46]). The relation between Rényi +entanglement entropy and charge fluctuations is a general aspect of quadratic theories (not only +free fermions, as our approach could also be generalized to free bosons), as in such theories, the +branch-point twist field can be written as a product of U(1) twist fields, which are then associated, +by the BFT, to the large-deviation theory of U(1) charge fluctuations. +Second, the methods we have developed show that the notion of quasi-particle used in inte- +grable systems to explain the behaviour of entanglement, should in fact be replaced by that of +hydrodynamic mode. Indeed, the BFT, which describes the exponential behaviour of twist field +correlation functions, is purely based on the Euler hydrodynamic data of the microscopic model. +In free fermion models, and in integrable models, it turns out that hydrodynamic modes are in +one-to-one correspondence with quasi-particles (see e.g. the review [47]); and in particular, the +single-mode twist fields we have introduced, are twist fields associated with such hydrodynamic +modes. But beyond these situations, hydrodynamic modes are the more general objects at play in +the large-scale dynamics of many-body systems. +Third, our calculations explain why it is possible to specify the growth and saturation of en- +tanglement after quenches from pair-production states in a simple way in terms of the long-time +GGE. This is not based on the conventional physical picture of entanglement produced by entan- +gled pairs of particles. But rather, it is based on the study of long-range correlations that such +pairs give rise to after quenches. It has not been appreciated until now that quenches give rise to +long-range spatial correlations , of the type found recently in non-equilibrium, long-wavelength +states [35, 38]. These long-range correlations are generically seen by observables supported on +regions of space that are large enough; specifically, with a ballistic scaling of the region’s length +x with respect to the time t since the quench. Thus, for such observables, the state is not a GGE. +This is important, as twist fields are semi-local with respect to the fermions, thus the semi-locality +branch is affected by such correlations. +The fact that the long-time GGE can be used to describe not only the saturation of the Rényi +entanglement entropies but also their growth in a simple way, is because of the particular structure +of long-range correlations in integrable pair-production quenches. Indeed, in such quenches, for +every choice of x/t, one can always choose a path in space-time which avoids all long-range cor- +relations. This follows from a simple geometric analysis of trajectories in space-time. Intuitively, +long-range correlations occur between positions in space-time where pairs of correlated particles +lie, a picture that we fully support by simple calculations of correlation functions of conserved den- +sities and currents and their asymptotic behaviours. Once the path avoids long-range correlation, +30 + +SciPost Physics +Submission +it only perceives the long-time GGE. +Note that it is obvious that the entanglement growth can be described purely in terms of the +final GGE (with no further information from the initial state needed), because of the one-to-one +correspondence between initial squeezed state (pair-production state) and GGEs, see Eq. (77) +(so no more information about the initial state is present at all). The structure of long-range +correlations however allow a simple description, in terms of fluctuations within the long-time +GGE. In general, the one-to-one correspondence is lost in quenches from more complicated states, +and the universality of the entanglement growth is also lost [32,33]. +Importantly, we find that the branch-point twist field can be decomposed into tensor factors – +the single-mode twist fields – that act on each small momentum interval. Each small momentum in- +terval is associated with a family of quasi-local operators, with respect to which branch-point twist +fields can be defined12. Each such twist field is only semi-local with respect to the fermions per- +taining to the single momentum interval, and, pairing intervals of opposite momenta, allowed us +to separate the two-point function of branch-point twist field (used to evaluate the Rényi entangle- +ment entropy) into a product of contributions on different momentum intervals. For each factor, +the semi-locality branch can be chosen in order to avoid long-range correlations corresponding to +the pair it perceives. We provided extensive calculations of correlation functions of single-mode +densities and currents that support this physical picture. +From these considerations we fully reproduced the long space-time dynamics of the Rényi +entanglement entropy that had been obtained by pair-entanglement argument. +Many extensions of our work are possible. Most importantly, we believe the derivation we +have provided, and many of the conclusions, can be extended to interacting integrable models, +and potentially to interacting non-integrable models. +In interacting integrable models, it would be interesting to reproduce, and provide a better +understanding of, the recent result [22]. This was obtained using crossing symmetry of relativistic +quantum field theory, in order to relate Rényi entanglement entropy growth in time to the linear +scaling of Rényi entanglement entropy in space, much like one can evaluate currents by crossing +from conserved densities [43]. However, as we have argued, the more general understanding of +time-extensive behaviours is via hydrodynamic modes. Thus, it is likely that hydrodynamic ideas +will provide a more first-principle derivation. +Technically, in interacting models, it is not possible to factorise the branch-point twist fields, +in such a simple way as in [7], into U(1) twist fields, a trick that we have used here. We believe +this difficulty can be surmounted as follows. First, it is still possible to diagonalise the twist field +action, at the price of making the resulting S-matrix non-diagonal, see App. C. The branch-point +twist field is then associated with a symmetry that has diagonal action on the new asymptotic +particles, and hydrodynamic modes can be constructed from these particles (by constructing the +corresponding nested thermodynamic Bethe ansatz). Having twist fields associated to charges +that are diagonal in the particle basis, the results of the BFT for generic intergable models can in +principle be applied. +It is also immediate that single-mode twist fields exist as well in interacting integrable models, +which would be interesting to study, independently form their applications to entanglement. +Another avenue is to use the ballistic macroscopic fluctuation (BMFT) theory developed re- +cently [35]. This is a more general construction which does not involve a flow on GGEs (by +constrast to the BFT). This would allow us to apply the principles introduced here – the relation +12In our calculation, we used the explicit free-fermion basis to first write the branch-point twist fields in terms of +U(1) twist fields by the standard arguments of [7], which we then factorised into single-mode twist fields. But the +concept remains valid without the mapping to U(1) twist fields. +31 + +SciPost Physics +Submission +between hydrodynamic fluctuations and entanglement entropy using twist fields – beyond homo- +geneous quantum quenches, and beyond the simple particle-pair quenches, as the BMFT is appli- +cable to inhomogeneous situations and for generic long-rance correlation structures. In particular, +it would be interesting to account for the long-range correlations not by choosing paths that avoid +them, but by evaluating directly their influence on the fluctuations. This would be important, as +initial states that are inhomogeneous generically produce long-range correlations [35, 38] that +are not necessarily of particle-pair type. This could also access quenches where multiple-particle +processes are involved [32,33]. +Beyond integrability, perhaps the main results are those based on a notion of “surface tension" +underlying entanglement entropy in chaotic systems [48]. It would be interesting to apply our +methods, based on hydrodynamics, to such situations. +A natural further extension is to introduce the effects of diffusion, in particular using the exact +results in integrable models [49], and potentially the effects of dispersion [50]. +Finally, there is no reason to restrict ourselves to entanglement entropy, or to quantum systems. +The methods we have introduced here are immediately applicable, for instance, to the large- +deviation theory of U(1) densities and currents after quenches in interacting models, both at +and away from integrability, and both for quantum and classical systems. This would be a very +interesting direction to investigate. +Acknowledgements +We thank Vincenzo Alba and Olalla Castro-Alvaredo for discussions and collaborations on closely +related subject. +Funding information +The work of BD was supported by the Engineering and Physical Sciences +Research Council (EPSRC) under grants EP/W000458/1 and EP/W010194/1. +A +Remarks on notions of locality and twist fields +There is a lot more that one can say about the twist fields we introduced in Sec. 2.4, as well as the +notions of locality we briefly discussed. Here we collect a number of remarks in order to provide +a brief and rough guide to this wide subject. +A.1 +Unbounded observables and topological charges +As is made clear from the height field definition of twist fields, a twist field may exist as soon +as there is an observable, say ϕ(x, t), that is “unbounded": that takes values in a non-compact +space which are not bounded by the dynamics. When this happens, the field can grow from a +value it has at x, t to another well separated value at x′, t′. If this growth is linear, then from +this we can construct a twist field as done in Sec. 2.4, with a large-deviation theory described by +the BFT. This usually happens when there is a non-compact symmetry13, such as the � symmetry +group of the sine-Gordon model if the sine-Gordon field is taken in �, or the � symmetry group +13This is essentially a “gauge symmetry": non-compactness is the main aspect of gauge invariance that makes is +different from oridnary symmetries. +32 + +SciPost Physics +Submission +of the real free massless Boson. In such cases, ϕ(∞) − ϕ(−∞) is a “topological charge", it is +an extensive conserved quantity with density q(x, t) = ∂xϕ(x, t). The vertex operators e−iηϕ(x,t) +are the associated twist fields, and the extensive charge ϕ(∞)−ϕ(−∞) should be considered as +part of the space of extensive charges Qi used to construct GGEs. It will appear after appropriate +quenches via (generalised) thermalisation. +A.2 +Descendant twist fields and semilocality sectors +The exchange relations (31) are a good way of characterising twist fields. However, they char- +acterise not a single field, but a family of fields. Indeed, clearly, the identification (34) is not the +unique choice satisfying (31). For instance, +T(x, t) = a(x, t)T−iη(x, t) +will also work, for any local observable a(x, t). The choice (34) may be seen as a “highest-weight" +twist field, and the above are usually referred to as “descendant twist fields" (these notions make +full sense, for instance, in quantum or conformal field theory, using the concept of dimension). All +such descendants are in the same “semilocality sector" T defined by the exchange relation (31). +One application of the BFT to descendant twist fields is explained in [41] in the context of the XX +quantum chain. +A.3 +Non-abelian semilocality +Given two “local enough" symmetry transformations σ and σ′, that is a(x, t) �→ σa(x, t) and +a(x, t) �→ σ′a(x, t), we can define two semilocality sectors Tσ and Tσ′. In general, if the transfor- +mations do not commute, one has that if T ′ ∈ Tσ′, then σT ′ ∈ Tσ◦σ′◦σ−1. Further, if T ∈ Tσ and +T ′ ∈ Tσ′, then the exchange relation takes the form +T ′(x, t)T(y, t) = +� +T(y, t)(σ−1T ′)(x, t) +(y ≪ x) +(σ′T)(y, t) T ′(x, t) +(y ≫ x). +(116) +A.4 +Twist fields in the literature +It is difficult to give a full account of the literature on twist fields. As a guidance we mention that +twist fields and their semilocality have been discussed extensively in various contexts, including: +phase transitions in classical and quantum statistical models [51–53] (see the review [54]); vertex +operators, Yangians, parafermions and orbifolds in conformal and integrable quantum field the- +ory [55–59]; tau-functions and Painlevé equations [60–67]; and entanglement entropy in quan- +tum field theory and in quantum spin chains [7,23,68,69]. Twist fields have also been considered +in higher dimensions [70]. In most works, the focus is on ultra-local “internal" symmetries, that +strictly factorise in space, usually part of a symmetry group such as �n, U(1), SU(n), permutations, +etc. Note that for ultra-local symmetries, large inequalities ≪, ≫ can be replaced by ordinary in- +equalities <, > in (31) for finite distances between the supports of the observables involved. This +is the usual way of writing the exchange relations. More recently, twist fields associated to space- +time boost transformations in QFT, that are not ultra-local, have been considered [71]; this is an +example where the hamiltonian density is not preserved by the transformation, ˜h(x, t) ̸= h(x, t). +33 + +SciPost Physics +Submission +A.5 +Concepts of locality in the literature +The concept of “locality" has been discussed widely in the literature, under a variety of definitions. +In relativistic QFT, local fields are those that commute at space-like distances with the energy- +momentum tensor. It is important to remark that under this general definition, local fields include +twist fields associated to internal symmetries. This definition can in fact be used in any quantum +model, be it a field theory, spin chain or model of interacting particles. In fact, one defines “locality +sectors" containing families of local fields that commute with each other at space-like distances; +and a distinguished locality sector is that containing the energy density. These considerations are +at the basis of orbifold conformal field theory [55]. +In spin chains, the most naïve concept is that of operators supported on finitely many sites. In +the C∗ algebra formulation, this is completed with respect to the operator norm [36]; importantly, +the property of operators commuting in the limit of large separations is preserved by this comple- +tion. Part of these C∗ algebra elements are the “quasi-local operators" that have been introduced +in order to describe generalised thermalisation in integrable models [45]. These are elements +of the C∗ algebra for which one can still define a finite support, but only up to corrections of +exponentially decaying norm. +But much like in QFT, twist fields, which are semilocal with respect to generic observables +but may be local with respect to some family of observables including the energy density, can +also be adjoined to the C∗ algebra, as is natural to do for instance in the context of Jordan- +Wigner transformations [72]. Then, either from the C∗ algebra, or from some potentially smaller +algebra of observables deemed local (for instance with appropriate decay of correlation functions, +and which, again, may include twist fields), other completions are possible, and sometimes more +physically relevant. For instance, the Gelfand-Naimark-Segal Hilbert space with respect to a given +state, and its space of bounded operators, both are usually larger than the C∗ algebra. Another +Hilbert space completion is that based on susceptibilities [37]. This gives the concept of “extensive" +quantities, a generalisation of quantities written as sums over space of local operators (or “local +densities"). These form a Hilbert space, a priori without any algebraic structure. In particular, +it has been shown [37] that extensive conserved quantities are in bijection with “pseudolocal +charges", roughly defined by their extensivity property 〈Q2〉c ∝ L in a system of length L [45]. +Extensive conserved charges form a complete set which has been rigorously shown to fully describe +the linearised Euler hydrodynamics [40], and they may be used to more formally and precisely +construct GGEs (addressing convergence issues) [37]. +Despite all these studies, the full relation between extensive conserved quantities, twist fields, +local symmetry transformations and GGEs is still not fully unexplored. +B +Correlations after a quench from an initial state with pair structure +In this appendix we provide supporting argument for the choice of the integration path for eval- +uating the SCGF made in the main text. We recall that the choice of path is dictated by two main +ideas: +• The production of pairs of quasi-particles by the initial state by the after-quench dynamics +gives rise to long-range correlations: points reached by a pair of quasi-particles with oppo- +site momenta are correlated. These quasi-particles have the interpretation of fluid modes, +and such long-range hydrodynamic correlations are akin to those seen in the Ballistic Macro- +scopic Fluctuation Theory (BMFT) [35,38]. +34 + +SciPost Physics +Submission +• BFT [24] is applicable only when multi-point correlation functions of the densities and cur- +rents, the integrand in (12), cluster fast enough along the chosen contour. Otherwise, de- +pending on the structure of such correlations, the SCGF and the cumulants may be divergent +under ballistic scaling, or the BFT result may receive additional contributions from these cor- +relations which have to be taken into account. The more general BMFT [35] in principle +provides the corrections. However, it is simpler to directly apply the BFT by choosing con- +tours that avoid such correlations, leading to the paths shown in Fig. 2. +Below, we explicitly show the presence/absence of such long-range correlations along the +different paths considered in the main text. +We use the notations a(x, t) and b(x, t) for free fermionic fields with the usual normalisation, +e.g. +{b†(x), b(y)} = δ(x − y), +{b† +θ, bα} = δ(θ − α), +b(x, t) = +� +dθ +� +2π +eixθ bθ(t) +(117) +where bθ(t) = e−iE(θ)t bθ. The global U(1) charge is +Q = +� +dθ b† +θ bθ. +(118) +Note that in the main text a variety of canonical free fermion fields were defined: the original +fields ψ(x, t), the replicated ones parametrised by a copy index ψi(x, t), and the fields obtained +from these by diagonalising in copy space, ψp(x, t). These all are canonical free fermion fields +(independent from each other for different copy number i, or for different diagonalised copy +number p). The calculations below therefore apply to any such choice of fields. +As we discuss quenches, in this appendix we use two consecutive letters for the canonical free +fermion fields: a(x, t) (for the pre-quench fermion) and b(x, t) (for the post-quench fermion). +B.1 +Global U(1) densities and currents and decay of correlations in GGEs +First, recall the definition of the generalised current as a line integral (12) +∆J(γ) = +� +ℓ +(j(x, t)dt − q(x, t)d x) +(119) +where j(x, t) and q(x, t) are the current and the density associated the U(1) conserved charge Q +in (118). We recall also that the above integral only depends upon the end points of the path ℓ +due to the conservation law relating current and density, ∂tq(x, t) + ∂x j(x, t) = 0. In free models +with global U(1) symmetry the fermionic Hamiltonian is of the form +H = +� +dθ E(θ)b† +θ bθ +(120) +where E(θ) is the dispersion relation (recall that E(θ) = E(−θ) is a strictly convex function). The +local charge density is given by +q(x, t) = b†(x, t)b(x, t) +(121) +and it can be easily verified by using the fermionic algebra that Q = +� +d x q(x, t) is conserved, +[Q, H] = 0. Let us find the associated local current. +35 + +SciPost Physics +Submission +By the conservation law we have (we suppress the time dependence as all fields are the same +time t) +∂x j(x) = −∂tq(x) = i[qx, H] = i +� +dθ E(θ) +� +b†(x)b(x), b† +θ bθ +� += i +� +dθ dk +� +2π +dk′ +� +2π +E(θ)e−ix(k−k′) � +b† +kbk′, b† +θ bθ +� += i +� +dθ dk +2π E(θ) +� +e−ix(k−θ)b† +kbθ − e−ix(θ−k)b† +θ bk +� += i +� +dθ dk +2πeix(k−θ) (E(k) − E(θ)) b† +θ bk. +(122) +Integrating with respect to x we find +j(x, t) = 1 +2π +� +dθdk eix(k−θ) +� E(k) − E(θ) +k − θ +� +b† +θ(t)bk(t) +(123) +where the x-independent integration constant is chosen in such a way that the result is a local +observable14. This is the known expression for the current in the case of a quadratic dispersion +relation in the continuum. Actually, restricting the integration over momenta in [−π,π] and +taking E(k) = cos(k) one reproduces also the current on the lattice; but here we keep θ, k ∈ � for +simplicity. +We now show that in a GGE, the connected correlation functions of densities decay fast enough +in space, and the correlation functions of currents decay fast enough in time, in such a way that +scaled cumulants are finite, thus making the BFT applicable. The former in fact is valid for all +local observables, while the latter only hold for the currents. +For simplicity, here and in the following subsections, we will concentrate solely on two-point +correlation functions – although all higher-point functions (and their respective cumulants) should +in principle be investigated similarly. +Let 〈·〉 be a GGE. Let us assume that the occupation function n(θ) characterising the GGE is +analytic in a neighbourhod of �. Using +〈b† +θ bθ ′〉 = δ(θ − θ ′)n(θ) +(124) +we have, on the one hand, +〈b†(x)b(0)〉 = +� +dθ +2π e−ixθ n(θ) +. +(125) +For x > 0 (resp. x < 0), contour deformation can be performed as θ �→ θ −iγ (resp. θ �→ θ +iγ) for +γ > 0 small enough, and we see that the resulting integral decays exponentially as |x| → ∞. This +implies exponential decay of all two-point connected correlation functions of local observables +formed out of sums of products of b(x), b†(x) and their derivatives, including U(1) densities. It +14This in fact fixes the result up to an overall term proportional to the identity operator 1; indeed there are no +x-independent homogeneous local operators, whose space-time translations are generated by the momentum and +Hamiltonian, other than 1. +36 + +SciPost Physics +Submission +also implies linear scaling of cumulants; for instance this would mean +� X +0 +d x +� X +0 +d x′ 〈b†(x)b(x′)〉 += +� X +0 +dx +� X +0 +d x′ +� +dθ +2π e−i(x−x′)θ n(θ) +∼ +� X +0 +dx +� X +0 +d x′e−γ|x−x′| ∼ X +(126) +where in the last line we have shifted θ �→ θ −isign(x − x′)γ and used sign(x)x = |x|. This is the +correct ballistic growth of the cumulant. +On the other hand, we find +〈b†(0, t)b(0,0)〉 = +� +dθ +2π eitE(θ)n(θ). +(127) +This has a stationary phase at θ∗ such that E′(θ∗) = 0; this point is unique by our assumption +of strict convexity (and θ∗ = 0 by symmetry, although we don’t make use of this fact in this +calculation), so a saddle point analysis gives +〈b†(0, t)b(0,0)〉 ∼ +� +i eitE(θ∗) n(θ∗) +� +2πt +. +(128) +Therefore, correlation functions of generic local observables o(x, t), o′(x, t) formed out of bilin- +ears of creation and annihilation operators have algebraic decay +〈o(0, t)o′(0,0)〉c = O +�1 +t +� +(t → ∞). +(129) +For such decay, cumulants of total time integrals do not grow linearly, +〈 +� T +0 +dt o(0, t) +� T +0 +dt′ o′(0, t′)〉 +c +≫ T +(T → ∞) +(130) +thus breaking the large-deviation principle at the basis of the BFT. However, an important re- +mark is that this generic behaviour of fermion bilinears does not hold in the case of currents, +o(x, t) = o′(x, t) = j(x, t). Indeed, using (123) with x = 0, we see that we must set θ = k = θ∗ +for the long-time limit of the current two-point function. From +E(k) − E(θ) +k − θ += E′(k) + O(k − θ) +(131) +we realise that E(k)−E(θ) +k−θ +�� +k=θ=θ∗ = 0. Therefore, the current two-point function decays faster than +1/t; in fact it decays as +〈j(0, t)j(0,0)〉c = O +� 1 +t3 +� +(t → ∞). +(132) +This guarantees the correct scaling of cumulants +〈 +� T +0 +dt j(0, t) +� T +0 +dt′ j(0, t′)〉c = O(T) +(T → ∞) +(133) +37 + +SciPost Physics +Submission +and thus the validity of the BFT. +A similar argument shows that the current perpendicular to the path ℓ – the integrand in (119) +– has a similar property along the path, thus guaranteeing that the clustering requirement (21) +holds. +B.2 +Quench protocol and initial state +In order to describe the quench protocol considered in the main text (for which we obtain predic- +tions on the dynamics of the entanglement entropy) we define the pre-quench and the post-quench +fermionic Hamiltonians respectively as +H0 = +� +dθ E0(θ)a† +θ aθ +(134) +H = +� +dθ E(θ)b† +θ bθ +(135) +where again the fermions satisfy {aθ, a† +θ ′} = δ(θ − θ ′), {bθ, b† +θ ′} = δ(θ − θ ′). According to +the protocol, the system is initialized in the ground state of H0 and then let evolve with H. This +corresponds to changing the whole dispersion relation (not only a parameter as in typical quenches +in the literature), see e.g. [44,73]. +The two set of fermions are related by a Bogolioubov-type transformation in the following way +� +aθ +a† +−θ +� += +� +fθ +gθ +g∗ +−θ +f ∗ +−θ +�� +bθ +b† +−θ +� +. +(136) +Imposing the validity of anticommutation relations one gets the following constraints on the func- +tions fθ, gθ +fθ g−θ + f−θ gθ = 0 +(137a) +|fθ|2 + |gθ|2 = 1. +(137b) +Note that the first of these is identically satisfied choosing fθ = f−θ and gθ = −g−θ or viceversa. +In our analysis we will keep these functions general. +The initial state |Ψ〉 is defined as +aθ |Ψ〉 = 0 +(138) +which, for instance, could be a filled Fermi sea so that operators aθ are to be interpreted as creating +excitations on top of these. It can be easily shown that in terms of post-quenches quantities this +is described by the following squeezed state +|Ψ〉 = 1 +N exp +� +1 +2 +� +dθ Kθ,−θ b† +θ b† +−θ +� +|0〉 +(139) +where |0〉 is the ground state of the post-quench Hamiltonian satisfying bθ |0〉 = 0 (this state has +the nice property of being gaussian so that Wick’s theorem applies). The function Kθ,θ ′ = −Kθ ′,θ +can be related directly to the functions fθ and gθ appearing in (136) using the fact that |Ψ〉 is +annihilated by aθ. In terms of post-quench operators, this condition reads +(fθ bθ + gθ b† +θ)exp +� +1 +2 +� +dθ ′Kθ ′,−θ ′ b† +θ ′ b† +−θ ′ +� +|0〉 = 0. +(140) +38 + +SciPost Physics +Submission +Using the BCH formula eABe−A = e[A,◦]B we obtain +exp +� +−1 +2 +� +dθ ′Kθ ′,−θ ′ b† +θ ′ b† +−θ ′ +� +bθ exp +� +1 +2 +� +dθ ′Kθ ′,−θ ′ b† +θ ′ b† +−θ ′ +� += bθ + 1 +2Kθ,−θ b† +−θ − 1 +2K−θ,θ b† +−θ += bθ + Kθ,−θ b† +−θ +(141) +which used in (140) in combination with bθ |0〉 = 0 gives +[fθ(bθ + Kθ,−θ b† +−θ) + gθ b† +−θ]|0〉 = 0 +(142) +so that the condition is +Kθ,−θ = − gθ +fθ +. +(143) +Note in particular that we must have, by the anti-symmetry Kθ,−θ = −K−θ,θ, +g0 = 0. +(144) +Finally, we may evaluate the predicted long-time GGE for the quench simply by evaluating the +post-quench conserved quantities in the pre-quench vacuum state |Ψ〉. Inverting (136), we write +bθ = +f ∗ +−θ aθ − gθ a† +−θ +fθ f ∗ +−θ − gθ g∗ +−θ +. +(145) +For this computation, it will be convenient to look at the values of the extensive conserved quanti- +ties; hence we take a finite system of length L. This is warranted, as the quench is homogeneous. +The above description of the quench stays valid with the discretisation θ ∈ �2π/L, and with the +usual canonical anti-commutation relations with regularisation δ(θ −θ ′) → δθ,θ ′ L +2π. We consider +b† +θ bθ, and obtain, after some algebra using Eqs. (137), +b† +θ bθ = a† +θ aθ|f−θ|2 + a−θ a† +−θ|gθ|2 . +(146) +Using 〈Ψ|a† +θ aθ|Ψ〉 = 0 and 〈Ψ|a−θ a† +−θ|Ψ〉 = +L +2π, we get +〈Ψ|b† +θ bθ|Ψ〉 = L +2π|gθ|2. +(147) +But also, in a GGE with density matrix ρw, see Sec. 2.5, we have 〈b† +θ bθ〉 = +L +2πn(θ), thus we identify +n(θ) = |gθ|2. +(148) +Again using Eqs. (137), we obtain +1 +|fθ|2 = +1 +|gθ|2 +|g−θ|2 +1 − |g−θ|2 +(149) +and therefore, from (37) and (148), +|Kθ,−θ|2 = |K−θ,θ|2 = |g−θ|2 +|f−θ|2 = +|gθ|2 +1 − |gθ|2 = e−w(θ). +(150) +39 + +SciPost Physics +Submission +Note that in our analysis of GGEs, we assume that n(θ), thus |g(θ)|2, has an analytic extension +to a neighbourhood of �. This analyticity property is not true of the function w(θ), which must +have a singularity (e.g. logarithmic) at θ = 0 because of Eq. (144). We also assume that n(θ) → 0 +as |θ| → 0, and thus this must also be true for g(θ). +Below we report for completeness all the relevant elementary correlation functions of fermionic +operators after the quench, and their symmetries. Those will be used in the following subsections, +where evaluating current and density correlations, which are bilinears in the fermions (so appli- +cation of Wick theorem requires only the knowledge of those). In real space we define +Gb†b +x y (t,s) = 〈Ψ| b†(x, t)b(y,s)|Ψ〉 +Gb†b† +x y (t,s) = 〈Ψ| b†(x, t)b†(y,s)|Ψ〉. +(151) +and similarly for their hermitian conjugates. Going to momentum space, these take the form +Gb†b +θθ ′(t,s) = 〈Ψ| b† +θ(t)bθ ′(s)|Ψ〉 = eiE(θ)(t−s)|gθ|2δ(θ − θ ′) +(152) +Gb†b† +θθ ′ (t,s) = 〈Ψ| b† +θ(t)b† +θ ′(s)|Ψ〉 = −eiE(θ)(t+s) fθ g∗ +θδ(θ + θ ′) +(153) +Gbb +θθ ′(t,s) = 〈Ψ| bθ(t)bθ ′(s)|Ψ〉 = −e−iE(θ)(t+s)gθ f ∗ +θ δ(θ + θ ′) +(154) +Gbb† +θθ ′(t,s) = 〈Ψ| bθ(t)b† +θ ′(s)|Ψ〉 = e−iE(θ)(t−s)|fθ|2δ(θ − θ ′) +(155) +where in particular |g(θ)|2 = n(θ) = 1 − |f (θ)|2. Note the following symmetries +Gb†b +θθ ′(t,s) = δ(θ − θ ′)eiE(θ)(t−s) − Gbb† +θ ′θ (s, t) +(156) +(Gb†b +x y (t,s))∗ = Gb†b +y x (s, t) +(157) +Gb†b +x y (t,s) = +� +dθ +2π e−iθ(x−y)+iE(θ)(t−s) − Gbb† +y x (s, t) +(158) +so that at equal times +Gb†b +x y (t, t) = δx y − Gbb† +y x (t, t) = δx y − (Gbb† +x y (t, t))∗ +(159) +and also +(Gb†b† +θθ ′ (t,s))∗ = Gbb +θ ′θ(s, t), +(Gb†b† +x y (t,s))∗ = Gbb +y x(s, t). +(160) +B.3 +Approach to the GGE +In the previous Sec. we have evaluated the GGE ρw corresponding to the initial state |Ψ〉 simply +by evaluating the averages of the mode occupation. Here we analyse a bit more in detail how the +GGE is approached in time. +We first note that +〈Ψ| b† +θ bθ ′ |Ψ〉 = 〈b† +θ bθ ′〉ρw = δ(θ − θ ′)n(θ). +(161) +Thus, by Wick’s theorem, the only difference between averages in |Ψ〉 and in 〈·〉ρw come from the +contraction +〈Ψ| bθ bθ ′ |Ψ〉 +(162) +and its complex conjugate. Thus we evaluate 〈Ψ| b(x, t)b(x′, t′)|Ψ〉 in three main situations that +are important for our analysis: t = t′, x ̸= x′ (for the cumulants of space-integrated conserved +40 + +SciPost Physics +Submission +densities), and x = x′, t ̸= t′ (for the cumulants of time-integrated currents) and x ̸= x′, t ̸= t′ +(for analysing the correlation between the spatially separated time-integrated currents). +In the first case, we have, using (145),(137), and the definition in (151) +Gbb +x x′(t, t) = − +� +dθ +2π ei(x−x′)θ−2itE(θ) f ∗ +−θ g−θ. +(163) +Consider t → ∞ with x, x′ fixed. Then there is a stationary phase at θ∗ : E′(θ∗) = 0, with a +resulting integral ∝ +1 +�t . Thus, this decays as t → ∞: for every two-point functions on intervals +that stay finite, the GGE is approached. We notice that as g−θ∗ = 0 (Eq. (144)), for fermion two- +point functions, the approach is proportional to 1/t3/2 instead of 1/�t; and for multilinears of +fermions, the approach is faster. +But we are interested in the scaling x, x′, t ∝ ℓ → ∞, (x − x′)/t = ξ, for which the expo- +nential has a stationary phase at θ∗ = θ∗(ξ) : E′(θ∗) = ξ/2, with a resulting integral ∝ 1/ +� +ℓ. +In charge-neutral fermion bilinears, such as those involved in densities and currents, two such +contractions will be multiplied with each other. Thus we have, for instance, +〈Ψ|q(x, t)q(x′, t)|Ψ〉c = 〈q(x, t)q(x′, t)〉c +ρw + C(ξ)(tℓ)−1 + O(ℓ−2), +(164) +thus the correction is O(1/ℓ). Then, for the cumulant we have +〈Ψ| +� ℓX +0 +d x q(x,ℓt) +� ℓX +0 +d x′ q(x′,ℓt)|Ψ〉c += +ℓ2 〈Ψ| +� X +0 +d x q(ℓx,ℓt) +� X +0 +d x′ q(ℓx′,ℓt)|Ψ〉c +∼ +〈 +� ℓX +0 +d x q(x,ℓt) +� ℓX +0 +d x′ q(x′,ℓt)〉c +ρw + O(ℓ) +where the correction O(ℓ) is ℓ +� X/t +−X/t dξ(X −2ξt)C(ξ). Therefore, the correction due to the quench +changes the linearly scaling part of the cumulant, hence modifies the scaled cumulant from its GGE +value (recall that the scaled cumulant is obtained by dividing by ℓX, and taking the large ℓ limit). +Here it would be possible to evaluate explicitly this modification, however it is not necessary for +our calculation. The modification due to the quench comes from pair productions – this will be +made much clearer when we study the single-mode densities and currents below. +In fact, there is one limit where it is useful to evaluate this correction term: the limit X/t → 0 +of ℓX-scaled spatially-integrated densities as above. The result for the correction is explicitly +lim +X/t→0 +t +X +� X/t +−X/t +dξ +�X +t − 2ξ +� +C(ξ) = 0 +(165) +as C(ξ) is bounded. Thus, in this limit we recover the GGE result. This is in agrement with taking +first the long-time limit of the finite-interval cumulant, then the limit of the scaled cumulant on a +long interval (this means that the limit X/t → 0 is in fact uniform in t). +In the second case, where we can set x = x′ = 0, we find, with E′(θ∗) = 0 and a saddle point +analysis (again remeber the definitions in (151)), +Gbb +00 (t, t′) = − +� +dθ +2π ei(t+t′)E(θ) f ∗ +−θ g−θ ∼ +� +i ei(t+t′)E(θ∗) f ∗ +−θ∗ g−θ∗ +� +2π(t + t′) +. +(166) +41 + +SciPost Physics +Submission +As E(θ) is symmetric, this is θ∗ = 0, and then, by Eq. (144), the result vanishes. Therefore, +Gbb +00 (t, t′) = O +� +1 +(t + t′)3/2 +� +. +(167) +Hence, the corrections to cumulants of charge-neutral bilinears involve +� T +1 +dt +� T +1 +dt′ +1 +(t + t′)2 = O +� 1 +T 3 +� +≪ T +(T → ∞) +(168) +(where the lower boundary does not matter for the large-T analysis). This correction is sublinear, +therefore the quench does not affect cumulants of equal-position time-integrated quantities: for +these, the GGE is reached quickly enough. The lack of modification due to the quench comes from +the lack of pairs of particles produced at equal (zero) momenta, due to the fermionic statistics. +We remark that if there were particles created at zero momenta (for instance, for bosonic +systems), then, still by a calculation similar to that of Eqs. (131)-(133), the correction due to the +quench would vanish for cumulants of total currents, which are in any case the objects of interest. +Therefore, the fact that pairs of particles of zero momenta are not produced, is not an essential +aspects of our calculation. +Finally, we may also analyse time-integrated currents at two different points in a similar way +as above, finding: +〈Ψ| +� ℓT +0 +dt j(ℓx, t) +� ℓT +0 +dt′ j(ℓx′, t′)|Ψ〉c +∼ +〈 +� ℓT +0 +dt j(ℓx, t) +� ℓT +0 +dt′ j(ℓx′, t′)〉c +ρw + O(ℓ) . +This case is necessary for the discussion in Sec. 4.5. With ξ = (x − x′)/(t + t′), the saddle point +leading to the O(ℓ) correction is at θ∗ : E′(θ∗) = ξ. Thus, the correction due to the quench again +changes the linearly scaling part of the two-point cumulant. Here, the limit ξ → ∞ is interesting, +and easy to evaluate: as ξ → ∞, the saddle point will be at θ∗ → ∞, and we only have to use +the fact that gθ → 0 as |θ| → ∞. Therefore, the correction vanishes as ξ → ∞, and we may use +the GGE result, where scaled cumulants of time-integrated currents become sums of cumulants at +x and at x′ in the GGE (which take the same values by translation invariance). +B.4 +Single-mode density and currents and decay of correlations in GGEs +In the main text, when studying the dynamics of the entanglement of an interval after a quench +(Sec. 4.5), we are interested in the single-mode twist fields. This requires constructing densi- +ties and currents not only for the global U(1) charge as done above, but also for the individual +conserved quantities b† +θ bθ. These conserved quantities are not extensive – they are not integrals +of local or quasi-local observables – however, as reviewed in [47] in the more general context of +integrable models, they form a scattering basis for such extensive quantities. Thus, integrations +over small θ-intervals give extensive conserved quantities. These are the single-mode conserved +quantities that we now investigate. +As we are working directly in the thermodynamic limit and, in the continuum of space, the +momenta fill the real axis [−∞,∞]. Let us write this as a union of disjoint intervals centered at +equispaced "target" momenta: ∪∞ +i=−∞Aθi where Aθ = [θ − ε/2,θ + ε/2) and θi = (i + 1/2)ε. We +can write the total charge as +Q = +� +d x b†(x)b(x) = +� +dθ b† +θ bθ = +∞ +� +i=−∞ +Qθi, +Qθ = +� +Aθ +dθ ′b† +θ ′ bθ ′. +(169) +42 + +SciPost Physics +Submission +Clearly each “regularised" (by ε) single-mode charge Qθ is conserved, [Qθ, H] = 0. It is also +extensive: in a GGE in a finite volume L, we have 〈Q2 +θ〉 +c ∝ L: +〈Q2 +θ〉 +c = +� θ+ε/2 +θ−ε/2 +dθ ′dθ ′′ δ(θ ′ − θ ′′)2n(θ ′)(1 − n(θ ′)) = L +2π +� θ+ε/2 +θ−ε/2 +dθ ′n(θ ′)(1 − n(θ ′)) . (170) +As mentioned, if we want to write a density in real space for each b† +θ bθ, we will get something +non local. However, Qθ’s have quasi-local densities. We seek a function fθ(x, y) such that +� +d xd y b†(x)b(y)fθ(x − y) = Qθ. +(171) +Going to Fourier space, one can show that (see (117)) +fθ(z) = +sin( εz +2 ) +πz +eiθz. +(172) +The corresponding regularised single-mode density, parametrised by the momentum, and one +choice of the density (the only hermitian and PT symmetric one), is given by +qθ(x, t) = +� +dz b†(x + z/2, t)b(x − z/2, t)fθ(z). +(173) +In terms of Fourier modes, this takes the form +qθ(x, t) = +� +dkdk′ +2π +eix(k′−k)ϑ +�ε +2 − +���k + k′ +2 +− θ +��� +� +b† +k(t)bk′(t). +(174) +As [Qθ, H] = 0, the density qθ(x, t) has an associated current satisfying a continuity equation and +by a calculation analogous to (123) one finds +jθ(x, t) = +� +dkdk′ +2π +eix(k′−k) +� E(k′) − E(k) +k′ − k +� +ϑ +�ε +2 − +���k + k′ +2 +− θ +��� +� +b† +k(t)bk′(t) +(175) +where ϑ(x) is the Heaviside theta function. This is basically the same as (123) with a restriction +on the q integration around the target mode θ. +For convenience, in fact we will consider the momentum-pair densities and currents +q|θ|(x, t) = qθ(x, t) + q−θ(x, t), +j|θ|(x, t) = jθ(x, t) + j−θ(x, t) +(176) +associated to a pair of opposite momenta; these are the ones used in the twist field decomposition +(101). +We now analyse the behavior of two-point functions on GGEs following the same lines of +B.1. This is because, again, the original BFT was defined for expectation values on GGEs and, +eventually, we would like to replace with those the expectations on the initial state before the +quench. Later, we will study more carefully the approach to the GGE. In a GGE (characterized by +Boltzmann factor w(k)), the only surviving elementary correlator is +〈b† +k(t)bk′(s)〉 = eiE(k)(t−s)δ(k − k′)n(k) +(177) +with n(k) = (1 + ew(k))−1. +43 + +SciPost Physics +Submission +We now consider the connected correlation functions of densities and currents along the paths +of interest and study under which condition they decay fast enough in such GGE. The two paths of +interest are the horizontal path (0, t) → (x, t), and the piecewise linear path (0, t) → (0,0) → (x,0) → (x, t) +(cf. Fig. 2 in the main text). For the first case, we need to evaluate density-density correlations at +equal time and different spatial points. In the second case, instead, we need to evaluate current- +current correlations both between the two different vertical segments and within the segments +themselves (using the same arguments of Sec. 4.3, one can show that the contribution of the +horizontal segment vanishes, so we do not need to look at density-currents correlations as well). +The calculation is slightly different than in the case of B.1 because we have to deal with pair- +mode densities and currents. The global quantities are determined directly by the correlators +〈b†(x, t)b(y,s)〉, while here we have to analyse directly the density-density or current-current +correlations. +Let us start by considering the connected density-density correlation function on the horizontal +path. We focus on single-mode quantities, the pair-mode ones being just linear combination of +those, i.e., +〈q|θ|(x, t)q|θ|(0, t)〉c = 〈qθ(x, t)qθ(0, t)〉c + 〈qθ(x, t)q−θ(0, t)〉c ++ 〈q−θ(x, t)qθ(0, t)〉c + 〈q−θ(x, t)q−θ(0, t)〉c +(178) +Below, in the current subsection, all expectations 〈·〉 are on the GGE. Let η = ±1, we have +〈qθ(x, t)qηθ(x′, t)〉c += +� +dkdk′ +2π +dqdq′ +2π +eix(k′−k)ϑ +�ε +2 − +���k + k′ +2 +− θ +��� +� +× eix′(q′−q)ϑ +� +ε +2 − +���� +q + q′ +2 +− ηθ +���� +� +〈b† +k(t)bk′(t)b† +q(t)bq′(t)〉 +c += +� +dkdk′ +2π +dqdq′ +2π +eix(k′−k)ϑ +�ε +2 − +���k + k′ +2 +− θ +��� +� +× eix′(q′−q)ϑ +�ε +2 − +���q + q′ +2 +− ηθ +��� +� +〈b† +k(t)bq′(t)〉〈bk′(t)b† +q(t)〉 += +� +dkdk′ +2π +ei(x−x′)(k′−k)ϑ +�ε +2 − +���k + k′ +2 +− θ +��� +� +ϑ +�ε +2 − +���k + k′ +2 +− ηθ +��� +� +(1 − n(k′))n(k) +. +(179) +Note that all four correlators needed can be deduced from the this upon exploiting θ �→ −θ +transformation. Apart from the overall step function which constrains one of the integrals, with +the assumptiong on n(k) that we have (it is analytic on a neighbourhood of the real line, and +decays rapidly enough as |k| → ∞), we can change variables to k± = k′ ± k and perform the k− +integral in the complex plane by letting k− �→ k− −iγsign(x − x′) (and γ > 0) and we see that the +decay is exponential making cumulants scale as in (126). Therefore, BFT is applicable. +For the current-current correlator, we look at two generic points in space and time. However, +by making use of time-translational invariance of the GGE, we can set one time to zero. Analogous +manipulations give (we focus on x > 0) +〈jθ(x, t)jηθ(0,0)〉c = +� +dkdk′ +2π +eix(k′−k)+it(E(k)−E(k′))ϑ +�ε +2 − +���k + k′ +2 +− ηθ +��� +� +× ϑ +�ε +2 − +���k + k′ +2 +− θ +��� +�� E(k′) − E(k) +k′ − k +�2 +(1 − n(k′))n(k) +(180) +44 + +SciPost Physics +Submission +and we see that that in the ballistic scaling limit, with ζ = x/t fixed, the exponential has a saddle +point at ∂kE(k∗) = ∂k′E(k′ +∗) = ζ so that for ζ ̸= 0,+∞ (recall eq. (131) and the one below) +〈jθ(x, t)jηθ(0,s)〉c ∼ e−i(t−s)E(k∗(ζ)) +2π(t − s) +(∂kE(k∗(ζ)))2ϑ +�ε +2 − +���k∗(ζ) − θ +��� +� +ϑ +�ε +2 − +���k∗(ζ) − ηθ +��� +� +× (1 − n(k∗(ζ)))n(k∗(ζ)) +. +(181) +Let us take the case η = 1 : the two step functions square to one and the condition for it +to vanish is k∗(ζ) ≥ ε/2 + θ or k∗(ζ) ≤ −ε/2 + θ. Using ζ = ∂kE(k∗(ζ)) = v(k∗(ζ)) and the +monotonicity of the velocity, we can invert the above relation, leading to a vanishing step functions +whenever +ζ ≥ v(θ + ε/2) = v(θ) + O(ε) +or +ζ ≤ v(θ − ε/2) = v(θ) − O(ε) +(182) +For η = −1 we obtain the same condition because we are considering ζ ≥ 0. This means that, +under the condition (182), the leading term of correlation (181) (coming from the saddle point) +vanishes. +Correlations between the two vertical segments, on the contrary, arise only in a tiny cone (of +order ε) around the velocity v(θ). When those are present, the second cumulant resulting from +(181) grows as O(T log T) at large time T (to be contrasted with (133)). +Note that if we consider the special case t = s ( ζ = +∞), then (181) has no saddle point and +therefore its decay is exponentially fast. +In the above calculation we assumed x ̸= 0 (ζ ̸= 0), which corresponds at looking at corre- +lations between the two different aforementioned vertical paths (see again Fig. 2, main text). In +order to study the decay of current-current correlations along any of such vertical paths, instead, +we need to set equal space (and different times). Again, using translational invariance we just set +x = 0 in (180). In this case we can repeat the argument in (131) and below to show +〈j|θ|(0, t)j|θ|(0,0)〉c = O +� 1 +t3 +� +(t → ∞) +(183) +with the corresponding cumulant scaling linearly in time. +Finally, we recall again that the horizontal segment of the path (0, t) → (0,0) → (x,0) → (x, t) +does not contribute to the cumulants. This observation, together with (183), and when the con- +dition (182) is satisfied, allows to conclude that within a GGE: (i) the SCGF associated to the +two-point function of the pair-mode twist fields can be correctly evaluated along this path (i.e., +BFT is applicable); (ii) the total SCGF factorises into those associated to the two vertical cuts, +namely it is the sum of the two corresponding SCGFs (again we refer to Remark 3.3 in [24]), and, +in fact, in the main text, we make use of such factorization. +B.5 +Approach to the GGE +Finally in this subsection, following B.3, we study in more detail the approach in time to the +corresponding GGE values of the same single-mode densities and currents correlations considered +in B.4. In particular we want to understand how cumulants in the GGE’s are modified at large time +when taking into account correlations coming from the initial state |ψ〉 (cf. Eq. (139)). Below, +we denote by 〈·〉 expectation values on such initial state, while 〈·〉ρw the ones on the GGE atteined +at infinite time. +45 + +SciPost Physics +Submission +We start with the single-mode connected density-density (again the corresponding pair-mode +correlator is obtained via (178)). In momentum space we have (we focus on x > 0) +〈qθ(x, t)qηθ(0, t)〉c = +� +dkdk′ +2π +dqdq′ +2π +eix(k′−k)ϑ +� +ε +2 − +���� +k + k′ +2 +− θ +���� +� +ϑ +� +ε +2 − +���� +q + q′ +2 +− ηθ +���� +� +× 〈b† +k(t)bk′(t)b† +q(t)bq′(t)〉 +c += +� +dkdk′ +2π +dqdq′ +2π +eix(k′−k)ϑ +� +ε +2 − +���� +k + k′ +2 +− θ +���� +� +ϑ +� +ε +2 − +���� +q + q′ +2 +− ηθ +���� +� +× +� +〈b† +k(t)bq′(t)〉〈bk′(t)b† +q(t)〉 − 〈b† +k(t)b† +q(t)〉〈bk′(t)bq′(t)〉 +� +(184) +where η = ±1. The first piece is nothing but the GGE contribution and, as discussed in B.4 (see +Eq. (179) and below) is always well-behaved in the ballistic regime. +The other part gives +� +dkdk′ +(2π)2 eix(k′−k)+2iE(k)t−2iE(k′)tϑ +� +ε +2 − +���� +k + k′ +2 +− θ +���� +� +ϑ +� +ε +2 − +���� +k + k′ +2 +− ηθ +���� +� +(1 − n(k))n(k′) +(185) +In the ballistic limit, with ζ = x/t fixed, we can use again saddle point analysis, similarly to calcula- +tions in B.4. Now, however, the saddle points is given by 2v(k′)−ζ = 0 and 2v(k) = ζ (note the fac- +tor 2 of difference wrt the saddle point of the integral (180)), namely k∗(ζ) = k′∗(ζ) = v−1(ζ/2). +After application of saddle-point method, the ϑ function is evaluated at these points. For η = 1, +using ϑ2 = ϑ, the result of the saddle point of (185) vanishes unless +��� k∗(ζ)+k′∗(ζ) +2 +− θ +��� ≤ ε +2 or, +equivalently, |v−1(ζ/2) − θ| ≤ ε/2 that it precisely +v(θ − ε/2) ≤ ζ/2 ≤ v(θ + ε/2) . +(186) +Having assumed x>0, for η = −1 we recover the very same condition. +This is exactly the expected condition for two particles of opposite momentum ±θ, initially +forming an entangled pair, not to hit both the segment [0, x] at time t (as be easily understood +geometrically from Fig. 2 of the main text). Note that, for ζ fixed (and due to the above mentioned +factor 2), this condition is more severe then Eq. (182), so it also guarantees a fast enough decay +of same correlation within the GGE. +When condition (186) does not hold, from the saddle point contribution, we get that the inte- +gral (185) decays as t−1, giving a slow approach to the GGE. This, in fact, modifies the behaviour +of the second cumulant because it gives a correction to the GGE value which is not subleading (it +is actually faster than linear in time). Outside the range (186) the correction to the GGE decays +fast enough, so that the associated cumulant is not modified at leading order. +Let us briefly comment the situation in the case of correlation functions of the single-mode +current even though the main idea exactly follows the density-density case. The relevant quantity +in this case is +〈jθ(x, t)jηθ(0,s)〉c = +� +dkdk′ +2π +dqdq′ +2π +eix(k′−k)ϑ +� +ε +2 − +���� +k + k′ +2 +− θ +���� +� +ϑ +� +ε +2 − +���� +q + q′ +2 +− ηθ +���� +� +× +� E(k′) − E(k) +k′ − k +�2 � +〈b† +k(t)bq′(s)〉〈bk′(t)b† +q(s)〉 − 〈b† +k(t)b† +q(s)〉〈bk′(t)bq′(s)〉 +� +. +(187) +46 + +SciPost Physics +Submission +This time we cannot use time-translation invariance as done before when computing the expec- +tation on a GGE. The first part in the above expression is again the GGE contribution alayzed in +(181), while the second comes from quasi-particle pairs and gives rise to saddle points in the bal- +listic regime. It can be checked that the condition for the saddle point contribution not to vanish +is exactly the same as for the density, Eq.(186). Therefore, only when the condition (186) does +not hold, the contribution of the second term in (187) is subleading wrt to the GGE one, and the +associated cumulant is not shifted with respect to the leading GGE value. +Therefore, depending on the value of θ (apart for θ in a region of order ε, which is to trace +back to our regularization of the observables) either the correlations of single-mode densities or +those of single-mode currents show a fast approach to their GGE value, thus imposing the right +path to choose when using BFT. +C +S matrix in the α−copy theory +Consider S(θ,θ ′) to be the S matrix in the single-copy theory (we consider the diagonal case for +simplicity, but everything below can be generalized to non-diagonal S matrices), i.e. +|θ,θ ′〉 = S(θ,θ ′)|θ ′,θ〉 +(188) +(namely, it is the factor we get by exchanging θ,θ ′ in the two-particle state), then one can define +in the α−copies theory +S(α)(θ, i;θ ′, i′) = δii′ +mod(α)S(θ,θ ′) ± (1 − δii′ +mod(α)) +(189) +acting on the state |θ, i;θ ′, i′〉, where we introduced explicitly the dependence on the copy-index. +Importantly, the ± sign in (189) depends on the commutation relations of the fields among the +copies. +Note that Eq. (189) is the S-matrix associated to α independent copies, which only describes +the symmetry of the α-copies theory (i.e., it does not take into account the constraints of the fields +in different copies implemented by the twist fields exchange relations (cf. (45)-(46))). However +this is enough for our purposes. +By going to Fourier space in the replica index Fi→p, we have +|θ, p;θ ′, p′〉 = +� +p,p′ +S(α)(θ,θ ′; p, p′; k, k′)|θ ′, k′;θ, k〉 +(190) +with (by simple algebra) +S(α)(θ,θ ′; p, p′; k, k′) = δmod(α)(k + k′ − p − p′) +� +S(θ,θ ′) ± 1 +� +∓ δmod(α)(p − k)δmod(α)(p′ − k′) . +(191) +This can be written explicitly as a 2α×2α matrix S(α)(θ,θ ′)m,n for α ∈ �, with row and column +indices m = +� +p, p′� +and n = +� +k, k′� +respectively. For example, for α = 2, it takes the form +S(2)(θ,θ ′) = +� +�� +S(θ,θ ′) +0 +0 +S(θ,θ ′) ± 1 +0 +S(θ,θ ′) +S(θ,θ ′) ± 1 +0 +0 +S(θ,θ ′) ± 1 +S(θ,θ ′) +0 +S(θ,θ ′) ± 1 +0 +0 +S(θ,θ ′) +� +�� +(192) +47 + +SciPost Physics +Submission +Note that one can check that, as expected, this S matrix satisfy the Yang-Baxter equations for +general S(θ,θ ′). +Now, for the specific case of free fermions, consider Eq. (50) in the main text. In the first +basis, the fields ψi commutes among different copies: in this case above we will choose the + +sign in S(α) (cfr. Eq. (189)). However, after the SU(α) trasformation to the ψj basis, the fields +in different copies also anticommute, and this amounts to choosing the sign − in (189). Since for +free fermions we have S(θ,θ ′) = −1, then we see that in the ψj basis, S(α) becomes diagonal (cfr. +Eq. (192)). +References +[1] L. Amico, R. Fazio, A. Osterloh and V. 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Mitra, Quantum quench dynamics, Annual Review of Condensed Matter Physics 9(1), +245 (2018), doi:10.1146/annurev-conmatphys-031016-025451, https://doi.org/10.1146/ +annurev-conmatphys-031016-025451. +53 + diff --git a/otE0T4oBgHgl3EQfZwCz/content/tmp_files/load_file.txt b/otE0T4oBgHgl3EQfZwCz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf0f57288e5e6c0e7ab435c450d394c0ce0a75ae --- /dev/null +++ b/otE0T4oBgHgl3EQfZwCz/content/tmp_files/load_file.txt @@ -0,0 +1,1678 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf,len=1677 +page_content='SciPost Physics Submission Entanglement Rényi Entropies from Ballistic Fluctuation Theory: the free fermionic case Giuseppe Del Vecchio Del Vecchio1, Benjamin Doyon2, Paola Ruggiero3 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK January 9, 2023 Abstract The large-scale behaviour of entanglement entropy in finite-density states, in and out of equilibrium, can be understood using the physical picture of particle pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, the full theoretical origin of this picture is not fully established yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In this work, we clarify this picture by investigating entanglement entropy using its connection with the large-deviation theory for thermodynamic and hydrodynamic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We apply the universal frame- work of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, to correlation functions of branch-point twist fields, the starting point for computing Rényi entanglement entropies within the replica approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Focusing on free fermionic systems in order to illustrate the ideas, we show that both the equilibrium behavior and the dy- namics of Rényi entanglement entropies can be fully derived from the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, we emphasise that long-range correlations develop after quantum quenches, and account- ing for these explain the structure of the entanglement growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We further show that this growth is related to fluctuations of charge transport, generalising to quantum quenches the relation between charge fluctuations and entanglement observed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The general ideas we introduce suggest that the large-scale behaviour of entanglement has its origin within hydrodynamic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Contents 1 Introduction 2 2 Ballistic Fluctuation Theory and twist fields 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 General setting 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Large deviation theory of currents 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 F(λ) in MES: biased measure, flow equation 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Application of the BFT to twist fields 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Explicit expression of F(λ) in free fermionic theories 12 3 Entanglement and branch-point twist fields 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Replicas and branch-point twist fields 14 1giuseppe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='del_vecchio_del_vecchio@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='uk 2benjamin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='doyon@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='uk 3paola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='ruggiero@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='uk 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='02326v1 [quant-ph] 5 Jan 2023 SciPost Physics Submission 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Enhanced symmetry in free fermions: from Zα to U(α) 15 4 Entanglement entropies from BFT 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Rényi entropies of a finite interval in a GGE (charge fluctuations in space) 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Long-range correlations due to correlated particle pairs in homogeneous global quenches 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 Rényi entropies of half system after a quench (current fluctuations in time) 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Single-mode and pair-mode twist fields 23 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Rényi entropies of an interval after a quench (fluctuations of single-mode densities and currents) 25 5 Discussion and conclusion 29 A Remarks on notions of locality and twist fields 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Unbounded observables and topological charges 32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Descendant twist fields and semilocality sectors 33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 Non-abelian semilocality 33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Twist fields in the literature 33 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Concepts of locality in the literature 34 B Correlations after a quench from an initial state with pair structure 34 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Global U(1) densities and currents and decay of correlations in GGEs 35 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Quench protocol and initial state 38 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 Approach to the GGE 40 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Single-mode density and currents and decay of correlations in GGEs 42 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Approach to the GGE 45 C S matrix in the α−copy theory 47 References 48 1 Introduction The understanding of entanglement in quantum many-body systems received a considerable boost in the last decades, with the introduction and characterization of many different quantities which “measure” the amount of entanglement in a given quantum state [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' An important set of such measures are the so-called entanglement Rényi entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Given a quantum system described by a density matrix ρ and a subsystem A of the total system, with ¯A denoting its complement, consider the associated reduced density matrix ρA = tr¯Aρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, for any α ∈ �+, the α-Rényi entropy is defined as Sα = 1 1 − α logtrρα A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (1) They are good entanglement measures for all pure quantum states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' states of the form ρ = |Ψ〉〈Ψ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' They fully characterise the entanglement spectrum, and an important property is that in the limit 2 SciPost Physics Submission α → 1 they reduce to the famous entanglement Von Neumann entropy S = −tr(ρA logρA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (2) In the context of one-dimensional systems, which is the focus of this paper, several exact results are available for such quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For example, at equilibrium, Rényi and entanglement entropies or their asymptotic behaviours can be obtained in the ground state states of critical [5], gapped [6] and more general integrable [7] field theories, as well as beyond integrability [8] (note that for free theories results were first obtained in [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the case of critical systems described by a conformal field theory (CFT), such results are easily generalized to finite temperature states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', Gibbs ensembles) [5], and also results for generic thermodynamic macrostates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', generalized Gibbs ensembles [10]) have been obtained [11–13] in the context of integrable models relying on (thermodynamic) Bethe ansatz [14] methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' When moving to out-of-equilibrium scenarios, the situation is more complicated and available results are mainly qualitative or in the form of conjecture (an exception, however, is the exact result in [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For example, an imaginary time path-integral formulation, together with confor- mal invariance, has been used for a qualitative understanding of the ubiquitous linear growth of entanglement [16] observed after quantum quenches [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Moreover, the dynamics of the en- tanglement entropy (2) for a generic integrable system was understood in terms of a semiclassical “quasiparticle picture” (whose original version was proposed in [16]), complemented with the Bethe ansatz knowledge of the stationary state attained at late times, as conjectured in [19] (see also [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These results have been extensively verified numerically (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' An impor- tant point to stress is that the quasi-particle picture does not admit a generalization for describing, for generic α, the growth of Rényi entropies [21,22] (with the exception of free systems [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A common starting point for (most of) these results is the so-called “replica approach”, whose main idea is that trρα A (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (1)) can be computed by considering α copies (with α an integer) of the original model, ending up with a “replicated” theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Appropriate analytic continuation to α ∈ �+, gives the Rényi and the Von Neumann entropy (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, within this approach, powerful tools are the so-called branch point twist fields, T α and its hermitian conjugate ¯T α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Twist fields, in general, are special fields associated to a given symmetry of the theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' they exist, in a many-body system, for every symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The branch point twist fields are special kind of those: as the replicated theory is invariant un- der permutations of the copies, T α, ¯T α are the twist fields associated to the generator of cyclic permutations i �→ i + 1 mod α and its inverse, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The quantities trρα A can be related to correlation functions of such twist fields, as first pointed out in quantum field theory in [7] clarifying ideas from [6], and as shown in quantum chains in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In this work, we make use of the large-deviation theory for ballistic transport, dubbed ballis- tic fluctuations theory (BFT), introduced in [24, 25], in order to study the Rényi entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The BFT, which is based on hydrodynamic projection principles, gives access to the large- deviation theory for fluctuations of total 2-currents on arbitrary rays in space-time, in homoge- neous and stationary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It generalises in a natural fashion the specific free energy from ther- modynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By the relation between currents and twist fields, the BFT, as pointed out in [24], also gives access to two-point functions of twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Concentrating on (generic) free fermionic systems, we show that both the equilibrium and the dynamics of Rényi entropies at large scales of space and time can be obtained from large- deviation principles and the BFT as applied to branch-point twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The resulting form of the Rényi entanglement entropy growth and saturation agree with previous results based on counting 3 SciPost Physics Submission particle pairs, but the method is new, and brings out, we believe, important new physics underlying the entanglement entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The main two observations are: (1) We obtain an exact relation between the growth of the Rényi entanglement entropies af- ter a so-called integrable [26–28], pair-production quench, and static and dynamic “full counting statistics" in the final GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Consider N<,> := � |v(θ)|<,>ξ/2 dθ ψ† θψθ the conserved quantity giv- ing the total number of “slow" and “fast" fermionic modes ψθ, with speeds |v(θ)| < ξ/2 and |v(θ)| > ξ/2, respectively, where ξ = x/t is a spacetime ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us denote F<,ξ dyn (λ) = lim t→∞ t−1 log〈eλJN<(t)〉 the scaled cumulant generating function for the total current JN<(t) of slow modes passing through a point in the time interval [0, t] in the final GGE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' and F>,ξ stat (λ) = lim x→∞ x−1 log〈eλN>(x)〉 the scaled cumulant generating function for the total number N>(x) of fast modes lying on the spatial interval [0, x] in the final GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Consider for simplicity α to be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, as x, t → ∞ with x/t = ξ fixed, the Rényi entanglement entropy on the interval [0, x], at time t after the quench, has asymptotic form: Sα(x, t) ∼ 1 1 − α � 2t α/2 � q=−α/2+1 F<,ξ dyn (ih2q−1) + x α/2 � q=−α/2+1 F>,ξ stat (ih2q−1) � , hp = πp α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (3) This extends earlier observations of the connection between entanglement entropy and full count- ing statistics [29–31] to non-equilibrium quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Our calculations also provide a fundamental explanation of such relations in terms of twist fields and the large-deviation theory for their asymp- totic behaviours, which, as far as we know, has not been noticed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (2) We give a new exact derivation of the so-called quasi-particle picture in the case of free fermions with generic dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Our derivation is completely independent from the other exact result for the Ising model in [15], which was based instead on Toeplitz matrix rep- resentation and multidimensional phase methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, our method makes transparent how it is the simple structure of long-range correlations induced by particle pairs in integrable quenches that allows one to describe both the growth and saturation of entanglement in a simple and universal way in terms of the long-time GGE, as this structure allows the separation of the contributions of fast and slow modes as per (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The emphasis on the structure of long-range cor- relations also gives a clear understanding as to why for quenches starting from more complicated states, for instance producing correlated groups of more than two particles, more information about the initial state is needed to describe the entanglement growth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' in these case no simple formula exists (as showed for example in [32,33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We concentrate on free fermion models for simplicity and in order to most clearly illustrate the method and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, as the method is based on general large-deviation and hydro- dynamic principles, it is expected to be much more widely applicable, which we leave for future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, it suggests that hydrodynamic modes and hydrodynamic projections are the more accurate notions at the root of the large-scale behaviour of the entanglement dynamics, rather than particles and their productions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 is an introduction to BFT and its relation to twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3 we review the replica approach and the associated branch-point twist fields, 4 SciPost Physics Submission and we discuss the simplifications occurring in the free fermionic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4 is the core of the paper, where we derive an expression for correlation function of twist fields from BFT, both in- and out-of- equilibrium, and use them to obtain (3) and recover the known formulas for Rényi and entanglement entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A discussion of our method and results is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The appendices complement the main text with observations and details of the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A contains remarks on notions of locality and twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B contains all the details about the applicability of BFT in the different situations we consider, by explicitly computing correlations, and their long-range behaviour, after a quench from a state with pair structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Finally, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' C is about the structure of the S-matrix in the α-copy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 Ballistic Fluctuation Theory and twist fields The BFT [24, 25], detailed below, is a theory describing the large-scale, ballistic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is expected to apply to a large class of quantum and classical many-body, extensive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It applies to generic systems with space-translation invariant dynamics and interaction range that is short enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It has been developed originally for states that are spacetime stationary and clustering in space, but many of the ideas have been extended to more general situations [34,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In this paper, we use the BFT as originally developed in [24, 25],and show how, and under which assumptions, it can be applied to states that emerge after quantum quenches as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Quan- tum quenches give rise to states that are locally spacetime stationary, but present time-varying long-range correlations, as we will explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We will explain how simple ideas based on the principles explained in [24] allow us to nevertheless use the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We mention that long-range cor- relations can also, in principle, be accounted for directly by using the more sophisticated ballistic macroscopic fluctuation theory (BMFT) [35], which we leave for further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 General setting The main strength of the BFT is that it stipulates that only some emergent properties of the system are required in order to describe the large-scale, ballistic fluctuations: the data of its Euler hydro- dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We assume the system of interest to have a certain number N (which in our application to free fermions will be infinite, as the system is integrable) of conserved quantities Qi = � d x qi(x, t) (4) such that dQi/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' They are assumed to be hermitian (in quantum systems), or real (in classical systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' They include the Hamiltonian H = � d x h(x, t), which generates time trans- lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For simplicity of the discussion we assume these to be in involution, [Qi,Q j] = 0 for all i, j ∈ {1,··· , N}, however this is not necessary in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' They have associated conservation laws ∂tqi + ∂x ji = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The observables qi, ji are the corresponding charge density and current, assumed to be “local".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the present paper, locality of qi and ji simply means that Qi has appropriate ex- tensivity properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' we keep the general discussion formal in order to avoid technicalities, but see the remark about locality concepts in the literature in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Within such systems, we focus on states belonging to the manifold of maximal entropy states (MES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Each state is characterized in terms of a vector β = {β1,··· ,βN} of “Lagrange multipli- ers", with N components (there are as many number of components as the number of conserved 5 SciPost Physics Submission quantities Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Given such a vector, the density matrix defining the system reads4 ρβ ∝ e− � i βiQi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (5) These include the GGEs studied in integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We consider the system to be in infinite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Below, when no ambiguity occurs, expectation values on such states will be denoted simply as 〈·〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Importantly, we assume such states to be clustering strongly enough: connected correlations tend to zero quickly enough at large spatial separations, 〈a(x,0)b(y,0)〉c := 〈a(x,0)b(y,0)〉 − 〈a(x,0)〉〈b(y,0)〉 → 0 (|x − y| → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (6) Here and below, a(x, t) is a local observable at the position x and evolved to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As per basic principles of statistical mechanics, the averages of densities are generated by the free energy 〈qi〉 = ∂ ∂ βi f (β) (7) and the mapping 〈q〉 ↔ β (8) is bijective (in appropriate regions of values of 〈qi〉 and βi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As mentioned, states that are spacetime stationary for local observables, but with spacetime varying long-range correlations, arise naturally in quantum quenches even after long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is because for thermalisation to happen on large regions of the system takes a long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In such cases, the state is not described by the MES (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A precise description of states with long-range correlations is more involved, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' [35, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But the concept of MES is still useful in these situations, as, nevertheless, averages of all local observables, or observables supported on regions smaller than the correlation range, still are described by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We will explain how to use this fact in order to “avoid” long-range correlations and apply the results of the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Large deviation theory of currents Consider some conserved quantity Q = Qi∗ (for a given i∗ ∈ {1,··· , N}), with associated density q = qi∗ and current j = ji∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is instructive to start with a description of the large-deviation theory for extensive charges at equilibrium, before discussing currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In a given state ρβ, a natural question is to charac- terize the restriction ∆J(x) = − � x 0 d x′ q(x′,0) of the charge Q to a spatial interval [0, x], and its fluctuations within this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here x denotes the horizontal path from (0,0) to (x,0) (and the notation ∆J(x) is adapted to generalising to currents, as done below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The fluctuations are fully characterized by the cumulants of ∆J(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is a simple result that the cumulant generating function at large x is given by a difference of specific free energies f (β) of the system, 〈eλ∆J(x)〉 ≍ e−x∆f (λ), ∆f (λ) = f ({βi + δii∗λ}i) − f (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (9) Here and below, we use the notation A(s) ≍ B(s) with the meaning that lims→∞ logA(s) log B(s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore ∆f (λ) = − lim x→∞ x−1 log〈eλ∆J(x)〉, (10) 4In (5), an infinite-volume limit needs to be taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In quantum spin chains, it is shown that if the weight determining the density matrix, � i βiQi, is short-range, then this defines a state that is unique and exponentially clustering in space [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' More generally, one can construct states using the Hilbert space of extensive charges, see [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 6 SciPost Physics Submission and this generates the scaled cumulants cm, ∆f (λ) = ∞ � m=1 λm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' cm, cm = − lim x→∞ x−1〈 � ∆J(x) �m 〉c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (11) When studying transport, similarly, we are interested in characterizing the total current pass- ing by a given spatial point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', the origin) in a given interval of time [0, t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' One is therefore interested in the total transfer of Q in time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', ∆J(t) = � t 0 dt′ j(0, t′), where now t denotes the vertical path from (0,0) to (0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As argued in [24] using hydrodynamic principles, the structure parallels closely the equilibrium case in ballistic systems, such as those admitting many conserved charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' At large times t → ∞, a large-deviation principle holds generically for linear scaling with t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In fact, one can go further and consider the 2-current j = (j,q), and the integral along a more general path ℓ, starting in (0,0) and ending in (x, t), over its perpendicular component to the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This defines the following object ∆J(ℓ) = � (x,t) (0,0) j(x′, t′) ∧ dℓ (12) where dℓ = (d x′, dt′) and j ∧ dℓ = jdt′ − qd x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By current conservation, the integral (12) is in fact independent of the path chosen, and therefore the result only depends on the end-points (0,0) and (x, t) (for lightness of notation, we keep implicit the dependence on (0,0)) ∆J(ℓ) = ∆J(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (13) For example, we may choose to connect the initial and final points of the path via a segment of ray x t = tanγ, |γ| < π 2 , (14) as will be done below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us consider the Euclidean distance between the initial and final points, ℓ = � x2 + t2 (15) (we do not assume Euclidean spacetime symmetry, this is simply a convenient way of control- ling the scale of x and t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As mentioned, in ballistic systems a large-deviation principle holds generically for ∆J(x, t) ∝ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, as in the equilibrium case, we can define the scaled cumulant generating function (SCGF) F(λ) of ∆J(x, t) as 〈eλ ∆J(x,t)〉 ≍ eℓF(λ), F(λ) = lim ℓ→∞ℓ−1 log〈eλ ∆J(x,t)〉 = ∞ � m=1 λm m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' cm(γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (16) The function F(λ) also depends on the angle γ, which we keep implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The coefficients cm(γ) are the scaled cumulants of ∆J(x, t), cm(γ) = lim ℓ→∞ℓ−1〈 � ∆J(x, t) �m〉c, (17) which are m-point correlation functions of currents and densities integrated over the path ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The finiteness of scaled cumulants depends on the asymptotic behavior of density and current corre- lation functions at large spacetime separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' See App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 7 SciPost Physics Submission 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 F(λ) in MES: biased measure, flow equation From the explanations above, at γ = π/2 (in the spatial direction) we have that F(λ) = −∆f (λ), explicitly given in terms of thermodynamic quantities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' What is the corresponding quantity for finite values of γ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', involving the time direction)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It turns out that the answer is completely given using the data of the Euler hydrodynamics of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The Euler hydrodynamics controls the motion and correlations of the many-body system at large scales of space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For our purposes, it is sufficient to recall that it is completely fixed by the flux jacobian matrix, defined as (using the bijection (8)) Ai j = ∂ 〈ji〉 ∂ 〈qj〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (18) In integrable systems, the flux Jacobian is in fact an infinite dimensional matrix, or more precisely an integral operator [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' From the definition it is clear that Ai j is basis-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Its fundamental information is con- tained in its spectrum � veff k � k=1,···,N, which is composed of eigenvalues if N is finite, and which admits a continuum in integrable systems (where N = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The spectrum is interpreted as the set of effective velocities, or “generalized sound velocities,” associated to normal modes of the hydrodynamics [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In order to compute F(λ) in (16), the idea is to bias the measure ρβ defining the MES (where expectation values are considered), by multiplying it by the exponential of the integrated 2-current ∆J(x, t), as appears in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The state thus becomes λ−dependent, and it is in fact a MES, which we write as ρλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Associated to the segment of path from (0,0) to (x, t), with tanγ = x/t, one can derive a flow equation for ρλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='β, within the space of MES, starting from ρ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='β = ρβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Conve- niently, this can be written as a flow equation for the Lagrange multipliers β(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) themselves as follows [24] ∂ ∂ λβj(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) = −sgn[A(λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) − tanγ �N]i∗ j , βj(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) = βj (19) where we explicitly introduced the dependence on λ as well as on the path (through γ) in all quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The main result of BFT is that, solving the flow (19), one can get an expression of the SCGF directly in terms of the 2-current j evaluated along the flow, namely F(λ) = � λ 0 dλ′ (cosγ〈j〉λ′ − sinγ〈q〉λ′) (20) where 〈·〉λ denotes expectation values on ρλ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='β (recall that 〈·〉0 ≡ 〈·〉) and 〈j〉λ′ = 〈j(0,0)〉λ′, 〈q〉λ′ = 〈q(0,0)〉λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Crucially, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (20), one has that F(λ) is given in terms of thermodynamic and Euler hydrodynamic objects only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The result (20) with (19) follows from a large-deviation principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The full derivation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (19-20) is not reported here, but can be found in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The main assumption underlying the validity of (20) is that of “strong enough” clustering along the space-time ray of velocity x/t = tanγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Specifically, spatial clustering (6) is not enough: one needs vanishing of correlation functions of perpendicular currents j⊥(x, t) := j(x, t) ∧ dℓ, the integrand in (12), when the distance along the ray (0,0) → (x, t) goes to infinity, 〈j⊥(y,s)j⊥(y + ℓsinγ,s + ℓcosγ)〉c → 0 (ℓ → ∞, y/s = tanγ) (21) 8 SciPost Physics Submission and similar requirements for all multipoint functions of perpendicular currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The vanishing must be fast enough to make integrals defining the cumulants rapidly converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A crucial remark for our work, concerning the requirement of clustering, is Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 of [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Recall that ∆J(x, t) = ∆J(ℓ) in (12) is independent of the path ℓ, and only depends on the end- points (0,0) (which we have kept implicit) and (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' One may therefore hope to apply the general result of the BFT for each path element and obtain, in place of (20), the expression F(λ) = lim ℓ→∞ℓ−1 � λ 0 dλ′ � (x,t) (0,0) 〈j(x′, t′)〉λ′ ∧ dℓ, (22) for any path ℓ (with a well-defined large-scale limit ℓ → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As explained in [24, Rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3], this is expected to be correct if and only if there are no strong correlations between the perpendicular currents amongst different points on the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (22) is the main result from BFT which will be applied below to expectation values of ob- servables (specifically, of twist fields) both at equilibrium in states described by GGEs (in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1), and out-of-equilibrium in states emerging after quenches (in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 and the following ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the latter case, this can be done by choosing a path that “avoids" the long-range correlations that such states present, so that expectation values in the pre-quench state can be approximated with the ones in the corresponding long-time GGE (where correlations can be shown to decay fast enough).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Remark (clustering and the ballistic large-deviation principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' If there is no path that satisfies strong clustering, then typically the ballistic large deviation principle is broken, and the SCGF resulting from the BFT may be infinite or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Exponential clustering, which is sufficiently strong, is ex- pected to hold on rays away from the fluid velocities, tanγ ̸= veff i ∀i, in generic equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In GGEs of integrable systems at nonzero entropy density, the spectrum of the flux Jacobian con- tains a continuum, and clustering is in fact a power-law, with power 1/ℓ2 for the perpendicular currents, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (21), for all rays within this continuum, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 for the case of free fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is still strong enough for the BFT results to hold, as confirmed numerically for the hard rods [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By contrast, in non-integrable systems, where the flux Jacobian has a discrete spec- trum, clustering is too weak on rays along any of the eigenvalues of the spectrum of the flux Jacobian (fluid velocties), tanγ = veff i for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In these directions, the ballistic large-deviation principle is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' See the discussion in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, at zero temperatures when there is no gap, weak power-law clustering is seen and the ballistic large-deviation principle is also bro- ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' When the breaking of the large-deviation principle happens as we change a parameter (a (generalised) temperature, a coupling), this can be seen as a “dynamical phase transition".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Application of the BFT to twist fields One of the most important application of the BFT is to two-point correlation functions of so-called “twist fields”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is useful, because, as explained in the introduction, twist fields are probably the most efficient way of studying entanglement entropy, the main object of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' There are a number of ways of defining twist fields, and we will discuss two natural approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The first is natural in the context of the large-deviation theory as recalled above and based on the explicit knowledge of extensive conserved quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' it applies to classical and quantum systems alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The second is a more abstract formulation that does not require the explicit knowledge of extensive conserved quantities, but that is better adapted to quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 9 SciPost Physics Submission Consider as above an extensive conserved quantity Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Recall that Q has associated density and current q(x, t) and j(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is convenient to define the height field ϕ(x, t) via the relations q(x, t) = ∂xϕ(x, t), j(x, t) = −∂tϕ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (23) This ensures that the continuity equation ∂tq +∂x j = 0 is automatically satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The height field is unbounded (because the charge is extensive), and we note in particular that differences grow linearly, ϕ(0,0) − ϕ(x, t) = ∆J(x, t) ∝ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (24) The height field may be written as an integral over half space5, ϕ(x, t) = − � ∞ x d y q(y, t) + ϕ(∞, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (25) One finds that the boundary term at infinity is constant in time, ϕ(∞, t) = ϕ(∞) (which can be chosen to vanish).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Further, it is clear that the result is independent from the choice of path in spacetime thanks to the conservation laws, ϕ(x, t) = � (∞,t) (x,t) j ∧ dℓ + ϕ(∞), (26) and thus the height field ϕ(x, t) only depends on the point (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This justifies the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' One may also choose a different direction for the half-space integral, the difference being encoded within ϕ(∞) − ϕ(−∞) = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (27) Exponentials of ϕ(x, t), that is Tλ = eλϕ, (28) are known in general as “twist fields”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Because of the expression (25), they are not “local" in the naïve sense, but are usually referred to as “semilocal" in the literature (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A for an overview);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' this is made clearer below using exchange relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As an immediate result of the large-deviation analysis, using Tλ(0,0)T−λ(x, t) = eλ(ϕ(0,0)−ϕ(x,t)) (29) with (24), we get the leading exponential behavior of the two-point correlations of twist fields as 〈Tλ(0,0)T−λ(x, t)〉 ≍ eℓF(λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (30) That is, if the ballistic large-derivation principle holds for the charge Q, then the associated twist field shows an exponential behaviour at large space-time separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Because eλϕ is not bounded for λ ∈ �, Tλ should be referred to as an “unbounded twist field".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In quantum models, because of the lack of commutativity of observables, and a fortiori in quantum field theory (QFT), because, additionally, of the necessary renormalisation procedure applied to the twist fields, the relation (29) is not strictly valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, corrections do not affect the leading exponential behaviour of the two-point function, as argued for the XX quantum chain in [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 5This expression is somewhat formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In quantum spin chains, by applying a time derivative to (25), one can make the result mathematically rigorous using an appropriate Hilbert space of the Gelfand-Naimark-Segal type, as proved in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 10 SciPost Physics Submission A second viewpoint on twist fields is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A twist field T(x, t) is in general an opera- tor associated to a symmetry transformation that “acts locally enough".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is a transformation a(x, t) �→ ˜a(x, t) of the model, that maps local observables to local observables, and that pre- serves the operator algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' one usually also requires that it preserves the Hamiltonian density, ˜h(x, t) = h(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The property of an operator T(x, t) that makes it a twist field associated to such a symmetry, is the following equal-time exchange relation: T(x, t)a(y, t) = � ˜a(y, t)T(x, t) y ≫ x a(y, t)T(x, t) y ≪ x (31) for every local observable a(y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Corrections at larges distances |y − x| should be small enough, for instance exponentially decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This equal-time exchange relation is known in the literature as expressing the “semilocality" of the twist field6 T(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, symmetry transformations act via unitary operators U as ˜a(x, t) = Ua(x, t)U−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For instance, in quantum spin chains, one often can write U = � x∈� Ux where Ux acts non-trivially on a neighbourhood of x (possibly up to exponentially decaying corrections);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' this is indeed local enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In such instances, one can simple set T(x,0) = � y≥x Uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (32) How does the exchange-relation formulation (31) connect with the height-field formulation (28) of twist fields?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is from the general principle that every extensive conserved quantity gives rise to a continuous, one-parameter unitary group of symmetry transformations that act locally enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' That is, given an extensive Q (recall that it is assumed to be hermitian), we may consider the symmetry transformation ˜a(x, t) = eiηQa(x, t)e−iηQ (33) for a real parameter η ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is simple to see that this acts locally enough, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Indeed, by conservation we can replace Q by Q(t) in (33), and by locality of the density q(x, t), we have that � L −L d x [q(x, t), a(0, t)] approaches its limit as L → ∞ quickly enough, and gives in the limit a local observable supported at x = 0 at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, by using the Baker-Campbell- Hausdorff formula, at least for all η small enough, eiηQa(x, t)e−iηQ gives rise to a local observable at (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Using the same principles, it is then a simple matter to show that (31) holds with the choice (25) for the height field, and the identification in (28) T = T−iη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (34) Because e−iηϕ is bounded, we will referred to these as “bounded twist fields”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Bounded twist fields are the ones usually considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, because they are bounded, their two-point functions should decay in spacetime, ℜF(−iη) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (35) One may in fact argue that, viceversa, to every symmetry transformation that acts locally enough, we can associate an extensive conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In quantum field theory, Noether’s theorem shows 6If the twist field indeed preserves the hamiltonian density, it commutes with it at large distances, up to small (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' exponential) corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This has important implications, which justifies considering twist fields, for many pur- poses, on the same footing as local fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' for instance, the time-evolved twist field is analytic in the time variable at small enough times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 11 SciPost Physics Submission that, for continuous symmetry groups, we indeed have U = eiηQ for some conserved quantity Q associated to a conserved 2-current;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' again this is local enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, for any local enough transformations, one can identify, formally, an extensive charge Q with the operator −ilog U (thus taking (33) with η = 1), a formal construction that, we expect, could be used fruitfully within the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the present paper, we will consider a twist field associated to a discrete symmetry transformation, but this will be embedded within a continuous symmetry group thanks to the free-fermion structure, thus the charge Q will be explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Finally, we observe that applying the BFT for the bounded twist fields, using the identification (34), requires an analytic continuation of λ in the BFT formulae, to purely imaginary values −iη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is a subtle aspect, as for purely imaginary values of λ, the modified state by the flow equation is not strictly a MES (because the resulting linear functional on the algebra of observables is not necessarily positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We believe that if fluid velocities are well separated, as is typically the case in non-integrable systems, then the analytic continuation can be obtained meaningfully by keeping the sign of the eigenvalues constant in the flow equation (19) (as the analytic continuation will not “see" the jumps in eigenvalues), and integrating the flow in the complex λ-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In free fermion models, the analytic continuation can be performed directly on the explicit result for F(λ), as done in [41] and in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We will show below that the BFT indeed predicts decay of correlation functions in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We leave the discussion for interacting integrable systems to future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' See App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A for a brief discussion of notions of locality and twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Explicit expression of F(λ) in free fermionic theories Up to now, the theory was general, and all equations correctly give the ballistic part of large de- viations in general systems with the properties detailed in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' When considering integrable systems, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (20) gets further simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In fact, using the theory of generalized hydro- dynamics (GHD) [42,43], an explicit expression of the hydrodynamic quantities 〈q〉λ,〈j〉λ along the flow can be worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We now overview the simplified expression for F(λ) in the special case of free fermions in the continuum, arguably the easieast among 1D integrable systems, which is our focus in this paper (the corresponding results in the case of generic interacting integrable models can be found in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In order to keep the structure general, we simply assume that a fermionic, complex field ψ(x, t) exists with interactions that are quadratic and short-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Its Fourier modes are denoted ψθ, with anti-commutation relation {ψ† θ,ψθ ′} = δ(θ −θ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here θ represents the momentum, which we as- sume takes values in � for simplicity (for quantum chains, this would be a bounded subset instead, but the general ideas are not affected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We also denote the dispersion relation as E(θ), which we assume is strictly convex and symmetric E(θ) = E(−θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, under canonical normalisation, ψ(x, t) = 1 � 2π � dθ eiθ x−iE(θ)tψθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (36) As it is integrable, the model possesses an infinite number of conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A “scattering basis" for these is given by Qθ = ψ† θψθ, θ ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Strictly speaking, the Qθ’s are not linearly ex- tensive, but (for generic dispersion relation) any extensive conserved quantity can be obtained by a suitable “linear combination", or basis decomposition, Qi = � dθ hi(θ)Qθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here, hi(θ) is the one-particle eigenvalue of the extensive charge Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Examples are the number of fermions N = � dθ Qθ, the total momentum P = � dθ θQθ, and the total energy H = � dθ E(θ)Qθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 12 SciPost Physics Submission A typical GGE (5) takes the form ρw := ρβ ∝ e− � dθ w(θ)Qθ , where w(θ) = � i βihi(θ) is the generalised Boltzmann weight in the particle basis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For example, for a thermal state, we have ρw ∝ e−β(H−µN) = e− � dθ β(E(θ)−µ)Qθ , so w(θ) = β(E(θ) − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, the physical meaning of the Lagrange multipliers βi depends on the choice of the set of charges Qi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' on the choice of the set of one-particle eigenvalues hi(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But there is no need to choose any particular infinite set of charges Qi, or to write explcitly w(θ) in a basis decomposition w(θ) = � i βihi(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The function w(θ) fixes the GGE, and only few basic requirements constrain w(θ) for ρw to be a valid GGE (we will ask that it be positive and grow sufficiently fast as |θ| → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We note that the conserved charge densities take the standard form 〈qi〉 = � dθ/(2π) n(θ)hi(θ) in terms of the occupation function n(θ) = 1 1 + ew(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (37) and that, in a system of length L with periodic boundary conditions, we have 〈Qθ〉 = L 2πn(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In our calculations, we will assume that n(θ) has an analytic extension in a neighbourhood of �, and that n(θ) → 0 as |θ| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Consider the large-deviation problem for the charge Q = Qi∗, with one-particle eigenvalue hi∗(θ) = h(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For free fermions, F(λ) simplifies to F(λ) = − � dθ 2π |v(θ) cosγ − sinγ|[f (ελ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ)) − f (w(θ))] (38) where v(θ) = dE(θ)/dθ is the group velocity7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The function f (ε) is the fermionic free energy function (the free energy density per distance and per unit rapidity θ), f (ε) = −log(1 + e−ε), (39) and the function ελ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) is the Boltzmann weight along the flow (19) in the particle basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Its initial condition is ε0(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) = w(θ), and the corresponding flow equation (which simply follows from (19) in terms of βi) simplifies to ∂λελ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) = sgn(tanγ − v(θ))h(θ) , (40) which is explicitly solved as ελ(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) = w(θ) + λ sgn(tanγ − v(θ))h(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (41) As mentioned above, in order to apply the BFT to bounded twist fields associated to symmetry transformations, one needs to perform an analytic continuation in λ to the purely imaginary di- rection λ = −iη, η ∈ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In free Fermion systems this is simple to do, as the above formulae can be directly analytically continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The resulting F(λ) possesses, in general, both a real and an imag- inary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The real part describes the exponential decay of the two-point correlation functions of twist fields, while the imaginary part describes oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the following, we will not discuss oscillations, as their full description would require a more in-depth analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' we will concentrate on the exponential decay, hence the real part of F(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 7The effective velocity of GHD is just the group velocity in free particle models, veff(θ) = v(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 13 SciPost Physics Submission Correlation functions of twist fields are expected to be decaying at large spacetime distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is simple to show from (38) that indeed8, ℜF(−iη) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' One important remark is that the only information required about the current ∆J(x, t) whose SCGF is taken, is the one-particle eigenvalue h(θ) of the corresponding total charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, the BFT predicts that only a limited amount of information about the twist field is required in order to evaluate the exponential asymptotic of its two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note that this is true also in the interacting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3 Entanglement and branch-point twist fields In this section, we recall how entanglement entropies can be computed using a certain type of twist fields, called branch-point twist fields, associated to permutation symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We then recall that, in free fermionic theories, these can be re-written in terms of U(1) twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This will be used in the next section in order to apply the BFT to the calculation of entanglement entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Replicas and branch-point twist fields Within the replica method, in order to compute entanglement entropies (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (1)-(2)) in a given theory, one re-writes the quantity trρα A in terms of an appropriate expectation value in the replica model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is the model composed of α independent, commuting copies of the original model (α ∈ �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For a one-dimensional system in a state with density matrix ρ, and with the subsystem A being a single interval, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', A = [x1, x2], it is a simple matter to show [7, 23] that trρα A is exactly identified with the two-point function of branch-point twist fields, trρα A = 〈T α(x1,0)¯T α(x2,0)〉ρ⊗α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (42) The expectation value on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' is computed in the density matrix ρ⊗α = ⊗α i=1ρi, where ρi is the original density matrix, on copy i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Branch-point twist fields in the replica theory are twist fields associated to the symmetry under replica cyclic permutations of order α (which generate the group Zα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' They take the product form (32), involving on-site copy-permutation operators9 [23]: T α(x,0) = � y≥x Py (43) and ¯T α(x,0) = � T α(x,0) �†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here, denoting by ai(x) observables lying on (that is, acting non- trivially only on) copy i ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=',α} and position x, and identifying aα+1(x) ≡ a1(x), the permutation unitary is defined by Pxai(y)P−1 x = � ai+1(y) y = x ai(y) y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (44) This implies the equal-time exchange relations (see (31)) T α(x, t)ai(y, t) = � ai+1(y, t)T α(x, t) y ≥ x T α(x, t)ai(y, t) y < x (45) 8This is because in (38) one has e−ε−iη(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='γ) = ru where r = e−w(θ) > 0 and u is a pure phase, |u| = 1, and |1 + r| ≥ |1 + ru| for any r > 0 and any pure phase u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 9Here we omit any regularisation issue that may arise in models on a continuous space, which, as mention, do not affect exponential asymptotic behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 14 SciPost Physics Submission and ¯T α(x, t)ai(y, t) = � ai−1(y, t)¯T α(x, t) y ≥ x ¯T α(x, t)ai(y, t) y < x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (46) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (42), Rényi entanglement entropies can be simply obtained via Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (1)-(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the context of (1+1)-dimensional QFT, exchange relations of the form (45), (46) give the most appropriate formulation for working definitions of the branch-point twist field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is in this context that they were first introduced [7], as a way of evaluating partition functions on branched surfaces, taking inspiration from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We note that the action of branch-point twist fields can be diagonalized by going to the Fourier basis in the replica index, ap(x, t) = Fi→p[ai(x, t)] := 1 �α α � i=1 eiπpi/αai(x, t) (47) for p ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=',α − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This gives T α(x, t)ap(y, t) = � e−iπp/αap(y, t)T α(x, t) y ≥ x ap(y, t)T α(x, t) y < x (48) and similarly for ¯T α(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the next subsection we will use a similar construction, albeit in a different basis of the replica model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As will be explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4, for our purposes, the most general object we need to consider is the two-point correlation functions 〈T α(x1, t1)¯T α(x2, t2)〉ρ⊗α (49) at different spacetime points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Enhanced symmetry in free fermions: from Zα to U(α) Consider the special case of free fermions, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In this case, because of the quadratic nature of free fermion Hamiltonians, the Zα symmetry of the replicated theory turns out to be embedded into the larger symmetry group U(α), which accounts for not only permutation of replicas, but also rotations amongst them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, the branch-point twist field is a twist field associated to a particular symmetry transformation, part of a continuous symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4, using Noether’s theorem, this then allows one to write an explicit extensive charge associated to the twist field [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The U(α) symmetry is most clearly expressed in a different basis of the replica theory than that used above, obtained by “fermionising" the replica theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By the basic construction of the replica theory, different replicas commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, in order to extract the symmetry U(α), one needs fermions in different copies to anti-commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' One simply defines the replica theory by asserting that fermion fields anti-commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is of course natural, but changes the action of the branch-point twist field (the exchange relations (45), (46)) by introducing an extra minus sign, as worked out in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' From now on, we denote ψi(x, t) = 1 � 2π � dθ eiθ x−iE(θ)tψθ,i (50) 15 SciPost Physics Submission the Dirac fermion on the i-the copy in this new basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' thus ψi(x, t)ψj(x′, t′) = −ψj(x′, t′)ψi(x, t) if i ̸= j, and the canonical anti-commutation relations hold, {ψi(x, t),ψ† j(x′, t)} = δi jδ(x − x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We now recall the main arguments of [7] in order to obtain a useful form of the branch-point twist field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the new basis, the U(α) symmetry is explicitly a linear action on the fermions, in its fundamental representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Most importantly, in this basis, the cyclic permutation ψi → ψi+1 is a particular element of U(α), which is in fact an element of a U(1) subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The Fourier transform (47) can be performed in this basis, ψp(x, t) = 1 �α α � i=1 eiπpi/αψi(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (51) This diagonalises that U(1) subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' the action of the twist field is then diagonalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In fact, it turns out that the anti-commuting basis is also the one that guarantees that the Fourier transform operation keeps the S-matrix diagonal, see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus both the charge associated to the twist field, and the S-matrix, are diagonal in terms of the particles corresponding to ψp – this is at the root of the simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The fermion fields ψp(x, t) after Fourier transform are still independent free fermions with canonical anti-commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Each Fourier sector admits an independent U(1) symmetry, and, as shown in [7], the branch-point twist field can be written as a product of U(1) twist fields acting nontrivially on each Fourier sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Because of the extra minus sign in the twist field action, it is simpler to concentrate on the case of α even (the full dependence on α is obtained by analytic continuation), in which case the product goes over the following values of momenta: p ∈ Iα := {−α + 1,−α + 3,··· ,α − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (52) Specifically, it is found that [7] T α = � p∈Iα τα p = α/2 � q=−α/2+1 τα 2q−1 (53) with τα p being a U(1) twist field acting non-trivially only on ψp (as a phase), τα p(x, t)ψq(y, t) = � e−iπp/αψq(y, t)τα p(x, t) y ≥ x and p = q ψq(y, t)τα p(x, t) y < x or p ̸= q (54) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (48)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The decomposition (53) allows us to factorise the branch-point twist field two-point functions into products of U(1) twist field two-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This however only holds if the state can be likewise factorised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is nontrivial: the state ρ⊗α is naturally factorised in copy space, but not necessarily in the Fourier-copy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is a simple matter to verify that if ρ satisfies Wick theorem, then ρ⊗α also factorises as a tensor product of states ρ in Fourier-copy space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' this is because such states are completely determined by fermion two-point functions, which stay diagonal in Fourier- copy space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, we have, in Wick-theorem states ρ, 〈T α(0,0)¯T α(x, t)〉ρ⊗α = α/2 � q=−α/2+1 〈τα 2q−1(0,0)¯τα 2q−1(x, t)〉ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (55) Note how on the right-hand side, each factor is evaluated in the state ρ for the fermion ψ2q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 16 SciPost Physics Submission In the following we are going to apply the BFT machinery to each correlation function of the U(1) twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The crucial fact that makes it simple is that, for any given p, τα p(x, t) is the (bounded) twist field associated to the U(1)-charge Qp = πp α � d x ψ† p(x)ψp(x) = πp α � dθ ψ† θ,pψθ,p , (56) with explicit expressions as exponential of half-space integrals of charge densities, as per Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (28): τα p(x, t) = exp � i � ∞ x d x′ qp(x′, t) � , qp(x, t) = πp α ψ† p(x, t)ψp(x, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (57) Qp acts on the single-particle basis as Qp|θ,q〉 = hpδp,q|θ,q〉, with hp = πp α (58) (note that ψp(x) has Qp-charge −hp, in agreement with (54)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' With Q = Qp, the twist field τα p is identified with τα p = T−i in the notation of (34) (that is, with η = 1), acting on the sector p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Recall that the action of the charge on the single-particle basis is all we need to know in order to apply the BFT (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (38) and (40)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4 Entanglement entropies from BFT We arrived to a rewriting of the two-point function of the branch-point twist fields as product of two-point functions of U(1) twist fields, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' From there, using BFT, all such components can be accessed via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (30) specified to the twist fields τα p, which in the notation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (30) is identified with T−i, with the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' evaluated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (38), thus exploiting the free fermionic nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Considering different choices of the points in spacetime where the global fields T α, ¯T α are located, we are able to access Rényi entropies both at equilibrium and after a quench, as we are now going to discuss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Rényi entropies of a finite interval in a GGE (charge fluctuations in space) We start by considering the α−Rényi entropy of a finite interval A = [0, x] within a generic GGE ρw uniquely defined by the function w(θ) (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This means that we are interested in the following two-point function 〈T α(0,0)¯T α(x,0)〉ρ⊗α w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (59) From the BFT perspective, this is obtained by focusing on the purely spatial direction, namely, we consider an “horizontal path” by setting γ = π/2 in (38) (and we take h(θ) = hp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Each two-point function of U(1) twist fields in (55) reads 〈τα p(0,0)¯τα p(x,0)〉ρw ≍ exp � x Fp(−i) � , Fp(−i) = � dθ 2π log � 1 + eihp−w(θ) 1 + e−w(θ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (60) Then we consider the product in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (55), which turns into a sum in the exponent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', 〈T α(0,0)¯T α(x,0)〉ρ⊗α w ≍ exp{x Fα(−i)}, with Fα(−i) = α/2 � q=−α/2+1 F2q−1(−i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (61) 17 SciPost Physics Submission We may further evaluate those sums, by considering separately the part which depends and the part which does not depend on p (equivalently q,q′, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (53)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The latter is trivial and simply gives a contribution to Fα(−i) which is − � dθ/(2π) of 2 α/2 � q=1 log � 1 + e−w(θ)� = αlog � 1 + e−w(θ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (62) For the remaining part, let us start by focusing on half of the sum, the terms from q = 1 to α/2, in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By defining z = 2πi α , s = w + πi α , we get α/2 � q=1 log(1 + ezq−s) = ∞ � r=1 (−1)r+1 r e−r(s−z) � 1 − erzα/2 1 − erz � (63) where we used the Taylor expansion log(1+ x) = �∞ r=1(−1)r+1x r/r (which converges for w > 0), and we performed the sum over q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Next, we want to perform the sum in r in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' of (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' To do that, we substitute the values of z and w first: ∞ � r=1 (−1)r+1 r e−r(w− πi α ) � 1 − erπi 1 − er 2πi α � (64) where now we should consider separately three cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' r = αm for integer m: in this case r is even (as α is even), and we have ∞ � m=1 (−1)αm+1 αm e−αmw+mπi � rπi 2πir/α � = ∞ � m=1 (−1)m+1 2m e−αmw (65) = 1 2 log � 1 + e−αw� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (66) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' r even but r ̸= αm for any integer m: in this case each term of the sum (64) is zero due to the vanishing of the numerator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', (1 − erπi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' r odd: this gives � r odd 2 r e−rw � er πi α 1 − er 2πi α � = � r odd i r e−rw sin πr α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (67) The sum of the terms for q = −α/2 + 1 to 0 in (61) give exactly the complex conjugate of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus we get α/2 � q=−α/2+1 log(1 + ezq−s) = log(1 + e−αw) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (68) Putting everything together, Fα(−i) in (61) can be written as Fα(−i) = � dθ 2π � log � 1 + e−αw(θ)� − αlog � 1 + e−w(θ)�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (69) Finally, it is a matter of simple algebra to show that, in terms of the occupation function n(θ) (37), we get Fα(−i) = (1 − α) � dθ 2π Hα(θ) (70) 18 SciPost Physics Submission where we defined Hα(θ) = 1 1 − α log[n(θ)α + (1 − n(θ))α] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (71) The α−Rényi entropy is finally given by Sα(x) = 1 1 − α log〈T α(x,0)¯T α(0,0)〉ρ⊗α w ∼ x � dθ 2π Hα(θ) , (72) which coincides with the results obtained in [11,13] (there in the more general context of inter- acting integrable models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Long-range correlations due to correlated particle pairs in homogeneous global quenches We now review the main concepts underlying quantum quenches, restricting to “integrable" pair- producing initial states, and we explain how long-range correlations develop after such quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A quantum quench is an initial value problem for the many-body system where the initial state is the ground state of a different Hamiltonian than that used for the time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Typically, one imagines a sudden change of parameter, for instance of the mass parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In integrable models, certain quenches are known to be of “integrable" type [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In these cases, the initial state can be written explicitly in terms of the scattering states (or Bethe ansatz states) of the post-quench, evolution Hamiltonian, as a so-called “squeezed state": |Ψ〉 = 1 N exp � 1 2 � dθ Kθ,−θψ† θψ† −θ � |0〉 (73) for some (θ-dependent) factor Kθ,−θ, with Nθ denoting a normalization constant, and |0〉 being the ground state of the post-quench Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The squeezed state is generically a finite-density state, where the energy (of the post-quench Hamiltonian) is extensive with the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' See App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 for a discussion of such integrable initial states in free fermion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We will use later the fact there is a Bogolioubov transformation of the fermionic mode operators (a transformation between the post-quench and pre-quench fermions), ψ(x, t) ↔ ˜ ψ(x, t) (Bogolioubov) (74) such that the squeezed state satisfies (is defined by) ˜ ψ(x, t)|Ψ〉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (75) After a long time in a quench problem, the state locally approaches a GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In integrable quenches, there is a well-known relation between the squeezed-state representation of the ini- tial state, and the long-time GGE (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The statement of convergence to a GGE pertains only to local operators, or operators supported on finite intervals (that do not grow with time): 〈Ψ|a(x, t)|Ψ〉 → 〈a(x)〉ρw, t → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (76) The limit in (76) is expected to be valid everywhere in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The relation between initial state and long-time GGE in free fermions can be worked out explicitly (see (150)) e−w(θ) = |Kθ,−θ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (77) 19 SciPost Physics Submission Namely, we see that the map from squeezed states to GGEs is in fact one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Because of this one-to-one correspondence, it is clear that it is sufficient to know the long- time GGE in order to know the full behaviour of correlation functions in spacetime, as the GGE fixes the initial state uniquely (naturally under the condition that it be a pair-producing squeezed state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, this relation can be relatively complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Indeed, the statement of generalised thermalisation – that a GGE is reached – is true, generically, only on finite regions of space (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As is typically the case out of equilibrium, on large regions, say regions that grow linearly with the time after the quench, the state might not correctly be described by a GGE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' and this may even be true for all times!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Instead, the state may admit long-range spatial correlations, for instance correlations that have a large weight on distances that grow linearly with the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These are not present in GGEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' recall that, as discussed above, GGEs typically have correlations that decay quickly enough in space (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, even at long times, there may remain effects of the initial state that are not described by a GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the case of a squeezed state, such long-range correlations indeed exist, as we show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 (and also in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 for “single-mode densities and currents", introduced below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Their interpretation is that they are due to production of correlated pairs of opposite- momentum particles by the quench protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These particle pairs carry correlations to large dis- tances as they separate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We evaluate explicitly these long-range correlations for conserved den- sities and currents in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 (and App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For instance, we find, for the charge density q(x, t) = ψ†(x, t)ψ(x, t), that 〈Ψ|q(x, t)q(x′, t)|Ψ〉 − 〈Ψ|q(x, t)|Ψ〉〈Ψ|q(x′, t)|Ψ〉 exhibits strong, ballistic-scale correlations, in accordance with the picture according to which particle pairs are emitted at all velocities admitted by the dispersion relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In order to evaluate the Rényi entanglement entropy, as is clear from the calculation for GGEs in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1, we must evaluate the large-deviation theory for fluctuations of charges on large regions of space, and / or, as we will see below, fluctuations of current on large intervals of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The BFT allows us to do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, as mentioned, the BFT requires no long-range correlations along the path ℓ in (12), as the flow equation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (19)) assumes that the state along the path is a GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Long-range correlations may break the assumption that scaled cumulants are evaluated within a GGE – the effects of long-range correlations on the 2nd cumulants is illustrated in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Below, we take them into account by choosing appropriately the path ℓ in order to avoid such correlations!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, the knowledge of where such correlations exist, and the knowledge of the long-time GGE, is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The fact that there exists a path ℓ that avoids long-range correlations explains why, in pair- production states, the full behaviour of the Rényi entropies can be written in a simple and uni- versal way in terms of the long-time GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The same is not true when considering non-integrable quenches, namely quenches from more complicated states where groups of more than two cor- related particles are emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Without the constraint of pair-production, the one-to-one corre- spondence between the state and the GGE is lost, and long-range correlations carry additional information not present in the GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, from our approach, we see that a universal description in terms of the long-time GGE is lost because such an “avoiding path” does not exist in general anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 Rényi entropies of half system after a quench (current fluctuations in time) We now turn to the calculation of the α−Rényi entropy of a semi-infinite interval A = [0,∞) after a global homogeneous quantum quench, at long times t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is obtained from the 20 SciPost Physics Submission KING’S COLLEGE LONDON P AO L A R U G G I E RO : (for time-dependence) Correlations and breaking of LDT : further info BFT (“vertical” path) : Rényi entropy of semi-infinite system after a quench: A = [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' ∞] ⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t)⟩ ≃ ⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) ¯Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='0)⟩ Ψin⟩ = ∏ θ>0 1 Zθ eWθψ† θψ† −θ ∅⟩ ⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) ¯Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='0)⟩ ≍ exp {t Fα(1)} Sα(t) = t∫ dθ 2π |v(θ)|Hα(θ) : HALF SYSTEM AFTER A QUANTUM QUENCH Sα(t) t x Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) 0 t x Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) 0 ¯Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='0) −vθ vθ −vθ vθ [Alba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='Calabrese,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2017] Figure 1: Evolution of Rényi entropies of half system A = [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='∞] within BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Left: Ini- tial integration path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Because of initially entangled pairs, points along this path at time t will be correlated, which prevents us from applying BFT directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Right: Deformed integration path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Along this new path one can show that point are not correlated any- more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Moreover the only term contributing to the growth in time of entanglement is the vertical path from (0, t) to (0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' branch-point twist field one-point function 〈Ψα|T α(0, t)|Ψα〉 (78) in the state |Ψα〉 = |Ψ〉⊗α = �α i=1 |Ψi〉, the α-copy replica of (73), |Ψα〉 = α � i=1 1 N exp � 1 2 � dθ Kθ,−θψ† θ,iψ† −θ,i � |0, i〉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (79) As expressed in (76), one-point functions of local observables converge to averages within GGEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, as we discussed, twist-fields are “semi-local" observables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' from the point (0, t) emanates a branch cut, which is sensitive to the state where it passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The branch cut can be taken on the horizontal half-line {(x, t) : x ∈ [0,∞)} going from (0, t) to (∞, t), as done in the explicit construction of the field in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2, and analysed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, along this half-line, there exist long-range correlations due to coherent particle pairs emitted by the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This prevents us from applying the BFT along this path (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 (left)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Using path independence of twist fields correlation functions, we can deform the path, between its initial and final points, in a way to avoid such correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Specifically, we choose the piece- wise linear path joining the points (0, t) → (0,0) → (∞,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 1 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We note that as the final point is at spatial infinity, it can be displaced to time 0 – this in fact implements the correct physics of the entanglement entropy due to the single boundary at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, we may represent the one-point function as 〈Ψα|T α(0, t)|Ψα〉 ≍ 〈Ψα|T α(0, t)¯T α(0,0)T α(0+,0)|Ψα〉 (80) where the factors T α(0, t)¯T α(0,0) represent the segment of path (0, t) → (0,0), and the factor T α(0+,0), the segment (0,0) → (∞,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is valid as an asymptotic relation for large t, where the UV singularity due to the proximity of the fields ¯T α(0,0) and T α(0+,0) (which occurs because of renormalisation effects) is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We simplify the expression (80) in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' First, we note that the segment of path (0,0) → (∞,0) does not provide any contribution to the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is because we may re-write the branch-point twist field T α(0+,0) as is done in 21 SciPost Physics Submission Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3, but in the basis of the before-quench canonical free fermions of the replica theory, ˜ ψi(x,0), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The exchange relations (45) (here at t = 0) hold for any field ai(x,0), and in particular hold for ai(x,0) = ˜ ψi(x,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, by the same arguments, we obtain a decomposition as in (53), T α = � p∈Iα ˜τα p (81) but for different U(1) twist fields ˜τα p(x,0) = exp � i � ∞ x d x′ ˜qp(x′,0) � , ˜qp(x,0) = πp α ˜ ψ† p(x,0) ˜ ψp(x,0) (82) instead of (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By (75), we have ˜ ψi(x,0)|Ψj〉 = 0 for all i, j, so it is clear that ˜qp(x,0)|Ψα〉 = 0 , (83) therefore ˜τα p(x,0)|Ψα〉 = |Ψα〉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (84) Hence 〈Ψα|T α(0, t)|Ψα〉 ≍ 〈Ψα|T α(0, t)¯T α(0,0)|Ψα〉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (85) Second, we note that along the path (0, t) → (0,0), generic observables do not have long- range correlations coming from pair productions: correlations of generic observables approach those within the final GGE fast enough, in such a way that corrections due to the quench give only sublinear corrections to cumulants of time-integrated fermion bilinears (such as conserved den- sities and currents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is because particle pairs always create correlations between points at separate spatial coordinates: it is not possible to create two co-propagating fermions, with the same momentum (here they would be with vanishing momentum, as the total momentum has to be zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This fact is discussed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' the discussion there is for a single copy, but it extends immediately to the α-copy state |Ψα〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, rewriting the branch-point twist fields in terms of U(1) τα p with branches in the time direction, using (53), and expanding in cumulants of U(1) currents, we see that on the segment (0, t) → (0,0), for the purpose of the BFT, the state is correctly described by a GGE10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' An important consequence of these arguments is that we may evaluate the twist field one-point function after a quench, as an equal-space, different-time two-point function within the GGE repre- senting the final state, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (77): 〈Ψα|T α(0, t)|Ψα〉 ≍ 〈T α(0, t)¯T α(0,0)〉ρ⊗α w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (86) This is valid at long times, and omits small-time effects that occur before generalised thermalisa- tion (which do not affect the asymptotic regime we look at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In order to apply BFT to the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' of (86) a last observation is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As the GGE is a Wick- theorem state, we can use (55), thus we are interested in the separate two-point functions of U(1) twist fields τα p with branches in the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It turns out that, as emphasised in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, the currents jp(0, t′), t′ ∈ [0, t] in GGEs have time-correlations that decay fast enough so as to give only linearly growing cumulants: this is what allows the application of the BFT (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 for 10We remark that for bosonic system, this argument would break, as pairs of particles with equal, zero momenta are emitted with a finite density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, it turns out that this correction due to the quench would not affect the cumulants of time-integrated currents, as such pairs, being of zero momentum, do not carry any current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 22 SciPost Physics Submission a full discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We note that this is not true in general of other observables: in GGEs, generic fermion bilinears have cumulants that grow faster than linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But we are intersted in the currents only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We are now in position to apply standard BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This amounts to repeating the same calculation as above in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1, but now in the purely temporal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We use the general formula (38) for the “vertical” path connecting initial and final point by choosing γ = 0, and similarly get (note that the path is in opposite direction as that of formula (38), and thus we must take h(θ) = −hp) 〈τα p(0, t)¯τα p(0,0)〉ρw ≍ exp � t Fp(−i) � , Fp(−i) = � dθ 2π|v(θ)|log � 1 + eihp sgn(θ)−w(θ) 1 + e−w(θ) � (87) where we used sgn(v(θ)) = sgn(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Again, after performing the product all two-points functions of U(1) twist fields, we get 〈T α(0, t)¯T α(0,0)〉ρ⊗α w ≍ exp{t Fα(−i)}, Fα(−i) = � dθ 2π|v(θ)|log � 1 + e−αw(θ) � 1 + e−w(θ)�α � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (88) Using Hα(θ) as defined in (71), the α−Rényi entropy reads Sα(t) = 1 1 − α log〈T α(0, t)¯T α(0,0)〉ρ⊗α w ∼ t � dθ 2π|v(θ)|Hα(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (89) This is the result obtained both from exact calculation in [15] and within the quasi-particle picture in [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Single-mode and pair-mode twist fields We have discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 the conserved quantities Qθ = ψ† θψθ, forming a “scattering" or continuous basis for the extensive conserved quantities of the free fermion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2, we discussed the replica model with α copies, and the U(1) charges Qp, which are just the integration Qp = hp � dθ Qθ,p (with hp = πp α ) over all momenta θ of the continuous basis Qθ,p = ψ† θ,pψθ,p in the Fourier-copy p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' There, we have also discussed the twist fields τα p associated to these charges, which turned out to be useful in the computation of the Rényi entanglement entropies in Subsec- tions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A natural extension of these constructions is to the twist fields associated to each conserved quantity Qθ,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As we will see, these are indeed useful in evaluating the behaviour of Rényi entanglement entropies for intervals that grow linearly with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In order to simplify the notation, we consider a single copy of the fermion, and the scattering basis Qθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' the discussion immediately adapts to the Fourier-copy p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the study of the ballistic behaviours of many-body systems, and in particular in the BFT, it is essential that the conserved charge Q considered be extensive – scale linearly with the volume (typically one requires 〈Q2〉 c ∝ L [37,45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The charges Qθ are not extensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, as they form a continuous basis, integrals on small θ-intervals are extensive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' thus it is better to define, for ε > 0 as small as desired, Qθ = � θ+ε/2 θ−ε/2 dθ ′ ψ† θ ′ψθ ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (90) These act as Qθ |θ ′〉 = Θ(ε/2 − |θ ′ − θ|)|θ ′〉 (91) 23 SciPost Physics Submission hence have one-particle eigenvalues hθ(θ ′) = Θ(ε/2 − |θ ′ − θ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (92) We show in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 that such Qθ are indeed extensive in GGEs, and we evaluate explicitly their associated densities and currents qθ(x, t) and jθ(x, t), Qθ = � d x qθ(x, t), ∂tqθ(x, t) + ∂x jθ(x, t) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (93) From this, one can immediately construct the associated twist field τθ(x, t) = exp � i � ∞ x d x′ qθ(x′, t) � (94) and, for its correlation functions, apply the corresponding BFT based on the one-particle eigen- value (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In fact, we are interested in studying the squeezed state (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is clear that this state factorises into momentum intervals as follows: |Ψ〉 = � θ∈(�+ 1 2 )ε |Ψ|θ|〉 (95) where |Ψ|θ|〉 = 1 N|θ| exp �� θ+ε/2 θ−ε/2 dθ ′ Kθ,−θψ† θψ† −θ � |0|θ|〉 (96) and we write the ground state in a naturally factorised way as |0〉 = � θ∈(�+ 1 2 )ε |0|θ|〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Likewise, we will consider the pair-mode charges Q|θ| = Qθ + Q−θ and the associated densities q|θ|(x, t) = qθ(x, t) + q−θ(x, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (97) Both act trivially (as zero) on |Ψ|θ ′|〉 if θ ′ ̸= θ (θ,θ ′ ∈ (�+ 1 2)ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' From these, we get the pair-mode twist fields τ|θ|(x, t) = exp � i � ∞ x d x′ q|θ|(x′, t) � , (98) which acts trivially (as the identity) on |Ψ|θ ′|〉 if θ ′ ̸= θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These are still U(1) twist fields, for the sub-U(1) symmetry acting on the tensor factor of modes within [θ − ε/2,θ + ε/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note in particular that the global U(1) twist field τ(x, t) associated to the total charge Q = � dθ ψ† θψθ = � d x ψ†(x)ψ(x) can be factorised as τ(x, t) = � θ∈(�+ 1 2 )ε τ|θ|(x, t) (99) and that, by factorisation of its action on the state, we have 〈Ψ|τ(x, t)τ(x′, t′)|Ψ〉 = � θ∈(�+ 1 2 )ε 〈Ψ|θ||τ|θ|(x, t)τ|θ|(x′, t′)|Ψ|θ|〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (100) Clearly, as the pair-mode twist fields act trivially on other tensor factors in the state, we may also write, more simply, 〈Ψ|τ(x, t)τ(x′, t′)|Ψ〉 = � θ∈(�+ 1 2 )ε 〈Ψ|τ|θ|(x, t)τ|θ|(x′, t′)|Ψ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (101) 24 SciPost Physics Submission 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Rényi entropies of an interval after a quench (fluctuations of single-mode den- sities and currents) We finally extend the result for the entanglement growth after a quench to a finite, but ballistically growing interval A = [0, x], with x = ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' To this aim, we should consider the following two-point correlation function in a squeezed state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (73) (or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (79) in the replicated theory): 〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉, x = ξt, t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (102) The idea is the same as that used in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, that we need to deform the integration path in such a way that, everywhere along the path, all points remain uncorrelated (on large scales), thus enabling us to apply BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The choice of the path will now depend on the values of ξ, and in fact, we will need re-write the two-point function as a product of two-point functions of pair-mode twists fields, and to choose different paths for each such two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It will simplify the discussion to already re-write the two-point function in terms of U(1) twist field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As the squeezed state is a Wick-theorem state, we can directly use (55): 〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉 = α/2 � q=−α/2+1 〈Ψ2q−1|τα 2q−1(0, t)¯τα 2q−1(x, t)|Ψ2q−1〉 (103) where |Ψp〉 is the squeezed state |Ψ〉 for the fermions ψp(x),ψ† p(x) on Fourier-copy space p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We start by considering the two asymptotic regimes: At short times (more precisely in the limit ξ → ∞ of the scaled, long-time asymptotic be- haviour), entangled particle pairs coming out from the initial state will correlate points within the original integration path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' To apply BFT, then, we deform the straight path (0, t) → (x, t) to the piece-wise straight path (0, t) → (0,0) → (x,0) → (x, t), made of three segments (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 (left)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By the same arguments as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, the space-like segment will not contribute to the entanglement growth, and the time-like segments will give separated, identical contributions given by the long-time GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We are thus left with the contribution of the two, separate time-like segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The fact that the segments do not correlate with each other is thanks to the assumption that the GGE satisfies n(θ) → 0 as |θ| → ∞ (that is, the density of pairs produced at large momenta tends to zero), as is discussed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' At long enough times (either the limit ξ → 0 of the scaled, long-time asymptotic behaviour, or the long-time limit followed by the long-distance scaling), the particles generated from the initial state do not correlate points within the path (0, t) → (x, t): cumulants scaled by the distance x do not receive contributions from such particle pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Corrections terms to the GGE values of cumulants can only come from pairs of particles at infinitesimally small momenta, and, it turns out, such corrections become zero when the total number of correlated pairs on the interval [0, x] tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As there is at most a finite density of pairs produced per unit momenta, there remain no pairs on infinitesimally small momentum intervals11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus the asymptotic behaviour is that obtained from the long-time GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is discussed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Particle pairs of finite momenta would, of course, correlate points between the path segments (0, t) → (0,0) and (x,0) → (x, t) (also discussed in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3), 11In fact, as we are looking at fermionic models, the density tends to zero at zero momenta, but this is not required in this argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 25 SciPost Physics Submission KING’S COLLEGE LONDON P AO L A R U G G I E RO : Asymptotic regimes : “small” and “large” time ( ) Rényi entropies of finite interval after a quench : A = [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' x] ⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) ¯Tα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t)⟩ x/t → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 0 ⇒ ⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) ¯Tα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t)⟩ = { |⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) ¯Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='0)⟩|2 t ≪ x ⟨Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) ¯Tα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t)⟩ t ≫ x Sα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t) = 2t∫ dθ 2π |v(θ)|Hα(θ) t ≪ x x∫ dθ 2π Hα(θ) t ≫ x : FINITE INTERVAL AFTER A QUANTUM QUENCH Sα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t) t x Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) 0 ¯Tα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t) −vθ vθ t x Tα(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='t) 0 ¯Tα(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t) vθ −vθ Figure 2: Evolution of Rényi entropies of finite subsystem A = [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' x] within BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The integration path that we need to chose (continuous dark-gray line) in order for BFT to apply is different at short (left) and long (right) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The choice depends on which points in spacetime get correlated because of initially entangled pairs produced by the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' thus we must avoid the piece-wise straight path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore the correct way to use BFT is by using the original path (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' So we arrive to the following asymptotic results for x, t → ∞: 〈Ψα|T α(0, t)¯T α(x, t|Ψα〉 ≍ � 〈T α(0, t)¯T α(0,0)〉ρw〈T α(x,0)¯T α(x, t)〉ρw t ≪ x 〈T α(0, t)¯T α(x, t)〉ρw t ≫ x = ���〈T α(0, t)¯T α(0,0)〉ρw ��2 t ≪ x 〈T α(0, t)¯T α(x, t)〉ρw t ≫ x (104) where we used 〈T α(0, t)¯T α(0,0)〉∗ ρw = 〈T α(0,0)¯T α(0, t)〉ρw = 〈T α(x,0)¯T α(x, t)〉ρw (by the fact that T(x, t)† = ¯T(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This leads to Sα(x, t) = 1 1 − α log〈T α(0, t)¯T α(x, t)〉 ∼ � � � � � � � � � 2t � dθ 2π|v(θ)|Hα(θ) t ≪ x x � dθ 2π Hα(θ) t ≫ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (105) Namely, at short times (but much larger than microscopic times), the growth is described by the path in the purely temporal direction (89), and at long times, the system goes, uniformly as a function of the velocity x/t → 0, to the equilibrium GGE and there the result is the one of the purely spatial path (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These are, however, only asymptotic results in ξ, within the scaled regime x, t ∝ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It turns out that we can access all values of ξ = x/t within this regime, by using similar arguments, but now for the single-mode twist fields introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 (in fact, we need the pair-mode twist fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Effectively, using these, we will be able to take into account that the meaning of “short” and “long” time depend directly on the speed of the travelling particles v(θ) = E′(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We start with the decomposition of global U(1) twist fields into pair-mode twist fields (99), which we write in the replica model for each Fourier-copy p, and with single-particle charge eigen- 26 SciPost Physics Submission value hp = πp α instead of 1 in (99) (as done in (57)) τα p(x, t) = � θ∈(�+ 1 2 )ε τα |θ|,p(x, t) (106) where τα |θ|,p(x, t) = exp � i � ∞ x d x′ q|θ|,p(x′, t) � (107) and q|θ|,p(x, t) has the form (173) times hp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, by factorisation of two-point functions (101), we re-write (103) as 〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉 = � θ∈(�+ 1 2 )ε α/2 � q=−α/2+1 〈Ψ2q−1|τα |θ|,2q−1(0, t)¯τα |θ|,2q−1(x, t)|Ψ2q−1〉 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (108) Having made this re-writing, the analysis now follows that of the ξ → ∞ and ξ → 0 limits made above: there is an exact parallel for each individual two-point function 〈Ψp|τα |θ|,p(0,0)¯τα |θ|,p(x, t)|Ψp〉 (p = 2q − 1), with the only difference that it is not necessary to take the asymptotic limit in ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For each θ (and each p), the factor |Ψ|θ|,p〉 of the full state |Ψp〉, on which τα |θ|,q act non-trivially, correlates points (x, t), (x′, t′) only for |x − x′| |t + t′| ∈ [v(θ − ε/2), v(θ + ε/2)] (recall that θ ∈ (� + 1 2)ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The analysis of single-mode correlations is made in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, for ξ > 2v(θ + ε/2) correlations occur on the horizontal path (0, t) → (x, t), but no correlations occur on (0, t) → (0,0) → (x,0) → (x, t) (note that, again, the segment of path (0,0) → (x,0) does not contribute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus we must choose the latter path (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 (left)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' On the contrary, ξ < 2v(θ − ε/2), correlation occur between the segment of paths (0, t) → (0,0) and (x,0) → (x, t), but not on the horizontal path (0, t) → (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus we must choose the latter (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 (right)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In making these right choices, the correlation functions of pair-mode twist fields tend to their values in the long-time GGE, 〈Ψp|τα |θ|,p(0,0)¯τα |θ|,p(x, t)|Ψp〉 (109) ≍ � 〈τα |θ|,p(0, t)¯τα |θ|,p(0,0)τα |θ|,p(x,0)¯τα |θ|,p(x, t)〉ρw ξ > 2v(θ + ε/2) 〈τα |θ|,p(0,0)¯τα |θ|,p(x,0)〉ρw ξ < 2v(θ − ε/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note how in the first line, it is a four-point function that appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Now again, in order to apply the BFT, we need to consider the correlations between twist fields within GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We already argued in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 (with supporting calculations in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3) that no strong correlation occurs between local operators at equal times and different points in space, thus on the second line of (109) we may apply the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We also argued that no strong correlation occurs between current operators at equal space and different times, and in fact this also holds for single mode currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, in order to simplify the first line of (109), we need to address correlations between currents on the path segments (0, t) → (0,0) and (x,0) → (x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, local observables have strong correlations at different space-time points, due to hydrodynamic 27 SciPost Physics Submission modes propagating in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As we do not make any strong assumption about the disper- sion relation, all hydrodynamic velocities occur, hence correlations occur between generic local observables on these separate path segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, single-mode currents only produce hy- drodynamic modes at the corresponding velocities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' supporting calculations are found in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, as long as ξ > v(θ + ε/2), no correlation occurs between these paths for the single mode currents j±θ,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As v(θ) > 0, then ξ > 2v(θ + ε/2) ⇒ ξ > v(θ + ε/2), hence on the first line simplifies and we have 〈Ψp|τα |θ|,p(0,0)¯τα |θ|,p(x, t)|Ψp〉 ≍ � � � ���〈τα |θ|,p(0, t)¯τα |θ|,p(0,0)〉ρw ��� 2 ξ > 2v(θ + ε/2) 〈τα |θ|,p(0,0)¯τα |θ|,p(x,0)〉ρw ξ < 2v(θ − ε/2) (110) where the BFT can be applied for all two-point functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For the two-point functions with spatial separation 〈τα |θ ′|,p(0,0)¯τα |θ ′|,p(x,0)〉ρw, we can use the analysis of (60) made in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1, where we only have to replace, on the right-hand side of (60) inside the θ-integral, the constant one-particle eigenvalue hp by the piece-wise constant function hp � Θ(ε/2 − |θ − θ ′|) + Θ(ε/2 − |θ + θ ′|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus the same analysis goes through, but with the integral restricted to θ ∈ Iθ ′,ε := [θ ′ − ε/2,θ ′ + ε/2] ∪ [−θ ′ − ε/2,−θ ′ + ε/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Likewise for the two-point functions with temporal separation 〈τα |θ ′|,p(0, t)¯τα |θ ′|,p(0,0)〉ρw, with the analysis of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Putting the results together, we obtain 1 1 − α log � α/2 � q=−α/2+1 〈Ψ2q−1|τα |θ|,2q−1(0, t)¯τα |θ|,2q−1(x, t)|Ψ2q−1〉 � ∼ � Iθ,ε dθ ′ 2π min(x,2t|v(θ ′)|) Hα(θ ′) (111) which is valid for x/t < 2v(θ − ε/2) or x/t > 2v(θ + ε/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For the case of x/t within this excluded region, we do not have explicit results, but the scaled cumulants still are finite (as one can see by doing a calculation similar to App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, the result may be deemed valid as well within this excluded region, up to an error of order ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Taking the product over θ’s as per (108), 1 1 − α log〈Ψα|T α(0, t)¯T α(x, t)|Ψα〉 ≍ � dθ 2π min(x,2t|v(θ)|) Hα(θ) + O(ε) (112) and as this holds for all ε > 0, we can take the limit ε → 0 and we obtain Sα(x, t) = 1 1 − α log〈T α(0, t)¯T α(x, t)〉 ∼ � dθ 2π min(x,2t|v(θ)|) Hα(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (113) This is in full agreement with the quasiparticle picture [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 28 SciPost Physics Submission Finally, we note that the relation (3) between this formula for Rényi entanglement entropy growth, and the static and dynamic fluctuations, is directly obtained from the above discussion, by identifying JN<(t) = � t 0 dt′ � θ∈(�+ 1 2 )ε 2v(θ)(x) = � x 0 d x′ � θ∈(�+ 1 2 )ε 2v(θ)>x/t q|θ|(x′,0) (115) using the explicit expressions of pair-mode densities and currents (176) in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4, and taking the limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 5 Discussion and conclusion In this paper we have studied the Rényi entanglement entropy in GGEs and after quenches from integrable (pair-production) states in free fermion theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Although this has been relatively well studied in the literature, most results were based on specific ways of writing the Rényi entangle- ment entropy using the free fermion structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' in terms of determinants), and on the idea of entanglement due to engangled pairs produced by the quench [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A first-principle deriva- tion that generalises beyond free fermions was still largely missing, while it is known that the simple quasi-particle picture fails for α-Rényi entanglement entropies (with α ̸= 1) in interacting models [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We have proposed a new approach based on twist-field correlation functions and hydrody- namic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This uses the hydrodynamic theory for free fermions, which is a special case of generalised hydrodynamics (GHD), and the ballistic fluctuation theory (BFT), which relates the exponential decay of twist-field correlation functions to hydrodynamic large-deviation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Crucially, in order to have a full understanding of the quench dynamics, we have introduced a new concept: that of single-mode twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These are twist fields associated to the quasi-local charge counting the number of fermions within a small interval of momentum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' or more generally twist fields “acting" on the quasi-locality sector of observables supported on a small momentum interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The approach is potentially more general and more fundamental, as hydrodynamics, the BFT and single-mode twist fields – twist fields associated to individual hydrodynamic modes – are applicable much beyond free fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Perhaps most interestingly, it reveals the new physics of thermodynamic and hydrodynamic fluctuations behind the behaviour of the Rényi entanglement entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Three important concepts are brought forward: Entanglement is deeply connected to fluctuations, and the large-scale behaviour of entan- glement, both static and dynamic, is controlled by large-deviation and hydrodynamic prin- ciples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Hydrodynamic modes and projections onto such modes are more accurate and general no- tions which replace the idea of particle-pair productions used to understand entanglement dynamics in integrable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 29 SciPost Physics Submission The fact that the entanglement growth in quenches from “integrable” states can be written as a simple and universal function of the generalised Gibbs ensemble (GGE) reached at long times comes from the particularly simple structure of the long-range correlations that such states present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' To elaborate on these concepts: first we have confirmed that the Rényi entanglement entropy in GGEs is controlled by thermodynamic fluctuations, and related to (a simple analytic continua- tion of) a difference of thermodynamic free energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is in agreement with the observations, made earlier [29–31], that the large-deviation theory for charge fluctuations is closely related to the entanglement entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here, this relation appears naturally from completely general concepts: branch-point twist fields and the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Using these, in fact, the conclusion is pushed further: we show that the growth of Rényi entanglement entropy after a quench is controlled by hydrodynamic current fluctuations, and related to a dynamical free energy associated to the large-deviation the- ory for charge transport, as fully encoded in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (3) in the introduction (we mention that some qualitative arguments in this direction were already present in [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The relation between Rényi entanglement entropy and charge fluctuations is a general aspect of quadratic theories (not only free fermions, as our approach could also be generalized to free bosons), as in such theories, the branch-point twist field can be written as a product of U(1) twist fields, which are then associated, by the BFT, to the large-deviation theory of U(1) charge fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Second, the methods we have developed show that the notion of quasi-particle used in inte- grable systems to explain the behaviour of entanglement, should in fact be replaced by that of hydrodynamic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Indeed, the BFT, which describes the exponential behaviour of twist field correlation functions, is purely based on the Euler hydrodynamic data of the microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In free fermion models, and in integrable models, it turns out that hydrodynamic modes are in one-to-one correspondence with quasi-particles (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' the review [47]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' and in particular, the single-mode twist fields we have introduced, are twist fields associated with such hydrodynamic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But beyond these situations, hydrodynamic modes are the more general objects at play in the large-scale dynamics of many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Third, our calculations explain why it is possible to specify the growth and saturation of en- tanglement after quenches from pair-production states in a simple way in terms of the long-time GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is not based on the conventional physical picture of entanglement produced by entan- gled pairs of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But rather, it is based on the study of long-range correlations that such pairs give rise to after quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It has not been appreciated until now that quenches give rise to long-range spatial correlations , of the type found recently in non-equilibrium, long-wavelength states [35, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These long-range correlations are generically seen by observables supported on regions of space that are large enough;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' specifically, with a ballistic scaling of the region’s length x with respect to the time t since the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, for such observables, the state is not a GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is important, as twist fields are semi-local with respect to the fermions, thus the semi-locality branch is affected by such correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The fact that the long-time GGE can be used to describe not only the saturation of the Rényi entanglement entropies but also their growth in a simple way, is because of the particular structure of long-range correlations in integrable pair-production quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Indeed, in such quenches, for every choice of x/t, one can always choose a path in space-time which avoids all long-range cor- relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This follows from a simple geometric analysis of trajectories in space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Intuitively, long-range correlations occur between positions in space-time where pairs of correlated particles lie, a picture that we fully support by simple calculations of correlation functions of conserved den- sities and currents and their asymptotic behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Once the path avoids long-range correlation, 30 SciPost Physics Submission it only perceives the long-time GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note that it is obvious that the entanglement growth can be described purely in terms of the final GGE (with no further information from the initial state needed), because of the one-to-one correspondence between initial squeezed state (pair-production state) and GGEs, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (77) (so no more information about the initial state is present at all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The structure of long-range correlations however allow a simple description, in terms of fluctuations within the long-time GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, the one-to-one correspondence is lost in quenches from more complicated states, and the universality of the entanglement growth is also lost [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Importantly, we find that the branch-point twist field can be decomposed into tensor factors – the single-mode twist fields – that act on each small momentum interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Each small momentum in- terval is associated with a family of quasi-local operators, with respect to which branch-point twist fields can be defined12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Each such twist field is only semi-local with respect to the fermions per- taining to the single momentum interval, and, pairing intervals of opposite momenta, allowed us to separate the two-point function of branch-point twist field (used to evaluate the Rényi entangle- ment entropy) into a product of contributions on different momentum intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For each factor, the semi-locality branch can be chosen in order to avoid long-range correlations corresponding to the pair it perceives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We provided extensive calculations of correlation functions of single-mode densities and currents that support this physical picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' From these considerations we fully reproduced the long space-time dynamics of the Rényi entanglement entropy that had been obtained by pair-entanglement argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Many extensions of our work are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Most importantly, we believe the derivation we have provided, and many of the conclusions, can be extended to interacting integrable models, and potentially to interacting non-integrable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In interacting integrable models, it would be interesting to reproduce, and provide a better understanding of, the recent result [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This was obtained using crossing symmetry of relativistic quantum field theory, in order to relate Rényi entanglement entropy growth in time to the linear scaling of Rényi entanglement entropy in space, much like one can evaluate currents by crossing from conserved densities [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, as we have argued, the more general understanding of time-extensive behaviours is via hydrodynamic modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, it is likely that hydrodynamic ideas will provide a more first-principle derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Technically, in interacting models, it is not possible to factorise the branch-point twist fields, in such a simple way as in [7], into U(1) twist fields, a trick that we have used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We believe this difficulty can be surmounted as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' First, it is still possible to diagonalise the twist field action, at the price of making the resulting S-matrix non-diagonal, see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The branch-point twist field is then associated with a symmetry that has diagonal action on the new asymptotic particles, and hydrodynamic modes can be constructed from these particles (by constructing the corresponding nested thermodynamic Bethe ansatz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Having twist fields associated to charges that are diagonal in the particle basis, the results of the BFT for generic intergable models can in principle be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is also immediate that single-mode twist fields exist as well in interacting integrable models, which would be interesting to study, independently form their applications to entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Another avenue is to use the ballistic macroscopic fluctuation (BMFT) theory developed re- cently [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is a more general construction which does not involve a flow on GGEs (by constrast to the BFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This would allow us to apply the principles introduced here – the relation 12In our calculation, we used the explicit free-fermion basis to first write the branch-point twist fields in terms of U(1) twist fields by the standard arguments of [7], which we then factorised into single-mode twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But the concept remains valid without the mapping to U(1) twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 31 SciPost Physics Submission between hydrodynamic fluctuations and entanglement entropy using twist fields – beyond homo- geneous quantum quenches, and beyond the simple particle-pair quenches, as the BMFT is appli- cable to inhomogeneous situations and for generic long-rance correlation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, it would be interesting to account for the long-range correlations not by choosing paths that avoid them, but by evaluating directly their influence on the fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This would be important, as initial states that are inhomogeneous generically produce long-range correlations [35, 38] that are not necessarily of particle-pair type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This could also access quenches where multiple-particle processes are involved [32,33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Beyond integrability, perhaps the main results are those based on a notion of “surface tension" underlying entanglement entropy in chaotic systems [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It would be interesting to apply our methods, based on hydrodynamics, to such situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A natural further extension is to introduce the effects of diffusion, in particular using the exact results in integrable models [49], and potentially the effects of dispersion [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Finally, there is no reason to restrict ourselves to entanglement entropy, or to quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The methods we have introduced here are immediately applicable, for instance, to the large- deviation theory of U(1) densities and currents after quenches in interacting models, both at and away from integrability, and both for quantum and classical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This would be a very interesting direction to investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Acknowledgements We thank Vincenzo Alba and Olalla Castro-Alvaredo for discussions and collaborations on closely related subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Funding information The work of BD was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grants EP/W000458/1 and EP/W010194/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A Remarks on notions of locality and twist fields There is a lot more that one can say about the twist fields we introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4, as well as the notions of locality we briefly discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here we collect a number of remarks in order to provide a brief and rough guide to this wide subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Unbounded observables and topological charges As is made clear from the height field definition of twist fields, a twist field may exist as soon as there is an observable, say ϕ(x, t), that is “unbounded": that takes values in a non-compact space which are not bounded by the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' When this happens, the field can grow from a value it has at x, t to another well separated value at x′, t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' If this growth is linear, then from this we can construct a twist field as done in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4, with a large-deviation theory described by the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This usually happens when there is a non-compact symmetry13, such as the � symmetry group of the sine-Gordon model if the sine-Gordon field is taken in �, or the � symmetry group 13This is essentially a “gauge symmetry": non-compactness is the main aspect of gauge invariance that makes is different from oridnary symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 32 SciPost Physics Submission of the real free massless Boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In such cases, ϕ(∞) − ϕ(−∞) is a “topological charge", it is an extensive conserved quantity with density q(x, t) = ∂xϕ(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The vertex operators e−iηϕ(x,t) are the associated twist fields, and the extensive charge ϕ(∞)−ϕ(−∞) should be considered as part of the space of extensive charges Qi used to construct GGEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It will appear after appropriate quenches via (generalised) thermalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Descendant twist fields and semilocality sectors The exchange relations (31) are a good way of characterising twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, they char- acterise not a single field, but a family of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Indeed, clearly, the identification (34) is not the unique choice satisfying (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For instance, T(x, t) = a(x, t)T−iη(x, t) will also work, for any local observable a(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The choice (34) may be seen as a “highest-weight" twist field, and the above are usually referred to as “descendant twist fields" (these notions make full sense, for instance, in quantum or conformal field theory, using the concept of dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' All such descendants are in the same “semilocality sector" T defined by the exchange relation (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' One application of the BFT to descendant twist fields is explained in [41] in the context of the XX quantum chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 Non-abelian semilocality Given two “local enough" symmetry transformations σ and σ′, that is a(x, t) �→ σa(x, t) and a(x, t) �→ σ′a(x, t), we can define two semilocality sectors Tσ and Tσ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In general, if the transfor- mations do not commute, one has that if T ′ ∈ Tσ′, then σT ′ ∈ Tσ◦σ′◦σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Further, if T ∈ Tσ and T ′ ∈ Tσ′, then the exchange relation takes the form T ′(x, t)T(y, t) = � T(y, t)(σ−1T ′)(x, t) (y ≪ x) (σ′T)(y, t) T ′(x, t) (y ≫ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (116) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Twist fields in the literature It is difficult to give a full account of the literature on twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As a guidance we mention that twist fields and their semilocality have been discussed extensively in various contexts, including: phase transitions in classical and quantum statistical models [51–53] (see the review [54]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' vertex operators, Yangians, parafermions and orbifolds in conformal and integrable quantum field the- ory [55–59];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' tau-functions and Painlevé equations [60–67];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' and entanglement entropy in quan- tum field theory and in quantum spin chains [7,23,68,69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Twist fields have also been considered in higher dimensions [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In most works, the focus is on ultra-local “internal" symmetries, that strictly factorise in space, usually part of a symmetry group such as �n, U(1), SU(n), permutations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note that for ultra-local symmetries, large inequalities ≪, ≫ can be replaced by ordinary in- equalities <, > in (31) for finite distances between the supports of the observables involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is the usual way of writing the exchange relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' More recently, twist fields associated to space- time boost transformations in QFT, that are not ultra-local, have been considered [71];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' this is an example where the hamiltonian density is not preserved by the transformation, ˜h(x, t) ̸= h(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 33 SciPost Physics Submission A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Concepts of locality in the literature The concept of “locality" has been discussed widely in the literature, under a variety of definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In relativistic QFT, local fields are those that commute at space-like distances with the energy- momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is important to remark that under this general definition, local fields include twist fields associated to internal symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This definition can in fact be used in any quantum model, be it a field theory, spin chain or model of interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In fact, one defines “locality sectors" containing families of local fields that commute with each other at space-like distances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' and a distinguished locality sector is that containing the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These considerations are at the basis of orbifold conformal field theory [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In spin chains, the most naïve concept is that of operators supported on finitely many sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the C∗ algebra formulation, this is completed with respect to the operator norm [36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' importantly, the property of operators commuting in the limit of large separations is preserved by this comple- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Part of these C∗ algebra elements are the “quasi-local operators" that have been introduced in order to describe generalised thermalisation in integrable models [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These are elements of the C∗ algebra for which one can still define a finite support, but only up to corrections of exponentially decaying norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But much like in QFT, twist fields, which are semilocal with respect to generic observables but may be local with respect to some family of observables including the energy density, can also be adjoined to the C∗ algebra, as is natural to do for instance in the context of Jordan- Wigner transformations [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, either from the C∗ algebra, or from some potentially smaller algebra of observables deemed local (for instance with appropriate decay of correlation functions, and which, again, may include twist fields), other completions are possible, and sometimes more physically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For instance, the Gelfand-Naimark-Segal Hilbert space with respect to a given state, and its space of bounded operators, both are usually larger than the C∗ algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Another Hilbert space completion is that based on susceptibilities [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This gives the concept of “extensive" quantities, a generalisation of quantities written as sums over space of local operators (or “local densities").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These form a Hilbert space, a priori without any algebraic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular, it has been shown [37] that extensive conserved quantities are in bijection with “pseudolocal charges", roughly defined by their extensivity property 〈Q2〉c ∝ L in a system of length L [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Extensive conserved charges form a complete set which has been rigorously shown to fully describe the linearised Euler hydrodynamics [40], and they may be used to more formally and precisely construct GGEs (addressing convergence issues) [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Despite all these studies, the full relation between extensive conserved quantities, twist fields, local symmetry transformations and GGEs is still not fully unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B Correlations after a quench from an initial state with pair structure In this appendix we provide supporting argument for the choice of the integration path for eval- uating the SCGF made in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We recall that the choice of path is dictated by two main ideas: The production of pairs of quasi-particles by the initial state by the after-quench dynamics gives rise to long-range correlations: points reached by a pair of quasi-particles with oppo- site momenta are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These quasi-particles have the interpretation of fluid modes, and such long-range hydrodynamic correlations are akin to those seen in the Ballistic Macro- scopic Fluctuation Theory (BMFT) [35,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 34 SciPost Physics Submission BFT [24] is applicable only when multi-point correlation functions of the densities and cur- rents, the integrand in (12), cluster fast enough along the chosen contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Otherwise, de- pending on the structure of such correlations, the SCGF and the cumulants may be divergent under ballistic scaling, or the BFT result may receive additional contributions from these cor- relations which have to be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The more general BMFT [35] in principle provides the corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, it is simpler to directly apply the BFT by choosing con- tours that avoid such correlations, leading to the paths shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Below, we explicitly show the presence/absence of such long-range correlations along the different paths considered in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We use the notations a(x, t) and b(x, t) for free fermionic fields with the usual normalisation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' {b†(x), b(y)} = δ(x − y), {b† θ, bα} = δ(θ − α), b(x, t) = � dθ � 2π eixθ bθ(t) (117) where bθ(t) = e−iE(θ)t bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The global U(1) charge is Q = � dθ b† θ bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (118) Note that in the main text a variety of canonical free fermion fields were defined: the original fields ψ(x, t), the replicated ones parametrised by a copy index ψi(x, t), and the fields obtained from these by diagonalising in copy space, ψp(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These all are canonical free fermion fields (independent from each other for different copy number i, or for different diagonalised copy number p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The calculations below therefore apply to any such choice of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As we discuss quenches, in this appendix we use two consecutive letters for the canonical free fermion fields: a(x, t) (for the pre-quench fermion) and b(x, t) (for the post-quench fermion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 Global U(1) densities and currents and decay of correlations in GGEs First, recall the definition of the generalised current as a line integral (12) ∆J(γ) = � ℓ (j(x, t)dt − q(x, t)d x) (119) where j(x, t) and q(x, t) are the current and the density associated the U(1) conserved charge Q in (118).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We recall also that the above integral only depends upon the end points of the path ℓ due to the conservation law relating current and density, ∂tq(x, t) + ∂x j(x, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In free models with global U(1) symmetry the fermionic Hamiltonian is of the form H = � dθ E(θ)b† θ bθ (120) where E(θ) is the dispersion relation (recall that E(θ) = E(−θ) is a strictly convex function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The local charge density is given by q(x, t) = b†(x, t)b(x, t) (121) and it can be easily verified by using the fermionic algebra that Q = � d x q(x, t) is conserved, [Q, H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us find the associated local current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 35 SciPost Physics Submission By the conservation law we have (we suppress the time dependence as all fields are the same time t) ∂x j(x) = −∂tq(x) = i[qx, H] = i � dθ E(θ) � b†(x)b(x), b† θ bθ � = i � dθ dk � 2π dk′ � 2π E(θ)e−ix(k−k′) � b† kbk′, b† θ bθ � = i � dθ dk 2π E(θ) � e−ix(k−θ)b† kbθ − e−ix(θ−k)b† θ bk � = i � dθ dk 2πeix(k−θ) (E(k) − E(θ)) b† θ bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (122) Integrating with respect to x we find j(x, t) = 1 2π � dθdk eix(k−θ) � E(k) − E(θ) k − θ � b† θ(t)bk(t) (123) where the x-independent integration constant is chosen in such a way that the result is a local observable14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is the known expression for the current in the case of a quadratic dispersion relation in the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Actually, restricting the integration over momenta in [−π,π] and taking E(k) = cos(k) one reproduces also the current on the lattice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' but here we keep θ, k ∈ � for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We now show that in a GGE, the connected correlation functions of densities decay fast enough in space, and the correlation functions of currents decay fast enough in time, in such a way that scaled cumulants are finite, thus making the BFT applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The former in fact is valid for all local observables, while the latter only hold for the currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For simplicity, here and in the following subsections, we will concentrate solely on two-point correlation functions – although all higher-point functions (and their respective cumulants) should in principle be investigated similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let 〈·〉 be a GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us assume that the occupation function n(θ) characterising the GGE is analytic in a neighbourhod of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Using 〈b† θ bθ ′〉 = δ(θ − θ ′)n(θ) (124) we have, on the one hand, 〈b†(x)b(0)〉 = � dθ 2π e−ixθ n(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (125) For x > 0 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' x < 0), contour deformation can be performed as θ �→ θ −iγ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' θ �→ θ +iγ) for γ > 0 small enough, and we see that the resulting integral decays exponentially as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This implies exponential decay of all two-point connected correlation functions of local observables formed out of sums of products of b(x), b†(x) and their derivatives, including U(1) densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It 14This in fact fixes the result up to an overall term proportional to the identity operator 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' indeed there are no x-independent homogeneous local operators, whose space-time translations are generated by the momentum and Hamiltonian, other than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 36 SciPost Physics Submission also implies linear scaling of cumulants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' for instance this would mean � X 0 d x � X 0 d x′ 〈b†(x)b(x′)〉 = � X 0 dx � X 0 d x′ � dθ 2π e−i(x−x′)θ n(θ) ∼ � X 0 dx � X 0 d x′e−γ|x−x′| ∼ X (126) where in the last line we have shifted θ �→ θ −isign(x − x′)γ and used sign(x)x = |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is the correct ballistic growth of the cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' On the other hand, we find 〈b†(0, t)b(0,0)〉 = � dθ 2π eitE(θ)n(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (127) This has a stationary phase at θ∗ such that E′(θ∗) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' this point is unique by our assumption of strict convexity (and θ∗ = 0 by symmetry, although we don’t make use of this fact in this calculation), so a saddle point analysis gives 〈b†(0, t)b(0,0)〉 ∼ � i eitE(θ∗) n(θ∗) � 2πt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (128) Therefore, correlation functions of generic local observables o(x, t), o′(x, t) formed out of bilin- ears of creation and annihilation operators have algebraic decay 〈o(0, t)o′(0,0)〉c = O �1 t � (t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (129) For such decay, cumulants of total time integrals do not grow linearly, 〈 � T 0 dt o(0, t) � T 0 dt′ o′(0, t′)〉 c ≫ T (T → ∞) (130) thus breaking the large-deviation principle at the basis of the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, an important re- mark is that this generic behaviour of fermion bilinears does not hold in the case of currents, o(x, t) = o′(x, t) = j(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Indeed, using (123) with x = 0, we see that we must set θ = k = θ∗ for the long-time limit of the current two-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' From E(k) − E(θ) k − θ = E′(k) + O(k − θ) (131) we realise that E(k)−E(θ) k−θ �� k=θ=θ∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, the current two-point function decays faster than 1/t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' in fact it decays as 〈j(0, t)j(0,0)〉c = O � 1 t3 � (t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (132) This guarantees the correct scaling of cumulants 〈 � T 0 dt j(0, t) � T 0 dt′ j(0, t′)〉c = O(T) (T → ∞) (133) 37 SciPost Physics Submission and thus the validity of the BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' A similar argument shows that the current perpendicular to the path ℓ – the integrand in (119) – has a similar property along the path, thus guaranteeing that the clustering requirement (21) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='2 Quench protocol and initial state In order to describe the quench protocol considered in the main text (for which we obtain predic- tions on the dynamics of the entanglement entropy) we define the pre-quench and the post-quench fermionic Hamiltonians respectively as H0 = � dθ E0(θ)a† θ aθ (134) H = � dθ E(θ)b† θ bθ (135) where again the fermions satisfy {aθ, a† θ ′} = δ(θ − θ ′), {bθ, b† θ ′} = δ(θ − θ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' According to the protocol, the system is initialized in the ground state of H0 and then let evolve with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This corresponds to changing the whole dispersion relation (not only a parameter as in typical quenches in the literature), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' [44,73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The two set of fermions are related by a Bogolioubov-type transformation in the following way � aθ a† −θ � = � fθ gθ g∗ −θ f ∗ −θ �� bθ b† −θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (136) Imposing the validity of anticommutation relations one gets the following constraints on the func- tions fθ, gθ fθ g−θ + f−θ gθ = 0 (137a) |fθ|2 + |gθ|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (137b) Note that the first of these is identically satisfied choosing fθ = f−θ and gθ = −g−θ or viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In our analysis we will keep these functions general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The initial state |Ψ〉 is defined as aθ |Ψ〉 = 0 (138) which, for instance, could be a filled Fermi sea so that operators aθ are to be interpreted as creating excitations on top of these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It can be easily shown that in terms of post-quenches quantities this is described by the following squeezed state |Ψ〉 = 1 N exp � 1 2 � dθ Kθ,−θ b† θ b† −θ � |0〉 (139) where |0〉 is the ground state of the post-quench Hamiltonian satisfying bθ |0〉 = 0 (this state has the nice property of being gaussian so that Wick’s theorem applies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The function Kθ,θ ′ = −Kθ ′,θ can be related directly to the functions fθ and gθ appearing in (136) using the fact that |Ψ〉 is annihilated by aθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In terms of post-quench operators, this condition reads (fθ bθ + gθ b† θ)exp � 1 2 � dθ ′Kθ ′,−θ ′ b† θ ′ b† −θ ′ � |0〉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (140) 38 SciPost Physics Submission Using the BCH formula eABe−A = e[A,◦]B we obtain exp � −1 2 � dθ ′Kθ ′,−θ ′ b† θ ′ b† −θ ′ � bθ exp � 1 2 � dθ ′Kθ ′,−θ ′ b† θ ′ b† −θ ′ � = bθ + 1 2Kθ,−θ b† −θ − 1 2K−θ,θ b† −θ = bθ + Kθ,−θ b† −θ (141) which used in (140) in combination with bθ |0〉 = 0 gives [fθ(bθ + Kθ,−θ b† −θ) + gθ b† −θ]|0〉 = 0 (142) so that the condition is Kθ,−θ = − gθ fθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (143) Note in particular that we must have, by the anti-symmetry Kθ,−θ = −K−θ,θ, g0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (144) Finally, we may evaluate the predicted long-time GGE for the quench simply by evaluating the post-quench conserved quantities in the pre-quench vacuum state |Ψ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Inverting (136), we write bθ = f ∗ −θ aθ − gθ a† −θ fθ f ∗ −θ − gθ g∗ −θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (145) For this computation, it will be convenient to look at the values of the extensive conserved quanti- ties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' hence we take a finite system of length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is warranted, as the quench is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The above description of the quench stays valid with the discretisation θ ∈ �2π/L, and with the usual canonical anti-commutation relations with regularisation δ(θ −θ ′) → δθ,θ ′ L 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We consider b† θ bθ, and obtain, after some algebra using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (137), b† θ bθ = a† θ aθ|f−θ|2 + a−θ a† −θ|gθ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (146) Using 〈Ψ|a† θ aθ|Ψ〉 = 0 and 〈Ψ|a−θ a† −θ|Ψ〉 = L 2π, we get 〈Ψ|b† θ bθ|Ψ〉 = L 2π|gθ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (147) But also, in a GGE with density matrix ρw, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5, we have 〈b† θ bθ〉 = L 2πn(θ), thus we identify n(θ) = |gθ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (148) Again using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (137), we obtain 1 |fθ|2 = 1 |gθ|2 |g−θ|2 1 − |g−θ|2 (149) and therefore, from (37) and (148), |Kθ,−θ|2 = |K−θ,θ|2 = |g−θ|2 |f−θ|2 = |gθ|2 1 − |gθ|2 = e−w(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (150) 39 SciPost Physics Submission Note that in our analysis of GGEs, we assume that n(θ), thus |g(θ)|2, has an analytic extension to a neighbourhood of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This analyticity property is not true of the function w(θ), which must have a singularity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' logarithmic) at θ = 0 because of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (144).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We also assume that n(θ) → 0 as |θ| → 0, and thus this must also be true for g(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Below we report for completeness all the relevant elementary correlation functions of fermionic operators after the quench, and their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Those will be used in the following subsections, where evaluating current and density correlations, which are bilinears in the fermions (so appli- cation of Wick theorem requires only the knowledge of those).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In real space we define Gb†b x y (t,s) = 〈Ψ| b†(x, t)b(y,s)|Ψ〉 Gb†b† x y (t,s) = 〈Ψ| b†(x, t)b†(y,s)|Ψ〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (151) and similarly for their hermitian conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Going to momentum space, these take the form Gb†b θθ ′(t,s) = 〈Ψ| b† θ(t)bθ ′(s)|Ψ〉 = eiE(θ)(t−s)|gθ|2δ(θ − θ ′) (152) Gb†b† θθ ′ (t,s) = 〈Ψ| b† θ(t)b† θ ′(s)|Ψ〉 = −eiE(θ)(t+s) fθ g∗ θδ(θ + θ ′) (153) Gbb θθ ′(t,s) = 〈Ψ| bθ(t)bθ ′(s)|Ψ〉 = −e−iE(θ)(t+s)gθ f ∗ θ δ(θ + θ ′) (154) Gbb† θθ ′(t,s) = 〈Ψ| bθ(t)b† θ ′(s)|Ψ〉 = e−iE(θ)(t−s)|fθ|2δ(θ − θ ′) (155) where in particular |g(θ)|2 = n(θ) = 1 − |f (θ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note the following symmetries Gb†b θθ ′(t,s) = δ(θ − θ ′)eiE(θ)(t−s) − Gbb† θ ′θ (s, t) (156) (Gb†b x y (t,s))∗ = Gb†b y x (s, t) (157) Gb†b x y (t,s) = � dθ 2π e−iθ(x−y)+iE(θ)(t−s) − Gbb† y x (s, t) (158) so that at equal times Gb†b x y (t, t) = δx y − Gbb† y x (t, t) = δx y − (Gbb† x y (t, t))∗ (159) and also (Gb†b† θθ ′ (t,s))∗ = Gbb θ ′θ(s, t), (Gb†b† x y (t,s))∗ = Gbb y x(s, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (160) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 Approach to the GGE In the previous Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' we have evaluated the GGE ρw corresponding to the initial state |Ψ〉 simply by evaluating the averages of the mode occupation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here we analyse a bit more in detail how the GGE is approached in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We first note that 〈Ψ| b† θ bθ ′ |Ψ〉 = 〈b† θ bθ ′〉ρw = δ(θ − θ ′)n(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (161) Thus, by Wick’s theorem, the only difference between averages in |Ψ〉 and in 〈·〉ρw come from the contraction 〈Ψ| bθ bθ ′ |Ψ〉 (162) and its complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus we evaluate 〈Ψ| b(x, t)b(x′, t′)|Ψ〉 in three main situations that are important for our analysis: t = t′, x ̸= x′ (for the cumulants of space-integrated conserved 40 SciPost Physics Submission densities), and x = x′, t ̸= t′ (for the cumulants of time-integrated currents) and x ̸= x′, t ̸= t′ (for analysing the correlation between the spatially separated time-integrated currents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the first case, we have, using (145),(137), and the definition in (151) Gbb x x′(t, t) = − � dθ 2π ei(x−x′)θ−2itE(θ) f ∗ −θ g−θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (163) Consider t → ∞ with x, x′ fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then there is a stationary phase at θ∗ : E′(θ∗) = 0, with a resulting integral ∝ 1 �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, this decays as t → ∞: for every two-point functions on intervals that stay finite, the GGE is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We notice that as g−θ∗ = 0 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (144)), for fermion two- point functions, the approach is proportional to 1/t3/2 instead of 1/�t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' and for multilinears of fermions, the approach is faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' But we are interested in the scaling x, x′, t ∝ ℓ → ∞, (x − x′)/t = ξ, for which the expo- nential has a stationary phase at θ∗ = θ∗(ξ) : E′(θ∗) = ξ/2, with a resulting integral ∝ 1/ � ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In charge-neutral fermion bilinears, such as those involved in densities and currents, two such contractions will be multiplied with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus we have, for instance, 〈Ψ|q(x, t)q(x′, t)|Ψ〉c = 〈q(x, t)q(x′, t)〉c ρw + C(ξ)(tℓ)−1 + O(ℓ−2), (164) thus the correction is O(1/ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Then, for the cumulant we have 〈Ψ| � ℓX 0 d x q(x,ℓt) � ℓX 0 d x′ q(x′,ℓt)|Ψ〉c = ℓ2 〈Ψ| � X 0 d x q(ℓx,ℓt) � X 0 d x′ q(ℓx′,ℓt)|Ψ〉c ∼ 〈 � ℓX 0 d x q(x,ℓt) � ℓX 0 d x′ q(x′,ℓt)〉c ρw + O(ℓ) where the correction O(ℓ) is ℓ � X/t −X/t dξ(X −2ξt)C(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, the correction due to the quench changes the linearly scaling part of the cumulant, hence modifies the scaled cumulant from its GGE value (recall that the scaled cumulant is obtained by dividing by ℓX, and taking the large ℓ limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here it would be possible to evaluate explicitly this modification, however it is not necessary for our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The modification due to the quench comes from pair productions – this will be made much clearer when we study the single-mode densities and currents below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In fact, there is one limit where it is useful to evaluate this correction term: the limit X/t → 0 of ℓX-scaled spatially-integrated densities as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The result for the correction is explicitly lim X/t→0 t X � X/t −X/t dξ �X t − 2ξ � C(ξ) = 0 (165) as C(ξ) is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, in this limit we recover the GGE result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is in agrement with taking first the long-time limit of the finite-interval cumulant, then the limit of the scaled cumulant on a long interval (this means that the limit X/t → 0 is in fact uniform in t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the second case, where we can set x = x′ = 0, we find, with E′(θ∗) = 0 and a saddle point analysis (again remeber the definitions in (151)), Gbb 00 (t, t′) = − � dθ 2π ei(t+t′)E(θ) f ∗ −θ g−θ ∼ � i ei(t+t′)E(θ∗) f ∗ −θ∗ g−θ∗ � 2π(t + t′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (166) 41 SciPost Physics Submission As E(θ) is symmetric, this is θ∗ = 0, and then, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (144), the result vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, Gbb 00 (t, t′) = O � 1 (t + t′)3/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (167) Hence, the corrections to cumulants of charge-neutral bilinears involve � T 1 dt � T 1 dt′ 1 (t + t′)2 = O � 1 T 3 � ≪ T (T → ∞) (168) (where the lower boundary does not matter for the large-T analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This correction is sublinear, therefore the quench does not affect cumulants of equal-position time-integrated quantities: for these, the GGE is reached quickly enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The lack of modification due to the quench comes from the lack of pairs of particles produced at equal (zero) momenta, due to the fermionic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We remark that if there were particles created at zero momenta (for instance, for bosonic systems), then, still by a calculation similar to that of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (131)-(133), the correction due to the quench would vanish for cumulants of total currents, which are in any case the objects of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, the fact that pairs of particles of zero momenta are not produced, is not an essential aspects of our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Finally, we may also analyse time-integrated currents at two different points in a similar way as above, finding: 〈Ψ| � ℓT 0 dt j(ℓx, t) � ℓT 0 dt′ j(ℓx′, t′)|Ψ〉c ∼ 〈 � ℓT 0 dt j(ℓx, t) � ℓT 0 dt′ j(ℓx′, t′)〉c ρw + O(ℓ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This case is necessary for the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' With ξ = (x − x′)/(t + t′), the saddle point leading to the O(ℓ) correction is at θ∗ : E′(θ∗) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, the correction due to the quench again changes the linearly scaling part of the two-point cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Here, the limit ξ → ∞ is interesting, and easy to evaluate: as ξ → ∞, the saddle point will be at θ∗ → ∞, and we only have to use the fact that gθ → 0 as |θ| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, the correction vanishes as ξ → ∞, and we may use the GGE result, where scaled cumulants of time-integrated currents become sums of cumulants at x and at x′ in the GGE (which take the same values by translation invariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 Single-mode density and currents and decay of correlations in GGEs In the main text, when studying the dynamics of the entanglement of an interval after a quench (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5), we are interested in the single-mode twist fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This requires constructing densi- ties and currents not only for the global U(1) charge as done above, but also for the individual conserved quantities b† θ bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These conserved quantities are not extensive – they are not integrals of local or quasi-local observables – however, as reviewed in [47] in the more general context of integrable models, they form a scattering basis for such extensive quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Thus, integrations over small θ-intervals give extensive conserved quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' These are the single-mode conserved quantities that we now investigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' As we are working directly in the thermodynamic limit and, in the continuum of space, the momenta fill the real axis [−∞,∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us write this as a union of disjoint intervals centered at equispaced "target" momenta: ∪∞ i=−∞Aθi where Aθ = [θ − ε/2,θ + ε/2) and θi = (i + 1/2)ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We can write the total charge as Q = � d x b†(x)b(x) = � dθ b† θ bθ = ∞ � i=−∞ Qθi, Qθ = � Aθ dθ ′b† θ ′ bθ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (169) 42 SciPost Physics Submission Clearly each “regularised" (by ε) single-mode charge Qθ is conserved, [Qθ, H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It is also extensive: in a GGE in a finite volume L, we have 〈Q2 θ〉 c ∝ L: 〈Q2 θ〉 c = � θ+ε/2 θ−ε/2 dθ ′dθ ′′ δ(θ ′ − θ ′′)2n(θ ′)(1 − n(θ ′)) = L 2π � θ+ε/2 θ−ε/2 dθ ′n(θ ′)(1 − n(θ ′)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (170) As mentioned, if we want to write a density in real space for each b† θ bθ, we will get something non local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, Qθ’s have quasi-local densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We seek a function fθ(x, y) such that � d xd y b†(x)b(y)fθ(x − y) = Qθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (171) Going to Fourier space, one can show that (see (117)) fθ(z) = sin( εz 2 ) πz eiθz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (172) The corresponding regularised single-mode density, parametrised by the momentum, and one choice of the density (the only hermitian and PT symmetric one), is given by qθ(x, t) = � dz b†(x + z/2, t)b(x − z/2, t)fθ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (173) In terms of Fourier modes, this takes the form qθ(x, t) = � dkdk′ 2π eix(k′−k)ϑ �ε 2 − ���k + k′ 2 − θ ��� � b† k(t)bk′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (174) As [Qθ, H] = 0, the density qθ(x, t) has an associated current satisfying a continuity equation and by a calculation analogous to (123) one finds jθ(x, t) = � dkdk′ 2π eix(k′−k) � E(k′) − E(k) k′ − k � ϑ �ε 2 − ���k + k′ 2 − θ ��� � b† k(t)bk′(t) (175) where ϑ(x) is the Heaviside theta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is basically the same as (123) with a restriction on the q integration around the target mode θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For convenience, in fact we will consider the momentum-pair densities and currents q|θ|(x, t) = qθ(x, t) + q−θ(x, t), j|θ|(x, t) = jθ(x, t) + j−θ(x, t) (176) associated to a pair of opposite momenta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' these are the ones used in the twist field decomposition (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We now analyse the behavior of two-point functions on GGEs following the same lines of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is because, again, the original BFT was defined for expectation values on GGEs and, eventually, we would like to replace with those the expectations on the initial state before the quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Later, we will study more carefully the approach to the GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In a GGE (characterized by Boltzmann factor w(k)), the only surviving elementary correlator is 〈b† k(t)bk′(s)〉 = eiE(k)(t−s)δ(k − k′)n(k) (177) with n(k) = (1 + ew(k))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 43 SciPost Physics Submission We now consider the connected correlation functions of densities and currents along the paths of interest and study under which condition they decay fast enough in such GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The two paths of interest are the horizontal path (0, t) → (x, t), and the piecewise linear path (0, t) → (0,0) → (x,0) → (x, t) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For the first case, we need to evaluate density-density correlations at equal time and different spatial points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the second case, instead, we need to evaluate current- current correlations both between the two different vertical segments and within the segments themselves (using the same arguments of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, one can show that the contribution of the horizontal segment vanishes, so we do not need to look at density-currents correlations as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The calculation is slightly different than in the case of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='1 because we have to deal with pair- mode densities and currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The global quantities are determined directly by the correlators 〈b†(x, t)b(y,s)〉, while here we have to analyse directly the density-density or current-current correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us start by considering the connected density-density correlation function on the horizontal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' We focus on single-mode quantities, the pair-mode ones being just linear combination of those, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', 〈q|θ|(x, t)q|θ|(0, t)〉c = 〈qθ(x, t)qθ(0, t)〉c + 〈qθ(x, t)q−θ(0, t)〉c + 〈q−θ(x, t)qθ(0, t)〉c + 〈q−θ(x, t)q−θ(0, t)〉c (178) Below, in the current subsection, all expectations 〈·〉 are on the GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let η = ±1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' we have 〈qθ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t)qηθ(x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' t)〉c = � dkdk′ 2π dqdq′ 2π eix(k′−k)ϑ �ε 2 − ���k + k′ 2 − θ ��� � × eix′(q′−q)ϑ � ε 2 − ���� q + q′ 2 − ηθ ���� � 〈b† k(t)bk′(t)b† q(t)bq′(t)〉 c = � dkdk′ 2π dqdq′ 2π eix(k′−k)ϑ �ε 2 − ���k + k′ 2 − θ ��� � × eix′(q′−q)ϑ �ε 2 − ���q + q′ 2 − ηθ ��� � 〈b† k(t)bq′(t)〉〈bk′(t)b† q(t)〉 = � dkdk′ 2π ei(x−x′)(k′−k)ϑ �ε 2 − ���k + k′ 2 − θ ��� � ϑ �ε 2 − ���k + k′ 2 − ηθ ��� � (1 − n(k′))n(k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (179) Note that all four correlators needed can be deduced from the this upon exploiting θ �→ −θ transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Apart from the overall step function which constrains one of the integrals, with the assumptiong on n(k) that we have (it is analytic on a neighbourhood of the real line, and decays rapidly enough as |k| → ∞), we can change variables to k± = k′ ± k and perform the k− integral in the complex plane by letting k− �→ k− −iγsign(x − x′) (and γ > 0) and we see that the decay is exponential making cumulants scale as in (126).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, BFT is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For the current-current correlator, we look at two generic points in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, by making use of time-translational invariance of the GGE, we can set one time to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Analogous manipulations give (we focus on x > 0) 〈jθ(x, t)jηθ(0,0)〉c = � dkdk′ 2π eix(k′−k)+it(E(k)−E(k′))ϑ �ε 2 − ���k + k′ 2 − ηθ ��� � × ϑ �ε 2 − ���k + k′ 2 − θ ��� �� E(k′) − E(k) k′ − k �2 (1 − n(k′))n(k) (180) 44 SciPost Physics Submission and we see that that in the ballistic scaling limit, with ζ = x/t fixed, the exponential has a saddle point at ∂kE(k∗) = ∂k′E(k′ ∗) = ζ so that for ζ ̸= 0,+∞ (recall eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (131) and the one below) 〈jθ(x, t)jηθ(0,s)〉c ∼ e−i(t−s)E(k∗(ζ)) 2π(t − s) (∂kE(k∗(ζ)))2ϑ �ε 2 − ���k∗(ζ) − θ ��� � ϑ �ε 2 − ���k∗(ζ) − ηθ ��� � × (1 − n(k∗(ζ)))n(k∗(ζ)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (181) Let us take the case η = 1 : the two step functions square to one and the condition for it to vanish is k∗(ζ) ≥ ε/2 + θ or k∗(ζ) ≤ −ε/2 + θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Using ζ = ∂kE(k∗(ζ)) = v(k∗(ζ)) and the monotonicity of the velocity, we can invert the above relation, leading to a vanishing step functions whenever ζ ≥ v(θ + ε/2) = v(θ) + O(ε) or ζ ≤ v(θ − ε/2) = v(θ) − O(ε) (182) For η = −1 we obtain the same condition because we are considering ζ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This means that, under the condition (182), the leading term of correlation (181) (coming from the saddle point) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Correlations between the two vertical segments, on the contrary, arise only in a tiny cone (of order ε) around the velocity v(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' When those are present, the second cumulant resulting from (181) grows as O(T log T) at large time T (to be contrasted with (133)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note that if we consider the special case t = s ( ζ = +∞), then (181) has no saddle point and therefore its decay is exponentially fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the above calculation we assumed x ̸= 0 (ζ ̸= 0), which corresponds at looking at corre- lations between the two different aforementioned vertical paths (see again Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2, main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In order to study the decay of current-current correlations along any of such vertical paths, instead, we need to set equal space (and different times).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Again, using translational invariance we just set x = 0 in (180).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In this case we can repeat the argument in (131) and below to show 〈j|θ|(0, t)j|θ|(0,0)〉c = O � 1 t3 � (t → ∞) (183) with the corresponding cumulant scaling linearly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Finally, we recall again that the horizontal segment of the path (0, t) → (0,0) → (x,0) → (x, t) does not contribute to the cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This observation, together with (183), and when the con- dition (182) is satisfied, allows to conclude that within a GGE: (i) the SCGF associated to the two-point function of the pair-mode twist fields can be correctly evaluated along this path (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', BFT is applicable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (ii) the total SCGF factorises into those associated to the two vertical cuts, namely it is the sum of the two corresponding SCGFs (again we refer to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3 in [24]), and, in fact, in the main text, we make use of such factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='5 Approach to the GGE Finally in this subsection, following B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='3, we study in more detail the approach in time to the corresponding GGE values of the same single-mode densities and currents correlations considered in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In particular we want to understand how cumulants in the GGE’s are modified at large time when taking into account correlations coming from the initial state |ψ〉 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (139)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Below, we denote by 〈·〉 expectation values on such initial state, while 〈·〉ρw the ones on the GGE atteined at infinite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 45 SciPost Physics Submission We start with the single-mode connected density-density (again the corresponding pair-mode correlator is obtained via (178)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In momentum space we have (we focus on x > 0) 〈qθ(x, t)qηθ(0, t)〉c = � dkdk′ 2π dqdq′ 2π eix(k′−k)ϑ � ε 2 − ���� k + k′ 2 − θ ���� � ϑ � ε 2 − ���� q + q′ 2 − ηθ ���� � × 〈b† k(t)bk′(t)b† q(t)bq′(t)〉 c = � dkdk′ 2π dqdq′ 2π eix(k′−k)ϑ � ε 2 − ���� k + k′ 2 − θ ���� � ϑ � ε 2 − ���� q + q′ 2 − ηθ ���� � × � 〈b† k(t)bq′(t)〉〈bk′(t)b† q(t)〉 − 〈b† k(t)b† q(t)〉〈bk′(t)bq′(t)〉 � (184) where η = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The first piece is nothing but the GGE contribution and, as discussed in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (179) and below) is always well-behaved in the ballistic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The other part gives � dkdk′ (2π)2 eix(k′−k)+2iE(k)t−2iE(k′)tϑ � ε 2 − ���� k + k′ 2 − θ ���� � ϑ � ε 2 − ���� k + k′ 2 − ηθ ���� � (1 − n(k))n(k′) (185) In the ballistic limit, with ζ = x/t fixed, we can use again saddle point analysis, similarly to calcula- tions in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Now, however, the saddle points is given by 2v(k′)−ζ = 0 and 2v(k) = ζ (note the fac- tor 2 of difference wrt the saddle point of the integral (180)), namely k∗(ζ) = k′∗(ζ) = v−1(ζ/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' After application of saddle-point method, the ϑ function is evaluated at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For η = 1, using ϑ2 = ϑ, the result of the saddle point of (185) vanishes unless ��� k∗(ζ)+k′∗(ζ) 2 − θ ��� ≤ ε 2 or, equivalently, |v−1(ζ/2) − θ| ≤ ε/2 that it precisely v(θ − ε/2) ≤ ζ/2 ≤ v(θ + ε/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (186) Having assumed x>0, for η = −1 we recover the very same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This is exactly the expected condition for two particles of opposite momentum ±θ, initially forming an entangled pair, not to hit both the segment [0, x] at time t (as be easily understood geometrically from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' 2 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note that, for ζ fixed (and due to the above mentioned factor 2), this condition is more severe then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (182), so it also guarantees a fast enough decay of same correlation within the GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' When condition (186) does not hold, from the saddle point contribution, we get that the inte- gral (185) decays as t−1, giving a slow approach to the GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' This, in fact, modifies the behaviour of the second cumulant because it gives a correction to the GGE value which is not subleading (it is actually faster than linear in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Outside the range (186) the correction to the GGE decays fast enough, so that the associated cumulant is not modified at leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Let us briefly comment the situation in the case of correlation functions of the single-mode current even though the main idea exactly follows the density-density case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The relevant quantity in this case is 〈jθ(x, t)jηθ(0,s)〉c = � dkdk′ 2π dqdq′ 2π eix(k′−k)ϑ � ε 2 − ���� k + k′ 2 − θ ���� � ϑ � ε 2 − ���� q + q′ 2 − ηθ ���� � × � E(k′) − E(k) k′ − k �2 � 〈b† k(t)bq′(s)〉〈bk′(t)b† q(s)〉 − 〈b† k(t)b† q(s)〉〈bk′(t)bq′(s)〉 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (187) 46 SciPost Physics Submission This time we cannot use time-translation invariance as done before when computing the expec- tation on a GGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' The first part in the above expression is again the GGE contribution alayzed in (181), while the second comes from quasi-particle pairs and gives rise to saddle points in the bal- listic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' It can be checked that the condition for the saddle point contribution not to vanish is exactly the same as for the density, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='(186).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, only when the condition (186) does not hold, the contribution of the second term in (187) is subleading wrt to the GGE one, and the associated cumulant is not shifted with respect to the leading GGE value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Therefore, depending on the value of θ (apart for θ in a region of order ε, which is to trace back to our regularization of the observables) either the correlations of single-mode densities or those of single-mode currents show a fast approach to their GGE value, thus imposing the right path to choose when using BFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' C S matrix in the α−copy theory Consider S(θ,θ ′) to be the S matrix in the single-copy theory (we consider the diagonal case for simplicity, but everything below can be generalized to non-diagonal S matrices), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' |θ,θ ′〉 = S(θ,θ ′)|θ ′,θ〉 (188) (namely, it is the factor we get by exchanging θ,θ ′ in the two-particle state), then one can define in the α−copies theory S(α)(θ, i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='θ ′, i′) = δii′ mod(α)S(θ,θ ′) ± (1 − δii′ mod(α)) (189) acting on the state |θ, i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='θ ′, i′〉, where we introduced explicitly the dependence on the copy-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Importantly, the ± sign in (189) depends on the commutation relations of the fields among the copies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (189) is the S-matrix associated to α independent copies, which only describes the symmetry of the α-copies theory (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=', it does not take into account the constraints of the fields in different copies implemented by the twist fields exchange relations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (45)-(46))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However this is enough for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' By going to Fourier space in the replica index Fi→p, we have |θ, p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='θ ′, p′〉 = � p,p′ S(α)(θ,θ ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' p, p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' k, k′)|θ ′, k′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content='θ, k〉 (190) with (by simple algebra) S(α)(θ,θ ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' p, p′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' k, k′) = δmod(α)(k + k′ − p − p′) � S(θ,θ ′) ± 1 � ∓ δmod(α)(p − k)δmod(α)(p′ − k′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (191) This can be written explicitly as a 2α×2α matrix S(α)(θ,θ ′)m,n for α ∈ �, with row and column indices m = � p, p′� and n = � k, k′� respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' For example, for α = 2, it takes the form S(2)(θ,θ ′) = � �� S(θ,θ ′) 0 0 S(θ,θ ′) ± 1 0 S(θ,θ ′) S(θ,θ ′) ± 1 0 0 S(θ,θ ′) ± 1 S(θ,θ ′) 0 S(θ,θ ′) ± 1 0 0 S(θ,θ ′) � �� (192) 47 SciPost Physics Submission Note that one can check that, as expected, this S matrix satisfy the Yang-Baxter equations for general S(θ,θ ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Now, for the specific case of free fermions, consider Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (50) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' In the first basis, the fields ψi commutes among different copies: in this case above we will choose the + sign in S(α) (cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (189)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' However, after the SU(α) trasformation to the ψj basis, the fields in different copies also anticommute, and this amounts to choosing the sign − in (189).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Since for free fermions we have S(θ,θ ′) = −1, then we see that in the ψj basis, S(α) becomes diagonal (cfr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' (192)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Amico, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Fazio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/otE0T4oBgHgl3EQfZwCz/content/2301.02326v1.pdf'} +page_content=' Osterloh and V.' metadata={'source': 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a/sNFKT4oBgHgl3EQf1S6Z/content/tmp_files/2301.11919v1.pdf.txt b/sNFKT4oBgHgl3EQf1S6Z/content/tmp_files/2301.11919v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b1de33f0bf23029a26aff690b98d27bbad92592f --- /dev/null +++ b/sNFKT4oBgHgl3EQf1S6Z/content/tmp_files/2301.11919v1.pdf.txt @@ -0,0 +1,1948 @@ +INCORPORATING BACKGROUND KNOWLEDGE IN SYMBOLIC +REGRESSION USING A COMPUTER ALGEBRA SYSTEM +A PREPRINT +Charles Fox2, Neil Tran1, Nikki Nacion1, Samiha Sharlin1, and Tyler R. Josephson1,2 +1Department of Chemical, Biochemical, and Environmental Engineering, University of Maryland Baltimore County, +1000 Hilltop Circle, Baltimore, MD 21250 +2Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, +1000 Hilltop Circle, Baltimore, MD 21250 +ABSTRACT +Symbolic Regression (SR) can generate interpretable, concise expressions that fit a given dataset, +allowing for more human understanding of the structure than black-box approaches. The addition of +background knowledge (in the form of symbolic mathematical constraints) allows for the generation +of expressions that are meaningful with respect to theory while also being consistent with data. We +specifically examine the addition of constraints to traditional genetic algorithm (GA) based SR (PySR) +as well as a Markov-chain Monte Carlo (MCMC) based Bayesian SR architecture (Bayesian Machine +Scientist), and apply these to rediscovering adsorption equations from experimental, historical datasets. +We find that, while hard constraints prevent GA and MCMC SR from searching, soft constraints can +lead to improved performance both in terms of search effectiveness and model meaningfulness, with +computational costs increasing by about an order-of-magnitude. If the constraints do not correlate +well with the dataset or expected models, they can hinder the search of expressions. We find Bayesian +SR is better these constraints (as the Bayesian prior) than by modifying the fitness function in the GA. +1 +Introduction +1.1 +Symbolic Regression for Scientific Discovery +Symbolic Regression (SR) is our tool of choice, as it generates mathematical expressions that are optimized for +complexity and accuracy to a given dataset. Since John Koza pioneered the paradigm of programming by means of +natural selection, many applications for SR in scientific discovery have emerged [1]. Unlike other applications of +machine learning techniques, scientific research demands explanation and verification, both of which are made more +feasible by the generation of human-interpretable mathematical models (as opposed to fitting a model with thousands of +parameters) [2–4]. Furthermore, SR can be effective even with very small datasets (∼10 items) such as those produced +by difficult or expensive experiments which are not easily repeated. The mathematical expressions produced by SR can +easily be extrapolated to untested or otherwise unreachable domains within a dataset (such as extreme pressures or +temperatures). +For decades, SR has discovered interesting models from data in many unique applications including inferring process +models at the Dow Chemical company [5], rainfall-runoff modelling [6] and rediscovering equations describing double- +pendulum motion [7]. Symbolic regression has been applied across all scales of scientific investigation, including the +atomistic (interatomic potentials [8]), macroscopic (computational fluid dynamics [9]), and cosmological (dark matter +overdensity [10]) scales. Some techniques facilitate search through billions of candidate expressions, such as the space +of nonlinear descriptors of material properties [11]. While most applications of SR in science focus on identifying +empirical patterns in data, such "data-only" approaches do not account for potential insights from background theory. In +fact, some SR works emphasize their capabilities of discovery “without any prior knowledge about physics, kinematics, +or geometry” [7]. Nonetheless, we posit that prior knowledge need not be discarded, and in this work, we explore how +arXiv:2301.11919v1 [cs.LG] 27 Jan 2023 + +Background Knowledge in Symbolic Regression +A PREPRINT +theory may be incorporated into symbolic regression to demonstrate machine learning in the context of background +knowledge. +1.2 +Incorporating Background Knowledge into Symbolic Regression +One particularly important step towards effective use of SR in specific domains is the addition of prior knowledge. This +step has the potential to take a general purpose SR algorithm and use it to find novel models with physical meaning. For +example, AI-DARWIN is uses prior knowledge of chemical reaction mechanisms in the form of predefined functions +that a genetic algorithm may use in its search of equation space, ensuring that each generated model is mechanistically +meaningful [12]. This approach specifically encodes the prior knowledge in the form of functions available instead +of limitations on functions generated. In another recent example, Engle and Sahinidis use a deterministic symbolic +regression algorithm that constrains the space of possible equations, not to those constructed from a library of meaningful +function components, but to those functions that obey derivative constraints from theory. This improves the quality +of generated expressions for thermodynamic equations of state [13]. Another approach to incorporating background +knowledge in symbolic regression is the Bayesian Machine Scientist (BMS) [14]. BMS rigorously incorporates +background knowledge in the form of a Bayesian prior on symbolic expressions; expressions are a priori more likely if +their distribution of mathematical operators aligns with the distribution of operators in a corpus of prominent equations. +However, their approach to the Bayesian prior does not incorporate meaning from particular scientific domains. +Checking consistency of equations after the search is complete is also possible. Previously, we showed that generated +expressions can be compared to rich background knowledge (expressed as axioms for the environment under study), by +posing generated expressions as conjectures to an automated theorem prover (ATP) [15]. However, state-of-the-art +ATPs are too slow to incorporate this logical check as symbolic expressions are generated, and therefore cannot be +easily used to bias the search for equations in light of that background knowledge. Moreover, translating scientific +theories into a computer-interpretable form is not straightforward. +We address these specific drawbacks by combining symbolic regression systems (both genetic algorithm and Bayesian +approaches) with a computer algebra system that checks constraints as an equation search is conducted. This is similar +to Logic Guided Genetic Algorithms (LGGA), which uses “auxiliary truths” (ATs) corresponding to datasets in order to +weigh items in a dataset as well as augment it with more information [16]. LGGA follows an iterative approach of +training an arbitrary genetic algorithm with some dataset, augmenting that dataset with ATs, and training that algorithm +again with more informative data. An important distinction between our work and LGGA is that the dataset is not +altered in any way and the addition of extra information is performed during the execution of the GA. +1.3 +Adsorption +Adsorption, the phenomenon in which molecules bind to a surface, enables chemical processes including carbon capture, +humidity control, removal of harmful pollutants from water, and hydrogen production [17] [18] [19] [20]. Models +of adsorption enable prediction and design of engineered adsorption processes, and many have been proposed over +the years (selected equations are shown in Table 1) [21–24]. These models relate the amount adsorbed at equilibrium +as a function of pressure or concentration and are commonly expressed as equations that are either empirical or +derived from theory. For example, the Freundlich isotherm [25], is an empirical function designed to fit observed +data, the Langmuir [26] and BET [27] isotherms are derived from physical models, and the Sips [28] isotherm is +Langmuir-inspired with empirical terms added for fitting flexibility. We wonder, “What kinds of models could be +generated by a machine learning system, and what role can background knowledge play in the search for accurate and +meaningful expressions?” +2 + +Background Knowledge in Symbolic Regression +A PREPRINT +Table 1: Some well-known isotherms written as SR might find them, and their complexities. +Isotherm +Literature Expression +Symbolic Regression Form +SR Complexity +Langmuir [26] +qmaxKeqp +1+Keqp +c1p +c2+p +7 +Dual-Site Langmuir [26] +qa +maxKa +eqp +1+Ka +eqp + +qb +maxKb +eqp +1+Kbeqp +c1p +c2+p + +c3p +c4+p +15 +BET [27] +c1(p/p0) +(1−p/p0)(1−p/p0+c2(p/p0)) +c1p +c2+c3p+c3p2 +15 +Freundlich [25] +c1p +1 +n +c1pc2 +5 +Sips [28] +c1p +1 +n +1+c1p +1 +n +pc2 +c1+pc2 +9 +1.4 +Thermodynamic Constraints +We consider models to be more meaningful when they satisfy thermodynamic constraints on the functional forms +appropriate for modeling these phenomena. That is, a random equation that fits data, but does not approach zero loading +correctly, is less trustworthy outside the training data than an equation constrained to follow thermodynamics. We have +identified three constraints relevant for single-component adsorption [15]: +lim +p→0 f(p) = 0 +(1) +lim +p→0 f ′(p) < ∞ +(2) +∀p > 0 +f ′(p) ≥ 0 +(3) +Constraint 1 ensures that, in the limit of zero pressure, all molecules must desorb, and loading cannot be negative. +Constraint 2 requires that in the limit of zero pressure, the slope of the isotherm must be a positive finite constant. Talu +and Myers show that, as pressure approaches zero, the slope of the adsorption isotherm equals the adsorption second +virial coefficient B1S, which characterizes the interaction between one molecule and the surface, and must be a finite +positive number [29] [30]: +lim +p→0 +df +dp = B1S +RT = c +(4) +Constraint 3 requires that loading does not decrease with increasing pressure (the isotherm is monotonically non- +decreasing) for all (∀) positive values of pressure. Note that this does not hold for mixture adsorption (in which +competition plays a role), nor in BET adsorption, which exhibits a discontinuity at the saturation pressure, instead of a +monotonic increase. +1.5 +PySR: Symbolic Regression using Genetic Algorithms +PySR, Python for Symbolic Regression, is a Python library that uses a genetic algorithm for symbolic regression [31]. +PySR is a Python wrapper that calls a Julia library by the same author, SymbolicRegression.jl (SR.jl), for numerical +performance. Due to the nature of the modifications needed to the algorithm for this work, the base Julia library was +used, but all added functionality should be inherited by the Python wrapper library as well. +The basic premise is that one or more populations of models move towards more optimal solutions via random mutations. +At each generation, some members of a population are removed based on their fitness, age, or some other criteria (PySR +replaces the oldest members). Beneficial solutions are encouraged by having more optimal members of a population +mutate and reproduce. +3 + +Background Knowledge in Symbolic Regression +A PREPRINT +Figure 1: All mutations (except for random tree generation and simplification) in PySR in succession (read from left to +right, top to bottom). Changes from each previous expression tree are shown in orange. +Changes include mutating a single constant or operator, simplifying the expression, or performing crossover between +two expressions (Fig. 1 and Fig. 2). PySR uses multiple populations in a method similar to the island methodology [32]. +This aims to allow for specialization by separately evolving unique populations, occasionally allowing some members +to move between them to share that specialization. Specifically, PySR implements the so-called Hall of Fame (HOF), +which is a Pareto front built from the best members across each population. After a number of generations, each +population submits its top 10 best members (based on score) which are then compared and pared down via Pareto front. +Expressions that remain in the HOF are used for future mutations in each of the populations. +Figure 2: An example of the crossover mutation between two expression trees. +4 + +Mutate +Langmuir +Mutate +Constant +Operator +Isotherm +Ci *p +Ci *p +Ci *p +C2+p +C2 +p +C3 +p +Prepend Random +Delete Random +Append Random +Operator +Operator +Operator +C1 *p +C1 *p +p ++ +C5 +(c4 * p) +(c4 * p) +C3 +C3 +C3 - (c4 * p)C2 +p +p +p +p +C: +p +C2 +p +p +pBackground Knowledge in Symbolic Regression +A PREPRINT +1.6 +Bayesian Symbolic Regression +The Bayesian Machine Scientist (BMS) by Guimera et. al. [14] approaches symbolic regression from a Bayesian +perspective. Bayesian Symbolic Regression (BSR) frames the search for accurate, concise and informed models as +sampling the marginal posterior distribution of symbolic models with respect to a prior and fit to a dataset. Markov +chain Monte Carlo (MC) is used to generate new expression trees (Fig. 3), which are accepted or rejected based on +their likelihood. The authors define three MC moves: node replacement, root addition/removal, and elementary tree +replacement, which together enable construction of expression trees while maintaining detailed balance, ensuring proper +sampling of the posterior. +Figure 3: Illustrating the moves available to the BMS algorithm, as applied to adsorption equations. In contrast to the +mutations available in PySR, these transformations satisfy detailed balance [14]. +Specifically, the probability of some model given some data is defined as: +p(fi|D) = 1 +Z +� +Θi +dθip(D|fi, θi)p(θi|fi)p(fi) = exp[−L(fi)] +Z +(5) +where Z is the probability of the dataset p(D), Θi is the space of possible values for parameters θi and L is the +description length of the model. +A central idea in BSR is the inclusion of a prior to emphasize expressions that are a priori more likely than others, +regardless of the data. Guimera, et al. based their prior off of a corpus of 4080 mathematical expressions collected +from Wikipedia (from the “list of scientific equations named after people"), and assigned the prior likelihood using the +counts of each unique operator (no) in the corpus, by fitting parameters α and β like so: +EP = − log(p(fi)) = +� +o∈O +� +αono(fi) + βon2 +o(fi) +� +(6) +While this method leads to a distribution of expressions that resembles the corpus prior when run with no data, p(fi) +can also be set to a constant value so that there is no bias based on operators present in the search process. For our +problem, we crafted a prior especially for adsorption thermodynamics (see details in Methods). +2 +Methods +2.1 +Checking Thermodynamic Constraints +Three constraint checking functions for the thermodynamic constraints described in Section 1.4) were developed using +the Python library SymPy, an open-source computer algebra system [33]. Each function returns either true or false, +depending on if its constraint is met or not (if a time limit is exceeded, the constraint is returned as false). For both +PySR and BSR, we found that hard constraints (rejecting every expression that fails any constraint) severely hinder +the search process, cutting off intermediate expressions between better expressions that may also pass the constraints. +Consequently, we impose these as “soft” constraints, penalizing expressions for constraint violation, without outright +rejecting them. This approach (as implemented in PySR) is detailed in algorithm 1. +5 + +Background Knowledge in Symbolic Regression +A PREPRINT +Constraints 1 and 2 could be checked using SymPy’s limit and derivative functionality, but Constraint 3 was more +challenging. Though SymPy can check if an expression is strictly increasing in a given range, the check for monotonicity +returns false if any change in curvature (critical point) exists for the expression – thus preventing functions such as x3 +from being considered monotonically non-decreasing. To allow for zero slope, we implemented a custom monotonic +non-decreasing check function (see alg. 2). Instead of just checking the slope in one range, it checks the ranges between +all critical points (as well as to the start and end of the original range in question). +We hypothesize that the “equation space” explored by SR includes accurate, but not thermodynamically consistent +expressions that can be rejected through the incorporation of background knowledge, guiding the search to more +theory-informed expressions. +2.2 +PySR Modifications +In PySR, each member in a population has a score to be minimized, which combines the loss and complexity (defined +by total nodes in the expression tree). When a thermodynamic constraint is violated, we multiply the loss function by a +penalty, raising the score and making the expression less fit. This allows any number of constraints to be checked in any +order (as multiplication is commutative), and confers larger penalties to expressions that violate multiple constraints. +Loss: L = ℓR +2 ∗ +� +i=1,2,3 +cδi +i where δi = +� +1 +if constraint i passed +0 +if constraint i failed +� +(7) +Member Score: S = L + nnodes ∗ cl +(8) +The above equations detail how the loss and score are calculated in PySR. ℓR +2 is the L2 norm, ci is the penalty for +constraint i, δi indicates if constraint i is passed and cl is the penalty for the length / complexity of an expression. +PySR also has the option to take any operators defined in Julia or Python, including custom user-defined operators. For +this work only the operators +, −, ∗ and ÷ were used to manage the size of the search space. Expressions written in +their canonical form may use other operators such as exponents but these are only due to simplification of generated +expressions. +2.3 +BMS Modifications +The Bayesian prior used in the Bayesian Machine Scientist code Bayesian Machine Scientist code by Guimera et +al. [14] incorporates “background knowledge” in its equation search, through the use of a Bayesian prior based on +mathematical operation frequency among named equations in Wikipedia. Because the majority of these equations are +unrelated to adsorption, they may lead the search in a less optimal direction. Instead, we consider the thermodynamic +constraints described above to be our “prior knowledge,” and construct the following expression: +EP = +� +o∈O +� +copsno(fi) +� ++ +� +i=1,2 +ci ∗ δi where δi = +� +1 +if constraint i passed +0 +if constraint i failed +� +(9) +where cops is the constraint penalty for operators (analogous to the parsimony parameter in PySR), and no is the count +of each operator in expression fi. This expression directly replaces Eq. 6, changing the prior distribution. Note that +we checked all three constraints with PySR, and only the first two constraints with BSR (omitting the monotonic +non-decreasing check). +3 +Results +3.1 +Datasets +To examine the effects of adding constraints to SR during a search, four experimental adsorption datasets were identified: +adsorption of nitrogen and methane on mica [26], adsorption of isobutane in silicalite, [34] and adsorption of nitrogen +on Fe-Al203 catalyst [27]. The first and second datasets come from the landmark paper introducing the Langmuir +isotherm model [26]. This model assumes there are discrete loading "sites" that do not interact with each other, and that +each site can either be occupied or not. The isobutane dataset is well-described by a dual-site Langmuir model which +has two unique types of sites. The fourth dataset (referred to as the BET dataset) was used by the authors of BET theory +6 + +Background Knowledge in Symbolic Regression +A PREPRINT +to support their model for multilayer adsorption. These data and the respective ground truth model fits are shown in +Fig. 4. +Figure 4: Each dataset and corresponding ground truth model with constants fit using SciPy. Isobutane is shown with +log scaled pressure so that the two separate curves are visible. BET is shown with pressure increasing to 1 so the +asymptote is visible. +3.2 +Langmuir Datasets +The main results of this work are shown in two plot types. The left column contains Pareto fronts which show the best +expressions based on complexity and accuracy. In these, the horizontal axis shows increasing complexity (defined as +the total number of nodes in an expression tree), and the vertical axis shows loss, which is logarithmically scaled so the +trend of the Pareto front is more apparent. The best expression at each complexity is taken from each of 8 runs (gray +curves), with the overall Pareto front shown in orange. The “ground truth" expression for each dataset is also shown +in the form it would likely be expressed by SR, along with loss found using fit constants. The right column of each +figure shows the dataset and select expressions from the overall Pareto front for that test. Only some expressions are +shown so plots remain readable and because expressions longer than the ground truth are usually overfit and overlay the +ground truth expression too closely for distinction. The ground truth is plotted with a dotted line so that expressions +with similar accuracy can still be seen. Plotting the generated expressions on the data helps to illustrate how they may +or may not follow the thermodynamic constraints and how similar they are to the ground truth. +Figure 5 shows the results from both SR algorithms with constraints on and off on the Langmuir nitrogen dataset. The +first and second rows show BSR and PySR respectively with constraints off and clearly show that BSR finds the ground +truth while PySR does not. The expression that defines the corner at complexity 7 in the BSR Pareto front plot (Fig. 5a) +is indistinguishable from the ground truth (both written mathematically and drawn on the data) when viewed in the +isotherm plot (Fig. 5b). The BSR plot (Fig. 5a) has a much larger variance in terms of best Pareto fronts across 8 runs +(as shown by the grey lines) than PySR, but this may indicate longer time needed for the algorithm to converge. The +corresponding isotherm plots (the right column) show how expressions fit the data better as they become more complex, +following the general trend of the Pareto fronts. These plots also show how some expressions can fit the data reasonably +well while violating the constraints from theory, as is the case in the plot for PySR (Fig. 5d). In fact, only 2.2% of +7 + +Nitrogen Ground Truth +40 +Loading (q) +30 +20 +10 +Ground Truth: +(Ci * p) +(C2 + p) +0 +0 +10 +20 +30 +40 +50 +Pressure (p)Methane Ground Truth +140 +120 +100 +Loading (q) +80 +60 +40 +Ground Truth: +20 +(C1 * p) +(C2 + p) +0 +0 +25 +50 +75 +100 +125 +150 +175 +Pressure (p)Isobutane Ground Truth +2.0 - +Loading (q) +1.5 +1.0 - +0.5 +Ground Truth: +Ci*p +d× +(C2 + p) +(c4 + p) +0.0 +10-2 +10-1 +100 +101 +102 +Pressure (p)BET Ground Truth +400 +350 +300 +(b) +Loading ( +250 +200 +150 +100 +Ground Truth: +50 +(Ci * p/po) +(1 - p/po)*(1 - p/po + C3 *(p/po))) +0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pressure (p)Background Knowledge in Symbolic Regression +A PREPRINT +expressions generated by PySR (without enforcing constraints) pass the first constraint and only 33% pass the second +constraint (Table 2). Without constraints enforced, BSR finds more consistent expressions than PySR, with 37% of its +expressions passing the first constraint and 67% passing the second. +When the thermodynamic constraints are enabled, the effect is clearly shown in the Pareto fronts (bottom two rows). +Both SR methods find the ground truth and achieve the same or similar accuracy (accuracy is less for the same +expression when the constants were not optimized as thoroughly in the search). Datasets that are well represented by +the Langmuir isotherm show the effects of the constraints well because it is typically very accurate as well as being +concise. The isotherm plots show, as before, how the expressions fit the data better as they become more complex +but showing anything beyond a complexity of 7 is redundant as the ground truth is discovered and matches the pre-fit +ground truth almost perfectly. The trend of slightly more variation across BSR runs also continues here to some extent +and the variation across PySR runs appears roughly similar to with constraints disabled. Importantly, PySR sees a 5x +increase in expressions passing the first constraint (though still only 10%) and a marginal improvement across the other +two constraints (8% and 13%). The change is more stark in BSR where twice as many expressions now pass the first +constraint (up to 72%) and a significant portion pass the third constraint (up to 19% from 0.46%) even though it was not +included in the Bayesian prior. +While the results are mostly similar for the methane dataset, there are some important differences. Like with the +nitrogen dataset, BSR finds the ground truth without constraints enabled while PySR does not. This is apparent in +the Pareto fronts (Fig. 6a and Fig. 6c). In this case, PySR finds an expression with complexity 9 with more accuracy +than the ground truth, though with an extra constant in the numerator, it violates the thermodynamic constraints (Fig. +6d). Imposing the constraints penalized the loss for this expression relative to the ground truth, but not enough to +overcome the increased accuracy (Fig. 6h). As with nitrogen, BSR does a better job of finding expressions that pass the +constraints, even when they are not enabled, as it finds 33% passing the first and 51% passing the second (where PySR +finds 4.1% and 48% respectively). +3.3 +Isobutane Dataset +Unlike the methane and nitrogen datasets (Fig. 5 and 6) which are best modeled by the Langmuir isotherm, the +isobutane dataset (Fig. 7) is best modeled by the dual-site Langmuir isotherm, which has twice the complexity. Despite +this significant complexity, the dual-site Langmuir isotherm is not significantly more accurate than many expressions +shorter than it. This is best seen in Fig. 7c and 7g which show the Pareto fronts for PySR with constraints off and on +respectively. In both plots, expressions with half the complexity reach almost the same accuracy, creating a plateau from +complexity 7 onward. This is also shown well in the corresponding isotherm plots which show that the expressions +found at complexity 7 match the data as well as the ground truth. Importantly, these expressions do not satisfy the +thermodynamic constraints. +Unlike PySR, BSR does not find expressions with accuracy close to the ground truth until the same complexity. +For BSR, including constraints shifts the whole Pareto front down (Fig. 7a to Fig. 7e), indicating that more accurate +expressions were found at many complexity levels. While PySR did not find accurate expressions consistent with the +constraints, BSR did. In this case, BSR finds the ground truth expression while PySR does not. This is not apparent on +either the Pareto fronts or isotherm plots however, because the accuracy of the expression found is about 10x worse +than the fit ground truth and the best expressions found at that complexity. This is likely because, while the ground truth +is found, the form it was originally produced in (before being simplified) is much more complex. +In PySR, penalizing expressions that violate constraints actually led to populations of equations that violated constraints +two and three more often, with a decrease of about 10% in each case (see Table 2). This was surprising – we anticipated +that imposing penalties would lead to fewer violating expressions, but the opposite occurred. For BSR as well, including +constraints in the prior actually led to a decrease in expressions satisfying the second constraint (from 46% to 36%), +and a slight increase in the first and third constraints. +3.4 +BET Dataset +The BET dataset is unique because the ground truth expression diverges to infinity as the pressure approaches 1 (pressure +in this case is relative vapor pressure, p/psat; the vapor being adsorbed becomes a liquid as p/psat → 1. So in this +case, the third constraint (that it is monotonically non-decreasing) no longer holds for all pressure (seen in Fig. 8). +Nonetheless, we found that whether or not constraints were enabled, many of the most accurate expressions generated +by PySR for this dataset pass the third constraint (78.65% without constraints and 81.29% with), contrary to the ground +truth theory. Furthermore, PySR satisfies the first two constraints less frequently with constraints on compared to with +constraints off. One possible explanation for this behavior is that the dataset itself is more easily fit by expressions with +8 + +Background Knowledge in Symbolic Regression +A PREPRINT +expressions that are monotonically non-decreasing, at least from the perspective of the PySR algorithm. Overall, while +PySR can find accurate expressions for the BET dataset, it fails to find expressions that also follow the constraints, even +when they are enabled. +In contrast, BSR did not generate many expressions that were monotonically nondecreasing, and the incorporation +of constraints had a substantial effect on the search. Specifically, the second constraint is passed about 92% of the +time both with it enabled and disabled and the portion passing the first constraint increases dramatically from 16% to +85% once it is enabled. This leads to a large number of models which agree with the requisite constraints for BET, but +none of these are the ground truth rediscovered. Instead, many expressions with close to (or better than) the accuracy +of the ground truth are found by both algorithms in both cases, none of the isotherms plotted appear similar. The +asymptote at a partial pressure of 1 is not replicated by any similarly accurate expressions and the slight curve of the +ground truth in the middle of the dataset is also absent. These results together seem to indicate that the constraints, +while thermodynamically correct, do not provide enough information (or even provide contradictory information) for +rediscovering the BET ground truth expression. +Dataset +Constraints Active +BSR C1 +BSR C2 +BSR C3 +PySR C1 +PySR C2 +PySR C3 +Nitrogen +False +37% +67% +0.46% +2.2% +33% +46% +Nitrogen +True +72% +73% +19% +10% +41% +59% +Methane +False +33% +51% +0.51% +4.1% +48% +61% +Methane +True +59% +59% +2.5% +5.7% +54% +62% +Isobutane +False +24% +46% +0.45% +3.4% +65% +68% +Isobutane +True +36% +36% +1.3% +5.5% +56% +58% +BET +False +16% +92% +0.09% +6.2% +35% +79% +BET +True +85% +92% +2.4% +4.7% +30% +81% +Table 2: Percentage of expressions generated passing each of the three constraints. Results are shown across both SR +methods, all datasets and with constraints active and disabled. +9 + +Background Knowledge in Symbolic Regression +A PREPRINT +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 5: BSR and PySR on the Nitrogen dataset. The left column shows combined Pareto fronts across 8 runs and +the right column shows interesting isotherms found at the defining corners of those Pareto fronts. The constraints are +disabled in the top four subplots and enabled in the bottom four. The rows alternate between BSR and PySR. +10 + +PySR on Nitrogen - Constraints Off +103 +Pareto Fronts +Best Total Front +Ground Truth +102 +101 +LosS +100 +(C1 *p)/(C2 + p) +Loss: 0.122 +10-1 +T +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on Nitrogen - Constraints Off +40 +Loading (q) +30 +20 +[5]: (Ci *p + C2)/p +[7]: (Ci *p + C2)/(C3 +p) +[9]: (C1 + C2*p + C3 *p²)/p2 +10 +[13]: (C1*p+ C2*p² +C3 *p3 +C4)/p +GT [7]: (Ci *p)/(C2 +p) +0 +10 +20 +30 +40 +50 +Pressure (p)BSR on Nitrogen - Constraints On +103 +Pareto Fronts +Best Total Front +Ground Truth +102 +101 +Loss +100 +(C1 *p)/(C2 + p) +Loss: 0.122 +10-1 +Z +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on Nitrogen - Constraints On +40 +Loading (q) +30 +20 +[5]: C1 + C2 * p +10 +[7]: C1 *p/(C2 + p) +GT [7]: (C1 *p)/(C2 + p) +0 +10 +20 +30 +40 +50 +Pressure (p)PySR on Nitrogen - Constraints On +103 +Pareto Fronts +Best Total Front +Ground Truth +102 +101 +Loss +100 +(C1 *p)/(C2 + p) +Loss: 0.122 +10-1 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on Nitrogen - Constraints On +40 +Loading (q) +30 +20 +[5]: (Ci *p + C2)/p +[7]: C1 * p/(c2 + p) +10 +[15]: C1 *p + C2 - (C3*pC4 + c5)/p +GT [7]: (Ci*p)/(C2 + p) +0 +10 +20 +30 +40 +50 +Pressure (p)BSR on Nitrogen - Constraints Off +103 +Pareto Fronts +Best Total Front +Ground Truth +102 +101 +LosS +100 +(C1 *p)/(C2 + p) +Loss: 0.122 +10-1 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on Nitrogen - Constraints Off +40 +Loading (q) +30 +20 +[5]: C1 + C2 * p +10 +[7]: C1*p/(C2 + p) +GT [7]: (C1 *p)/(C2 + p) +0 +10 +20 +30 +40 +50 +Pressure (p)Background Knowledge in Symbolic Regression +A PREPRINT +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 6: BSR and PySR on the Methane dataset. The left column shows combined Pareto fronts across 8 runs and +the right column shows interesting isotherms found at the defining corners of those Pareto fronts. The constraints are +disabled in the top four subplots and enabled in the bottom four. The rows alternate between BSR and PySR. +11 + +BSR on Methane - Constraints Off +104 +Pareto Fronts +Best Total Front +Ground Truth +103 +LosS +102 +101 +(C1 *p)/(C2 + p) +L0Ss: 6.6747 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on Methane - Constraints Off +140 +120 +100 +Loading (q) +80 +60 +[5]: C1 + C2 *p +40 +[7]: C1*p/(C2 + p) +[13]: Ci*p²/(C2 + p² p) +20 +GT [7]: (Ci *p)/(C2 + p) +0 +25 +50 +75 +100 +125 +150 +175 +Pressure (p)PySR on Methane - Constraints Off +104 +Pareto Fronts +Best Total Front +Ground Truth +103 +LosS +102 +101 +(C1 *p)/(C2 + p) +L0ss: 6.6747 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on Methane - Constraints Off +140 +120 +100 +Loading (q) +80 +60 +[5]: (C1 + C2 *p)/p +40 +[7]: (C1 *p + C2)/(C3 +p) +[9]: (C1 + C2 *p)/(C3 + p) +20 +GT [7]: (C1 *p)/(C2 + p) +0 +25 +50 +75 +100 +125 +150 +175 +Pressure (p)BSR on Methane - Constraints On +104 +Pareto Fronts +Best Total Front +Ground Truth +103 +LosS +102 +101 +(C1 *p)/(C2 + p) +L0Ss: 6.6747 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on Methane - Constraints On +140 +120 +100 +Loading (q) +80 +60 +[5]: -3.0*p +40 +[7]: Ci*p/(C2 + p) +[13]: (Ci *p - 2.0*p2)/(C2 + p) +20 +GT [7]: (Ci *p)/(C2 + p) +0 +25 +50 +75 +100 +125 +150 +175 +Pressure (p)PySR on Methane - Constraints On +104 +Pareto Fronts +Best Total Front +Ground Truth +103 +LosS +102 +101 +(C1 *p)/(C2 + p) +Loss: 6.6747 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on Methane - Constraints On +140 +120 +100 +Loading (q) +80 +60 +[5]: (C1 + C2 *p)/p +40 +[7]: C1 *p/(C2 + p) +[9]: (C1 + C2 *p)/(C3 + p) +20 +GT [7]: (C1 *p)/(C2 + p) +0 +25 +50 +75 +100 +125 +150 +175 +Pressure (p)Background Knowledge in Symbolic Regression +A PREPRINT +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 7: BSR and PySR on the Isobutane dataset. The left column shows combined Pareto fronts across 8 runs and +the right column shows interesting isotherms found at the defining corners of those Pareto fronts. The constraints are +disabled in the top four subplots and enabled in the bottom four. The rows alternate between BSR and PySR. +12 + +BSR on Isobutane - Constraints Off +101 +Pareto Fronts +Best Total Front +100 +Ground Truth +10-1 +LosS +10-2 +10-3 +(C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) +Loss: 7.98e-4 +10-4 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on Isobutane - Constraints Off +[7]: (C1 *p + C2)/(C3 + p) +[11]: (Ci *p² + p)/(c2 +p) +2.0 - +[15]: (C1*p3 + C2*p)/(C3 +p²) +GT[15]: (Ci*p)/(C2 + p) +(C3 *p)/(C4 +p) +Loading (q) +1.5 +1.0 - +0.5 +0.0 : +10-2 +10-1 +100 +101 +102 +Pressure (p)PySR on Isobutane - Constraints Off +101 +Pareto Fronts +Best Total Front +100 +Ground Truth +10-1 +LosS +10-2 +10-3 +(C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) +Loss: 7.98e-4 +10-4 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on Isobutane - Constraints Off +[5]: C1 + C2 * p +[7]: (C1 *p + C2)/(C3 + p) +2.0 - +[15]: (C *p² + C2 + C3 *p)/(C4 *p + c5 +p²) +GT[15]: (C1 *p)/(C2 + p) + (C3 *p)/(C4 + p) +Loading (q) +1.5 +1.0 +0.5 +0.0 : +10-2 +10-1 +100 +101 +102 +Pressure (p)BSR on Isobutane - Constraints On +101 +Pareto Fronts +Best Total Front +100 +Ground Truth +10-1 +LosS +10-2 +10-3 +(C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) +Loss: 7.98e-4 +10-4 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on Isobutane - Constraints On +[7]: C1 * p/(C2 + p) +[11]: (Ci*p² +p)/(C2 +p) +2.0 - +[15]: (C1*p + C2*p²)/(C3 +p2) +GT[15]: (Ci*p)/(C2 + p) +(C3 *p)/(C4 +p) +Loading (q) +1.5 +1.0 - +0.5 +0.0 +10-2 +10-1 +100 +101 +102 +Pressure (p)PySR on Isobutane - Constraints On +101 +Pareto Fronts +Best Total Front +100 +Ground Truth +10-1 +LosS +10-2 +10-3 +(C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) +Loss: 7.98e-4 +10-4 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on Isobutane - Constraints On +[5]: C1 + C2 * p +[7]: (C1 *p + C2)/(C3 + p) +2.0 - +[15]: (Ci*p² + C2 +C3 *p)/(C4 + c5*p + p2) +GT [15]: (Ci*p)/(C2 + p) + (C3 *p)/(C4 +p) +Loading (q) +1.5 +1.0 +0.5 +0.0 : +10-2 +10-1 +100 +101 +102 +Pressure (p)Background Knowledge in Symbolic Regression +A PREPRINT +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +Figure 8: BSR and PySR on the BET dataset. The left column shows combined Pareto fronts across 8 runs and the right +column shows interesting isotherms found at the defining corners of those Pareto fronts. The constraints are disabled in +the top four subplots and enabled in the bottom four. The rows alternate between BSR and PySR. +13 + +BSR on BET - Constraints Off +Pareto Fronts +Best Total Front +104 +Ground Truth +103 +LosS +102 +101 +C1 * p/(C2*p + C3 + p2) +Loss: 24.06 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on BET - Constraints Off +400 +350 +300 +Loading (q) +250 +200 +150 +[5]: C1*p + C2 +100 +[13]: (C1 *p2 + C2 *p)/(C3 + p) +[17]: (C1 *p² + C2 *pC3 + C4 *p + c5*p3)/(c6 + p) +50 +GT[13]: C1 *p/(C2*p + C3 + p²) +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pressure (p)PySR on BET - Constraints Off +Pareto Fronts +Best Total Front +104 +Ground Truth +103 +LosS +102 +101 +C1* p/(C2* p + C3 +p²) +L0ss: 24.06 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on BET - Constraints Off +400 +350 +300 +(b) +Loading ( +250 +200 +[5]: C1 + C2 *p +150 +[9]: (C1 *p+ C2 + C3*p2)/p +100 +[13]: (Ci*p+ C2*p² + C3 + C4*p3)/p +[15]: (C1 *p²2 + C2*p+ C3 + C4 *p3)/p +50 +GT[13]: C1*p/(C2*p + C3 +p²) +-0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Pressure (p)BSR on BET - Constraints On +Pareto Fronts +Best Total Front +104 +Ground Truth +103 +LosS +102 +101 +C1* p/(C2* p + C3 + p2) +Loss: 24.06 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBSR on BET - Constraints On +400 +350 +300 +Loading (q) +250 +200 +150 +[5]: Ci*p + C2 +100 +[13]: (Ci*p² + C2 *p)/(C3 +p) +[17]: (C1*p+ C2*p3 + C3*p2)/(C4 +p) +50 +GT[13]: C1 *p/(C2*p + C3 + p2) +0 +0.2 +0.4 +0.6 +0.8 +1.0 +Pressure (p)PySR on BET - Constraints On +Pareto Fronts +Best Total Front +104 +Ground Truth +103 +LosS +102 +101 +C1* p/(C2* p + C3 +p²) +Loss: 24.06 +100 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityPySR on BET - Constraints On +400 +350 +300 +(b) +Loading ( +250 +200 +[9]: (C1*p2 + C2 + C3 *p)/p +150 +[11]: Ci*p+ C2*pC3 + C4 +100 +[13]: (C1*p3 + C2*p+ C3 + C4*p²)/p +[15]: C1/p+ C2 -p*(C3 *pC4 + c5) +50 +GT[13]: C1*p/(C2*p + C3 +p²) +0 - +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Pressure (p)Background Knowledge in Symbolic Regression +A PREPRINT +4 +Discussion +4.1 +Effectiveness +This work highlights that sometimes, a stochastic search through equation space finds equations that are superior to +ground truth expressions – in our case achieving comparable accuracy to the ground truth expressions while also being +less complex (shorter expression length). This is particularly observed in the case of isobutane (Fig. 7); both with +constraints on and off, PySR finds the expression c1p+c2 +c3+p , which fits the data well but diverges from the ground truth as +it approaches 0. But many of these expressions, while consistent with the data, are inconsistent with thermodynamics, +as they violate our tested constraints. We have demonstrated that accounting for these constraints in the search process +can guide the population or distribution of expressions toward thermodynamically consistent expressions, and aid in +the identification of the ground truth expression. Sometimes this leads to more consistent equations, and sometimes it +doesn’t improve the search at all. +4.2 +Exceptions To Constraints +While the three constraints presented in this work do follow from a broader thermodynamic theory, not all isotherm +models in this work satisfy all the constraints. Specifically, the BET isotherm does not satisfy the third constraint +because it reaches an asymptote as p/p0 approaches 1. While this does break the monotonicity constraint, it also +seems reasonable when considering what p0 represents (the pressure at which the adsorbate becomes a liquid and the +interaction fundamentally changes). This raises the question: what constraints to include and why? The decision to test +if expressions pass the third constraint necessarily excludes the BET model because, while it is theoretically grounded, +it only applies from in the range (0, p0). Furthermore, the data used does not extend past that range so expressions +that do pass the third constraint may not appear much different in terms of what is relevant. We recommend carefully +examining what constraints one may want to test and in what places they should actually be checked. Introducing +incorrect constraints may hinder the search with our biases, and prevent algorithms from discovering phenomena outside +our assumptions. +4.3 +Computational Complexity / Runtime +As seen in Fig. 10), consideration of constraints increases runtime by an order of magnitude; this is even after we +carefully integrated the computer algebra system into the SR algorithms to reduce overhead, and leveraged memory to +avoid redundant checks on expressions previously visited. On average, numerically checking models is much faster than +manipulating them symbolically (especially for larger expressions) – checking every new expression is quite expensive. +This is unfortunately necessary if the constraints are to be considered as an integral part of the search. If a cheaper +solution is needed, the search can be performed without constraints, and constraints checked after the fact. In fact, +this approach enables even more elaborate methods of considering background knowledge, such as comparing against +complex, multi-premise background theories using an automated theorem prover [15]. +4.4 +Challenges around complexity, simplification, and canonical form +In this work, simplification is necessary in order to identify whether a generated expression matches the ground truth +and to assign generated expressions an appropriate complexity. We augmented SymPy’s “simplify” function, to shorten +the numerous rational expressions we generated into a "canonical form" (details in the Supporting Information). While +some methods such as BSR attempt simplification during runtime, PySR does not because of the added computation +needed per expression. Generating a "canonical form" for expressions generated by PySR sometimes increases, and +sometimes decreases, the complexity. Some expressions are generated as complex expression trees that are much more +complex than their canonical forms (Fig. 9). +Simplification is a crucial challenge of this work because complexity plays a significant role in SR. After all, models +are compared via accuracy and complexity to make decisions during the search. A single model may very well take on +different scores / likelihoods because of how it is written, influencing not just its standing, but subsequent steps in the +search. Ideally, every model would always be written in the simplest form, but this is computationally intractable in +some circumstances [35]. Because of this, comparing functions based on behavior (symbolic constraints) may be more +appropriate, because limiting behavior is invariant to the numerous ways an expression can be written. +14 + +Background Knowledge in Symbolic Regression +A PREPRINT +Figure 9: The effect of the canonical form checking function on a Pareto front showing the results from PySR on the +Nitrogen dataset. Points on this plot are marked as “Passing" only if they pass all constraints checked. +4.5 +Reducing Underspecification Through Inductive Biases +Machine learning researchers at Google recently highlighted the role of underspecification in machine learning +pipelines [36]. They suggested that one way to combat underspecification is to use credible inductive biases to allow +for selection from otherwise similarly effective models, and that these constraints should have little negative effect on +accuracy if selected correctly. In this work, we find expressions that are roughly equivalent in terms of accuracy and +complexity but have different functional forms, leading to different behavior outside the range of the data – signatures +similar to those discussed in [36]. We find that adding thermodynamic constraints can help improve the search for good +expressions, but this doesn’t necessarily restrict the hypothesis space in the same way that inductive biases do; we +were unable to effectively search with hard constraints, and so our hypothesis space still included expressions that are +inconsistent with constraints. Instead, we can reduce the hypothesis space after the search is complete; by rejecting +accurate-but-inconsistent expressions using our background knowledge, we improve on the issues of underspecification. +Nonetheless, for datasets with reasonably complex behavior, there still exist multiple distinct thermodynamically- +consistent expressions of similar accuracy and complexity. The space of equations defined by the limited number of +operators considered here, even for one dimensional datasets, is just that vast! +5 +Conclusions +In this work, we couple a computer algebra system to two symbolic regression algorithms in order to check the +consistency of generated expressions with background knowledge. We find that including appropriate mathematical +constraints can improve search effectiveness or break the search entirely, depending on the dataset and implementation +details. Although computational costs increase by an order of magnitude, tightly integrating SR with a computer algebra +system is a practical way to check for constraints on each expression generated during the search. +We have shown that consideration of constraints helps in rediscovering ground-truth isotherm models from experimental +data, including the Langmuir and the dual-site Langmuir isotherms (though the dual-site Langmuir isotherm was +not identified on the Pareto front, it was present in the generated models). In contrast, the BET isotherm was not +rediscovered; more accurate and concise models were generated instead, and the most meaningful model (BET) was +consequently missed. We found that Bayesian Symbolic Regression is a more effective and intuitive platform for +15 + +PySR on Nitrogen +102 +101 +LosS +Original Vs Simplified Form +100 +Passing +Failing +Passing (Canonical) +Failing (Canonical) +10-1 +1 +3 +5 +7 +9 +11 +13 +15 +17 +19 +ComplexityBackground Knowledge in Symbolic Regression +A PREPRINT +incorporating symbolic constraints in a Bayesian prior, rather than by modifying the fitness function in traditional +genetic algorithms; the resulting populations of expressions were more attuned to the constraints with BSR. Finally, +though background knowledge can screen out accurate yet inconsistent solutions, symbolic regression pipelines remain +underspecified in our context, capable of generating multiple distinct solutions with similar performance and adherence +to constraints. +6 +Acknowledgements +We thank Marta Sales-Pardo and Roger Guimerà for discussions about the Bayesian Machine Scientist, and Miles +Cranmer for assistance with PySR. This material is based upon work supported by the National Science Foundation +under Grant No. (NSF #218938), as well as startup funds from the University of Maryland, Baltimore County. +7 +Supporting Information +The modified version of PySR and the code used to run it are both available on GitHub. The PySR code was forked +from the original repository on June 6th, 2020 and is available at https://github.com/CharFox1/SymbolicRegression.jl. +The +code +for +running +PySR, +parsing +its +output, +and +plotting +the +results, +and +is +available +at +https://github.com/ATOMSLab/pySR_adsorption. +The modified version of BMS code used in this paper is +available at https://github.com/ATOMSLab/BayesianSymbolicRegression. +The Supporting Information includes 1) further description of the adsorption models considered here, 2) further +discussion of the changes we implemented in PySR and BMS codes to implement thermodynamic constraint checking, +including pseudocode for new algorithms, 3) description of our pipeline for collecting and analyzing generated +expressions, 4) further discussion of the nuances around identifying of “interesting” expressions in automated pipelines +and algorithms for simplification and pattern-matching, 5) details of constant fitting, 6) experiments comparing runtime +of algorithms on different datasets, and 7) details of the testing environment on the UMBC supercomputer. +16 + +Background Knowledge in Symbolic Regression +A PREPRINT +References +[1] John R. 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Covert, Benoit Pasquier, Takafumi Arakaki, Alexey Stukalov, Andrew Clausen, Arno Strouwen, +and Benjamin Deonovic. JuliaNLSolvers/Optim.jl: v1.7.0, May 2022. +18 + +Background Knowledge in Symbolic Regression +A PREPRINT +[40] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni +Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua +Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C. J. +Carey, ˙Ilhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, +Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, +and Paul van Mulbregt. SciPy 1.0: fundamental algorithms for scientific computing in Python. Nature Methods, +17(3):261–272, March 2020. Number: 3 Publisher: Nature Publishing Group. +19 + +Background Knowledge in Symbolic Regression +A PREPRINT +8 +Supporting Information +8.1 +Langmuir +The Langmuir isotherm model was originally presented in 1918 and remains in common use today [26]. While written +and discovered in a simplified form in this work, the original model has a few specific terms with important meaning +for the materials they pertain to. +qe = qmKLp +1 + KLp or θA = +KLp +1 + KLp +(10) +The original Langmuir isotherm relates the volume adsorbed onto the surface (qe), the adsorption strength (KL), the +gas pressure p and the maximum adsorption capacity (qm) [24] [26]. The isotherm is often more simply written with +only the fractional adsorption θA (what fraction of the surface is occupied by adsorbed molecules). However, in order +to understand the expression that SR is likely to find, the isotherm should be rewritten as: +qe = qmKAP +1 + KAP → q = +c1 ∗ p +(c2 + p) +(11) +This is because the nitrogen and methane datasets used in this work relate pressure to volume adsorbed, not including +the total possible volume that could be adsorbed. Including that quantity in the numerator and dividing through by the +Langmuir Constant leads to two unique constants that an SR algorithm might try to fit. +The basic premise of the Langmuir model is to represent the adsorbent (the surface atoms or molecules are adsorbing +onto) as a simple surface with some number of free and full spots where the adsorbate could stick. The model is not +concerned with rough or porous surfaces, molecules that may take up multiple “sites" or otherwise interact with each +other after being adsorbed or, most importantly, any stacking of molecules. While this may seem somewhat idealistic +and restricted, the Langmuir model has been shown to apply well to various adsorbents (surfaces) and adsorbates +(gasses). Wang et. al. classify the Langmuir model as a chemical adsorption model because it is only concerned with +mono-layer adsorption in which a chemical bond is formed between the adsorbent and adsorbate [24]. +8.2 +Dual-Site +In some cases, the Langmuir model can be very simply extended to fit new phenomena. Specifically, if there are two +types of adsorption sites (with different properties) on a surface, adding another term of the Langmuir model may be +enough to describe the adsorption process again. This new model would be called a “dual-site Langmuir" model and +would be written as: +q = +qaKAp +(1 + KAp) + +qbKBp +(1 + KBp) → q = +c1 ∗ p +(c2 + p) + +c3 ∗ p +(c4 + p) +(12) +In this case, because there are two unique Langmuir terms, there are also two unique terms for both the maximum +volume that can be adsorbed (qa and qb) as well as the “Langmuir Constants" (KA and KB). Furthermore, this model +is considered to be the ground truth for the isobutane dataset because the adsorbent material, the MFI zeolite, has two +distinct adsorption sites for this adsorbate [37]. That is, the material that molecules are sticking to has two unique types +of places for them to stick. +8.3 +BET +Unlike the Langmuir model, the BET model is specifically designed with multi-layer adsorption in mind. Beyond the +first layer, the Van der Waals force is what pulls the adsorbate towards the surface (and itself). Because of this, Wang et. +al. classify the BET isotherm as a physical model as opposed to a chemical one [24]. The BET model can actually be +written in a few ways based on what the maximum number of layers that can be adsorbed is believed to be. At one layer, +it simplifies to the Langmuir model (which only models one layer) and at an arbitrary number of layers n it becomes +significantly more complex (and unlikely to be found via SR). Fortunately the form at n = ∞ is more concise and can +be simplified further: +20 + +Background Knowledge in Symbolic Regression +A PREPRINT +c1 ∗ (p/p0) +(1 − p/p0) ∗ (1 − p/p0 + c2(p/p0)) +(13) +Note that while the p0 term is written here, this constant is not provided to the SR algorithms, requiring that they fit a +third constant (if they were to find this functional form). +8.4 +Thermodynamic Constraint Functions +Three constraint checking functions (which follow from the three thermodynamic constraints introduced previously in +section 1.4) were developed using the Python library SymPy which is designed for symbolic math [33]. Each function +returns either true or false, depending on if its constraint is met or not. While these functions are useful for examining +the expressions generated after a run, simply discarding an expression for failing one or more constraint during a run +can severely hinder search potential by cutting off intermediary steps between better expressions that may also pass +the constraints. Because of this, each function has a corresponding weight that allows it to act as a “soft constraint". +Specifically, a constraint will not affect the score of an expression if it is passed but will multiply (worsen) that score if +it is not. This approach (as implemented in PySR) is detailed in algorithm 1. +Algorithm 1 Modified genetic algorithm +Input: tree, dataset, options +Output: score, loss +function SCOREFUNC(tree, dataset, options) +loss ← evalLoss(tree, dataset, options) +▷ Traditional error calculation +penalties ← options.penalties +if penalties not empty then +▷ Skip slow calculation if possible +expr, var ← parseTree(tree) +for p ∈ penalties and tf ∈ thermoFunctions do +if tf(expr, var) = False then +loss ← loss ∗ p +end if +end for +end if +score ← scoreFunc(loss, tree, options) +▷ Factor in complexity +return score, loss +end function +21 + +Background Knowledge in Symbolic Regression +A PREPRINT +Algorithm 2 Monotonic Non-decreasing Check +Input: expr, var, start, stop +Output: passing (If the expression is monotonically non-decreasing) +function MONTONICNONDECREASING(expr, var, start, stop) +if expr is constant then +return True +end if +turningPoints ← inflection points (zeros) of expr +▷ Calculated using SymPy +if turningPointsis empty then +if expr(stop) − expr(start) ≥ 0 then +▷ Measure slope +return True +else +return False +end if +end if +turningPoints ← [start, turningPoints, stop] +▷ Include bounds +for each sequential pair of points p, q ∈ turningPoints do +if expr(q) − expr(p) ≤ 0 then +▷ Measure slope between zeros +return False +end if +end for +return True +end function +8.5 +Implementation of Thermodynamic Constraints +In order to maintain parity between PySR and BMS, the Julia library PyCall was used. This allowed the same Python +code that checked thermodynamic constraints in BMS to be used within the Julia code of the SymbolicRegression.jl +library (the back-end of PySR). Beyond the concern of a fair comparison across platforms, PyCall showed itself to be +somewhat necessary for this work. As of writing, there is no Julia library close to as full-featured as SymPy (and SymPy +can have some serious limitations/difficulties). One method explored was using the Julia library MATLAB.jl to call +MATLAB code from within Julia in a similar way to how PyCall works with Python code. While this did eventually +work (and MATLAB does have robust symbolic math capabilities) it showed itself to be very difficult to set up due to +intricacies in how the two languages store data. In the end, using MATLAB was not significantly faster than Python in +this work (which is somewhat surprising). +One downside to the current structure of our version of SR.jl and the languages it relies on is that it must be run in +distributed mode instead of threaded mode (meaning multiple processes must be created, requiring more overhead). +This is due to the fact that PyCall expects only one Julia instance to attempt to use Python at once. With multiple +processes, a new instance of the PyCall Julia package is created for each process, thus avoiding memory access conflicts. +8.6 +PySR Data Collection +Although PySR handles many expressions, populations of those expressions and the Hall of Fame of the best expressions +across all populations so far, accessing that information has proven difficult. PySR’s goal each iteration is to produce a +Pareto front showing what it has found to be the most accurate expression at each complexity level. This means the vast +majority of expressions in a population are not shown (although some of them may be duplicates). While this is an +issue, on the other hand, if an expression never makes it the Pareto front at any point during the run, by definition of +the search it is not as important. That said, this limitation with output means that an expression such as the Langmuir +isotherm may be generated and never shown due to poor accuracy relative to its peers. +In order to transform a few hundred Pareto front printouts to usable data, the full output text is parsed using Python +and stored in a Pandas DataFrame [38]. The values collected from the raw text are the expression itself, score, loss, +complexity, runtime, iteration and run number. While this is a significant amount of data already, there are a few +interesting attributes yet to be calculated – specifically those relating to the simplified form of the expressions and those +related to the thermodynamic constraints. In order to make further parsing manageable, once the simplified form of +each expression is generated, only the most accurate expression with that form is retained. This step often reduces the +number of rows in the DataFrame by about 1000x. The simplified form of each expression is calculated both with +original (optimized/fit) and substituted constants. The new complexity of the simplified form may or may not be smaller +22 + +Background Knowledge in Symbolic Regression +A PREPRINT +than the original complexity so both attributes are kept. Finally, the thermodynamic constraints that may have been +used to guide the search are checked for each expression. +8.7 +Identifying “interesting" expressions +An important component of this work is identification of interesting or meaningful expressions generated by the SR +algorithms. An expression is most obviously meaningful if it is, or can be simplified to, the canonical form of an already +known expression such as the Langmuir or BET adsorption isotherms. Unfortunately, determining if one expression +is equivalent to another is a very difficult, and sometimes undecidable problem. In fact, while not true for this work +due to limits on operators and constants used, Richardson’s theorem shows that given the right set of operators and the +transcendental numbers π and e, it may be impossible to show that one expression is equivalent to another [35]. It is +also important to note that there is no incentive for SR to generate expressions that are easily read or understood – if it +even had a concept of what expressions fit those criteria. Even when ground truth expressions are found, they can often +be in unrecognisable forms. Furthermore, while SymPy does have a reasonably capable simplification function, it is +of course not perfect and may encounter expressions it cannot handle. Because of these facts, this work uses a more +complex function built on top of SymPy’s simplification function and also accepts that much analysis will have to be +done manually, at least for now. +8.8 +Simplification Function +While SymPy is capable of simplifying symbolic math expressions and equations, there are definitely expressions that it +may not simplify to their shortest form. Furthermore, while constants may be optimized per expression within both +PySR and BMS, the expressions themselves are the true object of the search so constants are often left symbolic (e.g. +c1, c2...). Because of this, in the interest of effectively simplifying without naively filling all symbolic constants with +random numbers, symbolic constants are substituted with prime numbers. This avoids situations where one constant can +completely remove another through multiplication or division. While this method has some advantages, there are also +reasons to simplify using fit constants. Primarily, some expressions such as the BET isotherm have the same constant in +multiple places, meaning it can be factored out. This should also be the case with well fit constants for newly generated +expressions (especially those that turn out to be the same structure as previous isotherms). The conversion from the +original BET isotherm to the form the simplification function finds is shown below: +c1 ∗ (p/p0) +(1 − p/p0) ∗ (1 − p/p0 + c2(p/p0)) → +c1 ∗ p +(c2 ∗ p + c3 + p2) +(14) +Once constants are decided (either fit or primes), ordinary SymPy simplification is used to compress the expression as +much as is reasonable. Then, if the resulting expression is rational (has only integer exponents and no division by zero), +it can be written as a fraction and possibly simplified further. Specifically, if there is a common factor between the +numerator and denominator because they are both degree 1 or larger, it may be possible to remove up to the difference +in their degrees (see example below). While this situation is not guaranteed, having the capability to simplify it is still +useful, especially when examining many generated expressions and differences between them. +3x +2x + 3x2 → +3 +2 + 3x +(15) +23 + +Background Knowledge in Symbolic Regression +A PREPRINT +Algorithm 3 Simplification Function +Input: expr, vars, pars +Output: can (Canonical form of expression) +function SIMPLIFY(expr, vars, pars) +expr ← expression +vars ← variables +pars ← parameters +for p in pars do +p ← prime number +▷ Substitute constants with unique primes +end for +Simplify using SymPy +if expr is rational then +num, denom ← expr as fraction +▷ Calculated using SymPy +if degree(num) > degree(denom) then +factor ← leading term of num +else +factor ← leading term of denom +end if +expr ← +num/factor +denom/factor +▷ Remove common factor +end if +return expr +end function +It is important to note that the simplification function will not always return the simplest possible form. While it does +obviously reduce complexity, the original goal of the function was to have a canonical form for each expression such +that whenever it is found, in whatever form, it will be simplified to that form. This effect is achieved by always writing +the expression as a rational function if possible (and not simplifying further from that point). Because of this, when +expressions are plotted on Pareto fronts (or their complexities are compared in general) the minimum complexity +between the generated form and the simplified form is used. +8.9 +Fitting Constants in PySR +By default, PySR uses Nelder-Mead optimization to fit constants for expressions as they are generated [31] [39]. +Nelder-Mead is well suited to optimizing parameters for arbitrary generated expressions because it does not require +any derivative or gradient information. Simply put, the algorithm evaluates the function at n + 1 points where n is the +number of parameters. Then the next point is selected by finding the point with the highest function evaluation and +looking to the opposite side of the rest of the points (a simplex). This iteratively moves the worst point to a likely better +location, eventually moving towards a local optimum regardless of how the function is evaluated. +One downside to this method is that it is not guaranteed (or even expected) to find global minima. This issue is usually +rectified by allowing for multiple starts from random locations, and selecting the best result. In PySR, each expression +gets 8 attempts by default, randomizing the parameters between each. This is typically enough but there are occasionally +cases where an expression should be much more accurate than it is due to poorly fitted constants. +To fit ground-truth expressions and check constants in post-processing, we also applied Nelder-Mead optimization +using SciPy [40]. In this case, many more iterations were allowed to ensure ground truths and interesting functional +forms had the best chance to show their effectiveness (or lack thereof). +8.10 +Runtime +An important consideration when examining the effectiveness of the thermodynamic constraints in guiding SR is the +impact on computation time. While not extreme, symbolic math (in SymPy) can be slow, especially compared to +the otherwise efficient and optimized Julia code running behind the PySR front-end. The following plot shows the +difference in iteration time (the time between one Pareto front and the next being printed by PySR) across different +datasets, and more importantly, across different constraint penalties. It shows that if SymPy can avoid being loaded, +runtime is not affected but that in all other cases, it is increased by about an order of magnitude. +24 + +Background Knowledge in Symbolic Regression +A PREPRINT +Figure 10: Average runtimes across all datasets and combinations of thermodynamic constraint penalties. Runs with all +penalties set to 1.0 are highlighted in orange. Standard deviation is shown by error bars at the top of each bar. +8.11 +Environment +Testing was done on both the batch and cpu2021 partitions of the UMBC High Performance Computing Facility +(https://hpcf.umbc.edu/). This allowed for the longer runtimes and larger parameter exploration necessitated by adding +thermodynamic constraints with variable penalties. It was also important to be able to run long term because BMS +requires some build-up time to converge to the desired distribution, since it is based off of MCMC. +While BMS is entirely based in Python, SymbolicRegression.jl (the backend for PySR) is entirely in Julia. Because of +the need to use SymPy, the Julia package PyCall.jl was used to allow Python code and libraries to be run by Julia (at the +time this work was completed, symbolic math libraries in Julia could not evaluate the constraints considered in this +work). To allow for parallel use of Python / SymPy from within Julia, PySR was run in distributed mode, necessitating +more overhead than threaded mode. +25 + +Average Runtime Per Iteration +Nitrogen [1.3, 1.3, 1.3] +Nitrogen [1.1, 1.1, 1.3] +Nitrogen [1.0, 1.0, 1.0] +Methane [1.3, 1.3, 1.3] +Methane [1.1. 1.1. 1.3] +Methane [1.0, 1.0. 1.01 +Isobutane [1.3, 1.3, 1.3] +Isobutane [1.1. 1.1, 1.3] +Isobutane [1.0, 1.0, 1.0] +1 +BET [1.3. 1.3. 1.31 +BET[1.3, 1.3, 1.0] +BET[1.1,1.1,1.3] +BET [1.0, 1.0, 1.0] +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +Iteration Time (in seconds) \ No newline at end of file diff --git a/sNFKT4oBgHgl3EQf1S6Z/content/tmp_files/load_file.txt b/sNFKT4oBgHgl3EQf1S6Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0f6d23240e674ad14f6500869d6bff6ae9f9f558 --- /dev/null +++ b/sNFKT4oBgHgl3EQf1S6Z/content/tmp_files/load_file.txt @@ -0,0 +1,850 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf,len=849 +page_content='INCORPORATING BACKGROUND KNOWLEDGE IN SYMBOLIC REGRESSION USING A COMPUTER ALGEBRA SYSTEM A PREPRINT Charles Fox2, Neil Tran1, Nikki Nacion1, Samiha Sharlin1, and Tyler R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Josephson1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 1Department of Chemical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Biochemical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' and Environmental Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' University of Maryland Baltimore County,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1000 Hilltop Circle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' MD 21250 2Department of Computer Science and Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' University of Maryland Baltimore County,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1000 Hilltop Circle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' MD 21250 ABSTRACT Symbolic Regression (SR) can generate interpretable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' concise expressions that fit a given dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' allowing for more human understanding of the structure than black-box approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The addition of background knowledge (in the form of symbolic mathematical constraints) allows for the generation of expressions that are meaningful with respect to theory while also being consistent with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We specifically examine the addition of constraints to traditional genetic algorithm (GA) based SR (PySR) as well as a Markov-chain Monte Carlo (MCMC) based Bayesian SR architecture (Bayesian Machine Scientist), and apply these to rediscovering adsorption equations from experimental, historical datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We find that, while hard constraints prevent GA and MCMC SR from searching, soft constraints can lead to improved performance both in terms of search effectiveness and model meaningfulness, with computational costs increasing by about an order-of-magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' If the constraints do not correlate well with the dataset or expected models, they can hinder the search of expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We find Bayesian SR is better these constraints (as the Bayesian prior) than by modifying the fitness function in the GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 Symbolic Regression for Scientific Discovery Symbolic Regression (SR) is our tool of choice, as it generates mathematical expressions that are optimized for complexity and accuracy to a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Since John Koza pioneered the paradigm of programming by means of natural selection, many applications for SR in scientific discovery have emerged [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Unlike other applications of machine learning techniques, scientific research demands explanation and verification, both of which are made more feasible by the generation of human-interpretable mathematical models (as opposed to fitting a model with thousands of parameters) [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Furthermore, SR can be effective even with very small datasets (∼10 items) such as those produced by difficult or expensive experiments which are not easily repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The mathematical expressions produced by SR can easily be extrapolated to untested or otherwise unreachable domains within a dataset (such as extreme pressures or temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For decades, SR has discovered interesting models from data in many unique applications including inferring process models at the Dow Chemical company [5], rainfall-runoff modelling [6] and rediscovering equations describing double- pendulum motion [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Symbolic regression has been applied across all scales of scientific investigation, including the atomistic (interatomic potentials [8]), macroscopic (computational fluid dynamics [9]), and cosmological (dark matter overdensity [10]) scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Some techniques facilitate search through billions of candidate expressions, such as the space of nonlinear descriptors of material properties [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While most applications of SR in science focus on identifying empirical patterns in data, such "data-only" approaches do not account for potential insights from background theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In fact, some SR works emphasize their capabilities of discovery “without any prior knowledge about physics, kinematics, or geometry” [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Nonetheless, we posit that prior knowledge need not be discarded, and in this work, we explore how arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='11919v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='LG] 27 Jan 2023 Background Knowledge in Symbolic Regression A PREPRINT theory may be incorporated into symbolic regression to demonstrate machine learning in the context of background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 Incorporating Background Knowledge into Symbolic Regression One particularly important step towards effective use of SR in specific domains is the addition of prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This step has the potential to take a general purpose SR algorithm and use it to find novel models with physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For example, AI-DARWIN is uses prior knowledge of chemical reaction mechanisms in the form of predefined functions that a genetic algorithm may use in its search of equation space, ensuring that each generated model is mechanistically meaningful [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This approach specifically encodes the prior knowledge in the form of functions available instead of limitations on functions generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In another recent example, Engle and Sahinidis use a deterministic symbolic regression algorithm that constrains the space of possible equations, not to those constructed from a library of meaningful function components, but to those functions that obey derivative constraints from theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This improves the quality of generated expressions for thermodynamic equations of state [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Another approach to incorporating background knowledge in symbolic regression is the Bayesian Machine Scientist (BMS) [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' BMS rigorously incorporates background knowledge in the form of a Bayesian prior on symbolic expressions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' expressions are a priori more likely if their distribution of mathematical operators aligns with the distribution of operators in a corpus of prominent equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' However, their approach to the Bayesian prior does not incorporate meaning from particular scientific domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Checking consistency of equations after the search is complete is also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Previously, we showed that generated expressions can be compared to rich background knowledge (expressed as axioms for the environment under study), by posing generated expressions as conjectures to an automated theorem prover (ATP) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' However, state-of-the-art ATPs are too slow to incorporate this logical check as symbolic expressions are generated, and therefore cannot be easily used to bias the search for equations in light of that background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Moreover, translating scientific theories into a computer-interpretable form is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We address these specific drawbacks by combining symbolic regression systems (both genetic algorithm and Bayesian approaches) with a computer algebra system that checks constraints as an equation search is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is similar to Logic Guided Genetic Algorithms (LGGA), which uses “auxiliary truths” (ATs) corresponding to datasets in order to weigh items in a dataset as well as augment it with more information [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' LGGA follows an iterative approach of training an arbitrary genetic algorithm with some dataset, augmenting that dataset with ATs, and training that algorithm again with more informative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' An important distinction between our work and LGGA is that the dataset is not altered in any way and the addition of extra information is performed during the execution of the GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3 Adsorption Adsorption, the phenomenon in which molecules bind to a surface, enables chemical processes including carbon capture, humidity control, removal of harmful pollutants from water, and hydrogen production [17] [18] [19] [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Models of adsorption enable prediction and design of engineered adsorption processes, and many have been proposed over the years (selected equations are shown in Table 1) [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' These models relate the amount adsorbed at equilibrium as a function of pressure or concentration and are commonly expressed as equations that are either empirical or derived from theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For example, the Freundlich isotherm [25], is an empirical function designed to fit observed data, the Langmuir [26] and BET [27] isotherms are derived from physical models, and the Sips [28] isotherm is Langmuir-inspired with empirical terms added for fitting flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We wonder, “What kinds of models could be generated by a machine learning system, and what role can background knowledge play in the search for accurate and meaningful expressions?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2 Background Knowledge in Symbolic Regression A PREPRINT Table 1: Some well-known isotherms written as SR might find them, and their complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Isotherm Literature Expression Symbolic Regression Form SR Complexity Langmuir [26] qmaxKeqp 1+Keqp c1p c2+p 7 Dual-Site Langmuir [26] qa maxKa eqp 1+Ka eqp + qb maxKb eqp 1+Kbeqp c1p c2+p + c3p c4+p 15 BET [27] c1(p/p0) (1−p/p0)(1−p/p0+c2(p/p0)) c1p c2+c3p+c3p2 15 Freundlich [25] c1p 1 n c1pc2 5 Sips [28] c1p 1 n 1+c1p 1 n pc2 c1+pc2 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 Thermodynamic Constraints We consider models to be more meaningful when they satisfy thermodynamic constraints on the functional forms appropriate for modeling these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' That is, a random equation that fits data, but does not approach zero loading correctly, is less trustworthy outside the training data than an equation constrained to follow thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We have identified three constraints relevant for single-component adsorption [15]: lim p→0 f(p) = 0 (1) lim p→0 f ′(p) < ∞ (2) ∀p > 0 f ′(p) ≥ 0 (3) Constraint 1 ensures that, in the limit of zero pressure, all molecules must desorb, and loading cannot be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Constraint 2 requires that in the limit of zero pressure, the slope of the isotherm must be a positive finite constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Talu and Myers show that, as pressure approaches zero, the slope of the adsorption isotherm equals the adsorption second virial coefficient B1S, which characterizes the interaction between one molecule and the surface, and must be a finite positive number [29] [30]: lim p→0 df dp = B1S RT = c (4) Constraint 3 requires that loading does not decrease with increasing pressure (the isotherm is monotonically non- decreasing) for all (∀) positive values of pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Note that this does not hold for mixture adsorption (in which competition plays a role), nor in BET adsorption, which exhibits a discontinuity at the saturation pressure, instead of a monotonic increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 PySR: Symbolic Regression using Genetic Algorithms PySR, Python for Symbolic Regression, is a Python library that uses a genetic algorithm for symbolic regression [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' PySR is a Python wrapper that calls a Julia library by the same author, SymbolicRegression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl (SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl), for numerical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Due to the nature of the modifications needed to the algorithm for this work, the base Julia library was used, but all added functionality should be inherited by the Python wrapper library as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The basic premise is that one or more populations of models move towards more optimal solutions via random mutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' At each generation, some members of a population are removed based on their fitness, age, or some other criteria (PySR replaces the oldest members).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Beneficial solutions are encouraged by having more optimal members of a population mutate and reproduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3 Background Knowledge in Symbolic Regression A PREPRINT Figure 1: All mutations (except for random tree generation and simplification) in PySR in succession (read from left to right, top to bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Changes from each previous expression tree are shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Changes include mutating a single constant or operator, simplifying the expression, or performing crossover between two expressions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' PySR uses multiple populations in a method similar to the island methodology [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This aims to allow for specialization by separately evolving unique populations, occasionally allowing some members to move between them to share that specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, PySR implements the so-called Hall of Fame (HOF), which is a Pareto front built from the best members across each population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' After a number of generations, each population submits its top 10 best members (based on score) which are then compared and pared down via Pareto front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Expressions that remain in the HOF are used for future mutations in each of the populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Figure 2: An example of the crossover mutation between two expression trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4 Mutate Langmuir Mutate Constant Operator Isotherm Ci *p Ci *p Ci *p C2+p C2 p C3 p Prepend Random Delete Random Append Random Operator Operator Operator C1 *p C1 *p p + C5 (c4 * p) (c4 * p) C3 C3 C3 - (c4 * p)C2 p p p p C: p C2 p p pBackground Knowledge in Symbolic Regression A PREPRINT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6 Bayesian Symbolic Regression The Bayesian Machine Scientist (BMS) by Guimera et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' [14] approaches symbolic regression from a Bayesian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Bayesian Symbolic Regression (BSR) frames the search for accurate, concise and informed models as sampling the marginal posterior distribution of symbolic models with respect to a prior and fit to a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Markov chain Monte Carlo (MC) is used to generate new expression trees (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3), which are accepted or rejected based on their likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The authors define three MC moves: node replacement, root addition/removal, and elementary tree replacement, which together enable construction of expression trees while maintaining detailed balance, ensuring proper sampling of the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Figure 3: Illustrating the moves available to the BMS algorithm, as applied to adsorption equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In contrast to the mutations available in PySR, these transformations satisfy detailed balance [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, the probability of some model given some data is defined as: p(fi|D) = 1 Z � Θi dθip(D|fi, θi)p(θi|fi)p(fi) = exp[−L(fi)] Z (5) where Z is the probability of the dataset p(D), Θi is the space of possible values for parameters θi and L is the description length of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' A central idea in BSR is the inclusion of a prior to emphasize expressions that are a priori more likely than others, regardless of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Guimera, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' based their prior off of a corpus of 4080 mathematical expressions collected from Wikipedia (from the “list of scientific equations named after people"),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' and assigned the prior likelihood using the counts of each unique operator (no) in the corpus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' by fitting parameters α and β like so: EP = − log(p(fi)) = � o∈O � αono(fi) + βon2 o(fi) � (6) While this method leads to a distribution of expressions that resembles the corpus prior when run with no data,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' p(fi) can also be set to a constant value so that there is no bias based on operators present in the search process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For our problem, we crafted a prior especially for adsorption thermodynamics (see details in Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2 Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 Checking Thermodynamic Constraints Three constraint checking functions for the thermodynamic constraints described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4) were developed using the Python library SymPy, an open-source computer algebra system [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Each function returns either true or false, depending on if its constraint is met or not (if a time limit is exceeded, the constraint is returned as false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For both PySR and BSR, we found that hard constraints (rejecting every expression that fails any constraint) severely hinder the search process, cutting off intermediate expressions between better expressions that may also pass the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Consequently, we impose these as “soft” constraints, penalizing expressions for constraint violation, without outright rejecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This approach (as implemented in PySR) is detailed in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5 Background Knowledge in Symbolic Regression A PREPRINT Constraints 1 and 2 could be checked using SymPy’s limit and derivative functionality, but Constraint 3 was more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Though SymPy can check if an expression is strictly increasing in a given range, the check for monotonicity returns false if any change in curvature (critical point) exists for the expression – thus preventing functions such as x3 from being considered monotonically non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' To allow for zero slope, we implemented a custom monotonic non-decreasing check function (see alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Instead of just checking the slope in one range, it checks the ranges between all critical points (as well as to the start and end of the original range in question).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We hypothesize that the “equation space” explored by SR includes accurate, but not thermodynamically consistent expressions that can be rejected through the incorporation of background knowledge, guiding the search to more theory-informed expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 PySR Modifications In PySR, each member in a population has a score to be minimized, which combines the loss and complexity (defined by total nodes in the expression tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' When a thermodynamic constraint is violated, we multiply the loss function by a penalty, raising the score and making the expression less fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This allows any number of constraints to be checked in any order (as multiplication is commutative), and confers larger penalties to expressions that violate multiple constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Loss: L = ℓR 2 ∗ � i=1,2,3 cδi i where δi = � 1 if constraint i passed 0 if constraint i failed � (7) Member Score: S = L + nnodes ∗ cl (8) The above equations detail how the loss and score are calculated in PySR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' ℓR 2 is the L2 norm, ci is the penalty for constraint i, δi indicates if constraint i is passed and cl is the penalty for the length / complexity of an expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' PySR also has the option to take any operators defined in Julia or Python, including custom user-defined operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For this work only the operators +, −, ∗ and ÷ were used to manage the size of the search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Expressions written in their canonical form may use other operators such as exponents but these are only due to simplification of generated expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3 BMS Modifications The Bayesian prior used in the Bayesian Machine Scientist code Bayesian Machine Scientist code by Guimera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' [14] incorporates “background knowledge” in its equation search, through the use of a Bayesian prior based on mathematical operation frequency among named equations in Wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because the majority of these equations are unrelated to adsorption, they may lead the search in a less optimal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Instead, we consider the thermodynamic constraints described above to be our “prior knowledge,” and construct the following expression: EP = � o∈O � copsno(fi) � + � i=1,2 ci ∗ δi where δi = � 1 if constraint i passed 0 if constraint i failed � (9) where cops is the constraint penalty for operators (analogous to the parsimony parameter in PySR), and no is the count of each operator in expression fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This expression directly replaces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6, changing the prior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Note that we checked all three constraints with PySR, and only the first two constraints with BSR (omitting the monotonic non-decreasing check).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 Datasets To examine the effects of adding constraints to SR during a search, four experimental adsorption datasets were identified: adsorption of nitrogen and methane on mica [26], adsorption of isobutane in silicalite, [34] and adsorption of nitrogen on Fe-Al203 catalyst [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The first and second datasets come from the landmark paper introducing the Langmuir isotherm model [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This model assumes there are discrete loading "sites" that do not interact with each other, and that each site can either be occupied or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The isobutane dataset is well-described by a dual-site Langmuir model which has two unique types of sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The fourth dataset (referred to as the BET dataset) was used by the authors of BET theory 6 Background Knowledge in Symbolic Regression A PREPRINT to support their model for multilayer adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' These data and the respective ground truth model fits are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Figure 4: Each dataset and corresponding ground truth model with constants fit using SciPy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Isobutane is shown with log scaled pressure so that the two separate curves are visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' BET is shown with pressure increasing to 1 so the asymptote is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 Langmuir Datasets The main results of this work are shown in two plot types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The left column contains Pareto fronts which show the best expressions based on complexity and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In these, the horizontal axis shows increasing complexity (defined as the total number of nodes in an expression tree), and the vertical axis shows loss, which is logarithmically scaled so the trend of the Pareto front is more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The best expression at each complexity is taken from each of 8 runs (gray curves), with the overall Pareto front shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The “ground truth" expression for each dataset is also shown in the form it would likely be expressed by SR, along with loss found using fit constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The right column of each figure shows the dataset and select expressions from the overall Pareto front for that test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Only some expressions are shown so plots remain readable and because expressions longer than the ground truth are usually overfit and overlay the ground truth expression too closely for distinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The ground truth is plotted with a dotted line so that expressions with similar accuracy can still be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Plotting the generated expressions on the data helps to illustrate how they may or may not follow the thermodynamic constraints and how similar they are to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Figure 5 shows the results from both SR algorithms with constraints on and off on the Langmuir nitrogen dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The first and second rows show BSR and PySR respectively with constraints off and clearly show that BSR finds the ground truth while PySR does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The expression that defines the corner at complexity 7 in the BSR Pareto front plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5a) is indistinguishable from the ground truth (both written mathematically and drawn on the data) when viewed in the isotherm plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The BSR plot (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5a) has a much larger variance in terms of best Pareto fronts across 8 runs (as shown by the grey lines) than PySR, but this may indicate longer time needed for the algorithm to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The corresponding isotherm plots (the right column) show how expressions fit the data better as they become more complex, following the general trend of the Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' These plots also show how some expressions can fit the data reasonably well while violating the constraints from theory, as is the case in the plot for PySR (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In fact, only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2% of 7 Nitrogen Ground Truth 40 Loading (q) 30 20 10 Ground Truth: (Ci * p) (C2 + p) 0 0 10 20 30 40 50 Pressure (p)Methane Ground Truth 140 120 100 Loading (q) 80 60 40 Ground Truth: 20 (C1 * p) (C2 + p) 0 0 25 50 75 100 125 150 175 Pressure (p)Isobutane Ground Truth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - Loading (q) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 Ground Truth: Ci*p d× (C2 + p) (c4 + p) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 10-2 10-1 100 101 102 Pressure (p)BET Ground Truth 400 350 300 (b) Loading ( 250 200 150 100 Ground Truth: 50 (Ci * p/po) (1 - p/po)*(1 - p/po + C3 *(p/po))) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 Pressure (p)Background Knowledge in Symbolic Regression A PREPRINT expressions generated by PySR (without enforcing constraints) pass the first constraint and only 33% pass the second constraint (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Without constraints enforced, BSR finds more consistent expressions than PySR, with 37% of its expressions passing the first constraint and 67% passing the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' When the thermodynamic constraints are enabled, the effect is clearly shown in the Pareto fronts (bottom two rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Both SR methods find the ground truth and achieve the same or similar accuracy (accuracy is less for the same expression when the constants were not optimized as thoroughly in the search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Datasets that are well represented by the Langmuir isotherm show the effects of the constraints well because it is typically very accurate as well as being concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The isotherm plots show, as before, how the expressions fit the data better as they become more complex but showing anything beyond a complexity of 7 is redundant as the ground truth is discovered and matches the pre-fit ground truth almost perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The trend of slightly more variation across BSR runs also continues here to some extent and the variation across PySR runs appears roughly similar to with constraints disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Importantly, PySR sees a 5x increase in expressions passing the first constraint (though still only 10%) and a marginal improvement across the other two constraints (8% and 13%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The change is more stark in BSR where twice as many expressions now pass the first constraint (up to 72%) and a significant portion pass the third constraint (up to 19% from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='46%) even though it was not included in the Bayesian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While the results are mostly similar for the methane dataset, there are some important differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Like with the nitrogen dataset, BSR finds the ground truth without constraints enabled while PySR does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is apparent in the Pareto fronts (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In this case, PySR finds an expression with complexity 9 with more accuracy than the ground truth, though with an extra constant in the numerator, it violates the thermodynamic constraints (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Imposing the constraints penalized the loss for this expression relative to the ground truth, but not enough to overcome the increased accuracy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' As with nitrogen, BSR does a better job of finding expressions that pass the constraints, even when they are not enabled, as it finds 33% passing the first and 51% passing the second (where PySR finds 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1% and 48% respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3 Isobutane Dataset Unlike the methane and nitrogen datasets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5 and 6) which are best modeled by the Langmuir isotherm, the isobutane dataset (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 7) is best modeled by the dual-site Langmuir isotherm, which has twice the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Despite this significant complexity, the dual-site Langmuir isotherm is not significantly more accurate than many expressions shorter than it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is best seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 7c and 7g which show the Pareto fronts for PySR with constraints off and on respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In both plots, expressions with half the complexity reach almost the same accuracy, creating a plateau from complexity 7 onward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is also shown well in the corresponding isotherm plots which show that the expressions found at complexity 7 match the data as well as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Importantly, these expressions do not satisfy the thermodynamic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Unlike PySR, BSR does not find expressions with accuracy close to the ground truth until the same complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For BSR, including constraints shifts the whole Pareto front down (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 7a to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 7e), indicating that more accurate expressions were found at many complexity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While PySR did not find accurate expressions consistent with the constraints, BSR did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In this case, BSR finds the ground truth expression while PySR does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is not apparent on either the Pareto fronts or isotherm plots however, because the accuracy of the expression found is about 10x worse than the fit ground truth and the best expressions found at that complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is likely because, while the ground truth is found, the form it was originally produced in (before being simplified) is much more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In PySR, penalizing expressions that violate constraints actually led to populations of equations that violated constraints two and three more often, with a decrease of about 10% in each case (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This was surprising – we anticipated that imposing penalties would lead to fewer violating expressions, but the opposite occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' For BSR as well, including constraints in the prior actually led to a decrease in expressions satisfying the second constraint (from 46% to 36%), and a slight increase in the first and third constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 BET Dataset The BET dataset is unique because the ground truth expression diverges to infinity as the pressure approaches 1 (pressure in this case is relative vapor pressure, p/psat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' the vapor being adsorbed becomes a liquid as p/psat → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' So in this case, the third constraint (that it is monotonically non-decreasing) no longer holds for all pressure (seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Nonetheless, we found that whether or not constraints were enabled, many of the most accurate expressions generated by PySR for this dataset pass the third constraint (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='65% without constraints and 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='29% with), contrary to the ground truth theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Furthermore, PySR satisfies the first two constraints less frequently with constraints on compared to with constraints off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' One possible explanation for this behavior is that the dataset itself is more easily fit by expressions with 8 Background Knowledge in Symbolic Regression A PREPRINT expressions that are monotonically non-decreasing, at least from the perspective of the PySR algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Overall, while PySR can find accurate expressions for the BET dataset, it fails to find expressions that also follow the constraints, even when they are enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In contrast, BSR did not generate many expressions that were monotonically nondecreasing, and the incorporation of constraints had a substantial effect on the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, the second constraint is passed about 92% of the time both with it enabled and disabled and the portion passing the first constraint increases dramatically from 16% to 85% once it is enabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This leads to a large number of models which agree with the requisite constraints for BET, but none of these are the ground truth rediscovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Instead, many expressions with close to (or better than) the accuracy of the ground truth are found by both algorithms in both cases, none of the isotherms plotted appear similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The asymptote at a partial pressure of 1 is not replicated by any similarly accurate expressions and the slight curve of the ground truth in the middle of the dataset is also absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' These results together seem to indicate that the constraints, while thermodynamically correct, do not provide enough information (or even provide contradictory information) for rediscovering the BET ground truth expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Dataset Constraints Active BSR C1 BSR C2 BSR C3 PySR C1 PySR C2 PySR C3 Nitrogen False 37% 67% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='46% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2% 33% 46% Nitrogen True 72% 73% 19% 10% 41% 59% Methane False 33% 51% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='51% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1% 48% 61% Methane True 59% 59% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='7% 54% 62% Isobutane False 24% 46% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='45% 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4% 65% 68% Isobutane True 36% 36% 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5% 56% 58% BET False 16% 92% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='09% 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2% 35% 79% BET True 85% 92% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='7% 30% 81% Table 2: Percentage of expressions generated passing each of the three constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Results are shown across both SR methods, all datasets and with constraints active and disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 9 Background Knowledge in Symbolic Regression A PREPRINT (a) (b) (c) (d) (e) (f) (g) (h) Figure 5: BSR and PySR on the Nitrogen dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The left column shows combined Pareto fronts across 8 runs and the right column shows interesting isotherms found at the defining corners of those Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The constraints are disabled in the top four subplots and enabled in the bottom four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The rows alternate between BSR and PySR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 10 PySR on Nitrogen - Constraints Off 103 Pareto Fronts Best Total Front Ground Truth 102 101 LosS 100 (C1 *p)/(C2 + p) Loss: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='122 10-1 T 3 5 7 9 11 13 15 17 19 ComplexityPySR on Nitrogen - Constraints Off 40 Loading (q) 30 20 [5]: (Ci *p + C2)/p [7]: (Ci *p + C2)/(C3 +p) [9]: (C1 + C2*p + C3 *p²)/p2 10 [13]: (C1*p+ C2*p² +C3 *p3 +C4)/p GT [7]: (Ci *p)/(C2 +p) 0 10 20 30 40 50 Pressure (p)BSR on Nitrogen - Constraints On 103 Pareto Fronts Best Total Front Ground Truth 102 101 Loss 100 (C1 *p)/(C2 + p) Loss: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='122 10-1 Z 3 5 7 9 11 13 15 17 19 ComplexityBSR on Nitrogen - Constraints On 40 Loading (q) 30 20 [5]: C1 + C2 * p 10 [7]: C1 *p/(C2 + p) GT [7]: (C1 *p)/(C2 + p) 0 10 20 30 40 50 Pressure (p)PySR on Nitrogen - Constraints On 103 Pareto Fronts Best Total Front Ground Truth 102 101 Loss 100 (C1 *p)/(C2 + p) Loss: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='122 10-1 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on Nitrogen - Constraints On 40 Loading (q) 30 20 [5]: (Ci *p + C2)/p [7]: C1 * p/(c2 + p) 10 [15]: C1 *p + C2 - (C3*pC4 + c5)/p GT [7]: (Ci*p)/(C2 + p) 0 10 20 30 40 50 Pressure (p)BSR on Nitrogen - Constraints Off 103 Pareto Fronts Best Total Front Ground Truth 102 101 LosS 100 (C1 *p)/(C2 + p) Loss: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='122 10-1 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on Nitrogen - Constraints Off 40 Loading (q) 30 20 [5]: C1 + C2 * p 10 [7]: C1*p/(C2 + p) GT [7]: (C1 *p)/(C2 + p) 0 10 20 30 40 50 Pressure (p)Background Knowledge in Symbolic Regression A PREPRINT (a) (b) (c) (d) (e) (f) (g) (h) Figure 6: BSR and PySR on the Methane dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The left column shows combined Pareto fronts across 8 runs and the right column shows interesting isotherms found at the defining corners of those Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The constraints are disabled in the top four subplots and enabled in the bottom four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The rows alternate between BSR and PySR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 11 BSR on Methane - Constraints Off 104 Pareto Fronts Best Total Front Ground Truth 103 LosS 102 101 (C1 *p)/(C2 + p) L0Ss: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6747 100 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on Methane - Constraints Off 140 120 100 Loading (q) 80 60 [5]: C1 + C2 *p 40 [7]: C1*p/(C2 + p) [13]: Ci*p²/(C2 + p² p) 20 GT [7]: (Ci *p)/(C2 + p) 0 25 50 75 100 125 150 175 Pressure (p)PySR on Methane - Constraints Off 104 Pareto Fronts Best Total Front Ground Truth 103 LosS 102 101 (C1 *p)/(C2 + p) L0ss: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6747 100 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on Methane - Constraints Off 140 120 100 Loading (q) 80 60 [5]: (C1 + C2 *p)/p 40 [7]: (C1 *p + C2)/(C3 +p) [9]: (C1 + C2 *p)/(C3 + p) 20 GT [7]: (C1 *p)/(C2 + p) 0 25 50 75 100 125 150 175 Pressure (p)BSR on Methane - Constraints On 104 Pareto Fronts Best Total Front Ground Truth 103 LosS 102 101 (C1 *p)/(C2 + p) L0Ss: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6747 100 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on Methane - Constraints On 140 120 100 Loading (q) 80 60 [5]: -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0*p 40 [7]: Ci*p/(C2 + p) [13]: (Ci *p - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0*p2)/(C2 + p) 20 GT [7]: (Ci *p)/(C2 + p) 0 25 50 75 100 125 150 175 Pressure (p)PySR on Methane - Constraints On 104 Pareto Fronts Best Total Front Ground Truth 103 LosS 102 101 (C1 *p)/(C2 + p) Loss: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6747 100 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on Methane - Constraints On 140 120 100 Loading (q) 80 60 [5]: (C1 + C2 *p)/p 40 [7]: C1 *p/(C2 + p) [9]: (C1 + C2 *p)/(C3 + p) 20 GT [7]: (C1 *p)/(C2 + p) 0 25 50 75 100 125 150 175 Pressure (p)Background Knowledge in Symbolic Regression A PREPRINT (a) (b) (c) (d) (e) (f) (g) (h) Figure 7: BSR and PySR on the Isobutane dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The left column shows combined Pareto fronts across 8 runs and the right column shows interesting isotherms found at the defining corners of those Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The constraints are disabled in the top four subplots and enabled in the bottom four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The rows alternate between BSR and PySR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 12 BSR on Isobutane - Constraints Off 101 Pareto Fronts Best Total Front 100 Ground Truth 10-1 LosS 10-2 10-3 (C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) Loss: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='98e-4 10-4 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on Isobutane - Constraints Off [7]: (C1 *p + C2)/(C3 + p) [11]: (Ci *p² + p)/(c2 +p) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - [15]: (C1*p3 + C2*p)/(C3 +p²) GT[15]: (Ci*p)/(C2 + p) +(C3 *p)/(C4 +p) Loading (q) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 : 10-2 10-1 100 101 102 Pressure (p)PySR on Isobutane - Constraints Off 101 Pareto Fronts Best Total Front 100 Ground Truth 10-1 LosS 10-2 10-3 (C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) Loss: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='98e-4 10-4 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on Isobutane - Constraints Off [5]: C1 + C2 * p [7]: (C1 *p + C2)/(C3 + p) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - [15]: (C *p² + C2 + C3 *p)/(C4 *p + c5 +p²) GT[15]: (C1 *p)/(C2 + p) + (C3 *p)/(C4 + p) Loading (q) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 : 10-2 10-1 100 101 102 Pressure (p)BSR on Isobutane - Constraints On 101 Pareto Fronts Best Total Front 100 Ground Truth 10-1 LosS 10-2 10-3 (C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) Loss: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='98e-4 10-4 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on Isobutane - Constraints On [7]: C1 * p/(C2 + p) [11]: (Ci*p² +p)/(C2 +p) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - [15]: (C1*p + C2*p²)/(C3 +p2) GT[15]: (Ci*p)/(C2 + p) +(C3 *p)/(C4 +p) Loading (q) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 10-2 10-1 100 101 102 Pressure (p)PySR on Isobutane - Constraints On 101 Pareto Fronts Best Total Front 100 Ground Truth 10-1 LosS 10-2 10-3 (C1 *p)/(C2 +p) + (C3 *p)/(C4 + p) Loss: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='98e-4 10-4 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on Isobutane - Constraints On [5]: C1 + C2 * p [7]: (C1 *p + C2)/(C3 + p) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 - [15]: (Ci*p² + C2 +C3 *p)/(C4 + c5*p + p2) GT [15]: (Ci*p)/(C2 + p) + (C3 *p)/(C4 +p) Loading (q) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 : 10-2 10-1 100 101 102 Pressure (p)Background Knowledge in Symbolic Regression A PREPRINT (a) (b) (c) (d) (e) (f) (g) (h) Figure 8: BSR and PySR on the BET dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The left column shows combined Pareto fronts across 8 runs and the right column shows interesting isotherms found at the defining corners of those Pareto fronts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The constraints are disabled in the top four subplots and enabled in the bottom four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The rows alternate between BSR and PySR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 13 BSR on BET - Constraints Off Pareto Fronts Best Total Front 104 Ground Truth 103 LosS 102 101 C1 * p/(C2*p + C3 + p2) Loss: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='06 100 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on BET - Constraints Off 400 350 300 Loading (q) 250 200 150 [5]: C1*p + C2 100 [13]: (C1 *p2 + C2 *p)/(C3 + p) [17]: (C1 *p² + C2 *pC3 + C4 *p + c5*p3)/(c6 + p) 50 GT[13]: C1 *p/(C2*p + C3 + p²) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 Pressure (p)PySR on BET - Constraints Off Pareto Fronts Best Total Front 104 Ground Truth 103 LosS 102 101 C1* p/(C2* p + C3 +p²) L0ss: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='06 100 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on BET - Constraints Off 400 350 300 (b) Loading ( 250 200 [5]: C1 + C2 *p 150 [9]: (C1 *p+ C2 + C3*p2)/p 100 [13]: (Ci*p+ C2*p² + C3 + C4*p3)/p [15]: (C1 *p²2 + C2*p+ C3 + C4 *p3)/p 50 GT[13]: C1*p/(C2*p + C3 +p²) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 Pressure (p)BSR on BET - Constraints On Pareto Fronts Best Total Front 104 Ground Truth 103 LosS 102 101 C1* p/(C2* p + C3 + p2) Loss: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='06 100 1 3 5 7 9 11 13 15 17 19 ComplexityBSR on BET - Constraints On 400 350 300 Loading (q) 250 200 150 [5]: Ci*p + C2 100 [13]: (Ci*p² + C2 *p)/(C3 +p) [17]: (C1*p+ C2*p3 + C3*p2)/(C4 +p) 50 GT[13]: C1 *p/(C2*p + C3 + p2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 Pressure (p)PySR on BET - Constraints On Pareto Fronts Best Total Front 104 Ground Truth 103 LosS 102 101 C1* p/(C2* p + C3 +p²) Loss: 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='06 100 1 3 5 7 9 11 13 15 17 19 ComplexityPySR on BET - Constraints On 400 350 300 (b) Loading ( 250 200 [9]: (C1*p2 + C2 + C3 *p)/p 150 [11]: Ci*p+ C2*pC3 + C4 100 [13]: (C1*p3 + C2*p+ C3 + C4*p²)/p [15]: C1/p+ C2 -p*(C3 *pC4 + c5) 50 GT[13]: C1*p/(C2*p + C3 +p²) 0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 Pressure (p)Background Knowledge in Symbolic Regression A PREPRINT 4 Discussion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 Effectiveness This work highlights that sometimes, a stochastic search through equation space finds equations that are superior to ground truth expressions – in our case achieving comparable accuracy to the ground truth expressions while also being less complex (shorter expression length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is particularly observed in the case of isobutane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' both with constraints on and off, PySR finds the expression c1p+c2 c3+p , which fits the data well but diverges from the ground truth as it approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' But many of these expressions, while consistent with the data, are inconsistent with thermodynamics, as they violate our tested constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We have demonstrated that accounting for these constraints in the search process can guide the population or distribution of expressions toward thermodynamically consistent expressions, and aid in the identification of the ground truth expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Sometimes this leads to more consistent equations, and sometimes it doesn’t improve the search at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 Exceptions To Constraints While the three constraints presented in this work do follow from a broader thermodynamic theory, not all isotherm models in this work satisfy all the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, the BET isotherm does not satisfy the third constraint because it reaches an asymptote as p/p0 approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this does break the monotonicity constraint, it also seems reasonable when considering what p0 represents (the pressure at which the adsorbate becomes a liquid and the interaction fundamentally changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This raises the question: what constraints to include and why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The decision to test if expressions pass the third constraint necessarily excludes the BET model because, while it is theoretically grounded, it only applies from in the range (0, p0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Furthermore, the data used does not extend past that range so expressions that do pass the third constraint may not appear much different in terms of what is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We recommend carefully examining what constraints one may want to test and in what places they should actually be checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Introducing incorrect constraints may hinder the search with our biases, and prevent algorithms from discovering phenomena outside our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3 Computational Complexity / Runtime As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 10), consideration of constraints increases runtime by an order of magnitude;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' this is even after we carefully integrated the computer algebra system into the SR algorithms to reduce overhead, and leveraged memory to avoid redundant checks on expressions previously visited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' On average, numerically checking models is much faster than manipulating them symbolically (especially for larger expressions) – checking every new expression is quite expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is unfortunately necessary if the constraints are to be considered as an integral part of the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' If a cheaper solution is needed, the search can be performed without constraints, and constraints checked after the fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In fact, this approach enables even more elaborate methods of considering background knowledge, such as comparing against complex, multi-premise background theories using an automated theorem prover [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 Challenges around complexity, simplification, and canonical form In this work, simplification is necessary in order to identify whether a generated expression matches the ground truth and to assign generated expressions an appropriate complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We augmented SymPy’s “simplify” function, to shorten the numerous rational expressions we generated into a "canonical form" (details in the Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While some methods such as BSR attempt simplification during runtime, PySR does not because of the added computation needed per expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Generating a "canonical form" for expressions generated by PySR sometimes increases, and sometimes decreases, the complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Some expressions are generated as complex expression trees that are much more complex than their canonical forms (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Simplification is a crucial challenge of this work because complexity plays a significant role in SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' After all, models are compared via accuracy and complexity to make decisions during the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' A single model may very well take on different scores / likelihoods because of how it is written, influencing not just its standing, but subsequent steps in the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Ideally, every model would always be written in the simplest form, but this is computationally intractable in some circumstances [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of this, comparing functions based on behavior (symbolic constraints) may be more appropriate, because limiting behavior is invariant to the numerous ways an expression can be written.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 14 Background Knowledge in Symbolic Regression A PREPRINT Figure 9: The effect of the canonical form checking function on a Pareto front showing the results from PySR on the Nitrogen dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Points on this plot are marked as “Passing" only if they pass all constraints checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 Reducing Underspecification Through Inductive Biases Machine learning researchers at Google recently highlighted the role of underspecification in machine learning pipelines [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' They suggested that one way to combat underspecification is to use credible inductive biases to allow for selection from otherwise similarly effective models, and that these constraints should have little negative effect on accuracy if selected correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In this work, we find expressions that are roughly equivalent in terms of accuracy and complexity but have different functional forms, leading to different behavior outside the range of the data – signatures similar to those discussed in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We find that adding thermodynamic constraints can help improve the search for good expressions, but this doesn’t necessarily restrict the hypothesis space in the same way that inductive biases do;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' we were unable to effectively search with hard constraints, and so our hypothesis space still included expressions that are inconsistent with constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Instead, we can reduce the hypothesis space after the search is complete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' by rejecting accurate-but-inconsistent expressions using our background knowledge, we improve on the issues of underspecification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Nonetheless, for datasets with reasonably complex behavior, there still exist multiple distinct thermodynamically- consistent expressions of similar accuracy and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The space of equations defined by the limited number of operators considered here, even for one dimensional datasets, is just that vast!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5 Conclusions In this work, we couple a computer algebra system to two symbolic regression algorithms in order to check the consistency of generated expressions with background knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We find that including appropriate mathematical constraints can improve search effectiveness or break the search entirely, depending on the dataset and implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Although computational costs increase by an order of magnitude, tightly integrating SR with a computer algebra system is a practical way to check for constraints on each expression generated during the search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We have shown that consideration of constraints helps in rediscovering ground-truth isotherm models from experimental data, including the Langmuir and the dual-site Langmuir isotherms (though the dual-site Langmuir isotherm was not identified on the Pareto front, it was present in the generated models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In contrast, the BET isotherm was not rediscovered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' more accurate and concise models were generated instead, and the most meaningful model (BET) was consequently missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' We found that Bayesian Symbolic Regression is a more effective and intuitive platform for 15 PySR on Nitrogen 102 101 LosS Original Vs Simplified Form 100 Passing Failing Passing (Canonical) Failing (Canonical) 10-1 1 3 5 7 9 11 13 15 17 19 ComplexityBackground Knowledge in Symbolic Regression A PREPRINT incorporating symbolic constraints in a Bayesian prior, rather than by modifying the fitness function in traditional genetic algorithms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' the resulting populations of expressions were more attuned to the constraints with BSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Finally, though background knowledge can screen out accurate yet inconsistent solutions, symbolic regression pipelines remain underspecified in our context, capable of generating multiple distinct solutions with similar performance and adherence to constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6 Acknowledgements We thank Marta Sales-Pardo and Roger Guimerà for discussions about the Bayesian Machine Scientist, and Miles Cranmer for assistance with PySR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This material is based upon work supported by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' (NSF #218938), as well as startup funds from the University of Maryland, Baltimore County.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 7 Supporting Information The modified version of PySR and the code used to run it are both available on GitHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The PySR code was forked from the original repository on June 6th, 2020 and is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='com/CharFox1/SymbolicRegression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The code for running PySR, parsing its output, and plotting the results, and is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='com/ATOMSLab/pySR_adsorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The modified version of BMS code used in this paper is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='com/ATOMSLab/BayesianSymbolicRegression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The Supporting Information includes 1) further description of the adsorption models considered here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 2) further discussion of the changes we implemented in PySR and BMS codes to implement thermodynamic constraint checking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' including pseudocode for new algorithms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3) description of our pipeline for collecting and analyzing generated expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 4) further discussion of the nuances around identifying of “interesting” expressions in automated pipelines and algorithms for simplification and pattern-matching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 5) details of constant fitting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 6) experiments comparing runtime of algorithms on different datasets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' and 7) details of the testing environment on the UMBC supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 16 Background Knowledge in Symbolic Regression A PREPRINT References [1] John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Koza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Genetic Programming, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' [2] Felipe Oviedo, Juan Lavista Ferres, Tonio Buonassisi, and Keith T Butler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Interpretable and explainable machine learning for materials science and chemistry.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' AIChE Journal, 68(6):e17457, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' _eprint: https://onlinelibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='com/doi/pdf/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1002/aic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='17457.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' SciPy 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0: fundamental algorithms for scientific computing in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Nature Methods, 17(3):261–272, March 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Number: 3 Publisher: Nature Publishing Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 19 Background Knowledge in Symbolic Regression A PREPRINT 8 Supporting Information 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1 Langmuir The Langmuir isotherm model was originally presented in 1918 and remains in common use today [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While written and discovered in a simplified form in this work, the original model has a few specific terms with important meaning for the materials they pertain to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' qe = qmKLp 1 + KLp or θA = KLp 1 + KLp (10) The original Langmuir isotherm relates the volume adsorbed onto the surface (qe), the adsorption strength (KL), the gas pressure p and the maximum adsorption capacity (qm) [24] [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The isotherm is often more simply written with only the fractional adsorption θA (what fraction of the surface is occupied by adsorbed molecules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' However, in order to understand the expression that SR is likely to find, the isotherm should be rewritten as: qe = qmKAP 1 + KAP → q = c1 ∗ p (c2 + p) (11) This is because the nitrogen and methane datasets used in this work relate pressure to volume adsorbed, not including the total possible volume that could be adsorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Including that quantity in the numerator and dividing through by the Langmuir Constant leads to two unique constants that an SR algorithm might try to fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The basic premise of the Langmuir model is to represent the adsorbent (the surface atoms or molecules are adsorbing onto) as a simple surface with some number of free and full spots where the adsorbate could stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The model is not concerned with rough or porous surfaces, molecules that may take up multiple “sites" or otherwise interact with each other after being adsorbed or, most importantly, any stacking of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this may seem somewhat idealistic and restricted, the Langmuir model has been shown to apply well to various adsorbents (surfaces) and adsorbates (gasses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Wang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' classify the Langmuir model as a chemical adsorption model because it is only concerned with mono-layer adsorption in which a chemical bond is formed between the adsorbent and adsorbate [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='2 Dual-Site In some cases, the Langmuir model can be very simply extended to fit new phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, if there are two types of adsorption sites (with different properties) on a surface, adding another term of the Langmuir model may be enough to describe the adsorption process again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This new model would be called a “dual-site Langmuir" model and would be written as: q = qaKAp (1 + KAp) + qbKBp (1 + KBp) → q = c1 ∗ p (c2 + p) + c3 ∗ p (c4 + p) (12) In this case, because there are two unique Langmuir terms, there are also two unique terms for both the maximum volume that can be adsorbed (qa and qb) as well as the “Langmuir Constants" (KA and KB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Furthermore, this model is considered to be the ground truth for the isobutane dataset because the adsorbent material, the MFI zeolite, has two distinct adsorption sites for this adsorbate [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' That is, the material that molecules are sticking to has two unique types of places for them to stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3 BET Unlike the Langmuir model, the BET model is specifically designed with multi-layer adsorption in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Beyond the first layer, the Van der Waals force is what pulls the adsorbate towards the surface (and itself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of this, Wang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' classify the BET isotherm as a physical model as opposed to a chemical one [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The BET model can actually be written in a few ways based on what the maximum number of layers that can be adsorbed is believed to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' At one layer, it simplifies to the Langmuir model (which only models one layer) and at an arbitrary number of layers n it becomes significantly more complex (and unlikely to be found via SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Fortunately the form at n = ∞ is more concise and can be simplified further: 20 Background Knowledge in Symbolic Regression A PREPRINT c1 ∗ (p/p0) (1 − p/p0) ∗ (1 − p/p0 + c2(p/p0)) (13) Note that while the p0 term is written here, this constant is not provided to the SR algorithms, requiring that they fit a third constant (if they were to find this functional form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4 Thermodynamic Constraint Functions Three constraint checking functions (which follow from the three thermodynamic constraints introduced previously in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='4) were developed using the Python library SymPy which is designed for symbolic math [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Each function returns either true or false, depending on if its constraint is met or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While these functions are useful for examining the expressions generated after a run, simply discarding an expression for failing one or more constraint during a run can severely hinder search potential by cutting off intermediary steps between better expressions that may also pass the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of this, each function has a corresponding weight that allows it to act as a “soft constraint".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, a constraint will not affect the score of an expression if it is passed but will multiply (worsen) that score if it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This approach (as implemented in PySR) is detailed in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Algorithm 1 Modified genetic algorithm Input: tree, dataset, options Output: score, loss function SCOREFUNC(tree, dataset, options) loss ← evalLoss(tree, dataset, options) ▷ Traditional error calculation penalties ← options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='penalties if penalties not empty then ▷ Skip slow calculation if possible expr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' var ← parseTree(tree) for p ∈ penalties and tf ∈ thermoFunctions do if tf(expr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' var) = False then loss ← loss ∗ p end if end for end if score ← scoreFunc(loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' tree,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' options) ▷ Factor in complexity return score,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' loss end function 21 Background Knowledge in Symbolic Regression A PREPRINT Algorithm 2 Monotonic Non-decreasing Check Input: expr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' var,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' start,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' stop Output: passing (If the expression is monotonically non-decreasing) function MONTONICNONDECREASING(expr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' var,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' start,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' stop) if expr is constant then return True end if turningPoints ← inflection points (zeros) of expr ▷ Calculated using SymPy if turningPointsis empty then if expr(stop) − expr(start) ≥ 0 then ▷ Measure slope return True else return False end if end if turningPoints ← [start,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' turningPoints,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' stop] ▷ Include bounds for each sequential pair of points p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' q ∈ turningPoints do if expr(q) − expr(p) ≤ 0 then ▷ Measure slope between zeros return False end if end for return True end function 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 Implementation of Thermodynamic Constraints In order to maintain parity between PySR and BMS, the Julia library PyCall was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This allowed the same Python code that checked thermodynamic constraints in BMS to be used within the Julia code of the SymbolicRegression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl library (the back-end of PySR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Beyond the concern of a fair comparison across platforms, PyCall showed itself to be somewhat necessary for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' As of writing, there is no Julia library close to as full-featured as SymPy (and SymPy can have some serious limitations/difficulties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' One method explored was using the Julia library MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl to call MATLAB code from within Julia in a similar way to how PyCall works with Python code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this did eventually work (and MATLAB does have robust symbolic math capabilities) it showed itself to be very difficult to set up due to intricacies in how the two languages store data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In the end, using MATLAB was not significantly faster than Python in this work (which is somewhat surprising).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' One downside to the current structure of our version of SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl and the languages it relies on is that it must be run in distributed mode instead of threaded mode (meaning multiple processes must be created, requiring more overhead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is due to the fact that PyCall expects only one Julia instance to attempt to use Python at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' With multiple processes, a new instance of the PyCall Julia package is created for each process, thus avoiding memory access conflicts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='6 PySR Data Collection Although PySR handles many expressions, populations of those expressions and the Hall of Fame of the best expressions across all populations so far, accessing that information has proven difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' PySR’s goal each iteration is to produce a Pareto front showing what it has found to be the most accurate expression at each complexity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This means the vast majority of expressions in a population are not shown (although some of them may be duplicates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this is an issue, on the other hand, if an expression never makes it the Pareto front at any point during the run, by definition of the search it is not as important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' That said, this limitation with output means that an expression such as the Langmuir isotherm may be generated and never shown due to poor accuracy relative to its peers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In order to transform a few hundred Pareto front printouts to usable data, the full output text is parsed using Python and stored in a Pandas DataFrame [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The values collected from the raw text are the expression itself, score, loss, complexity, runtime, iteration and run number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this is a significant amount of data already, there are a few interesting attributes yet to be calculated – specifically those relating to the simplified form of the expressions and those related to the thermodynamic constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In order to make further parsing manageable, once the simplified form of each expression is generated, only the most accurate expression with that form is retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This step often reduces the number of rows in the DataFrame by about 1000x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The simplified form of each expression is calculated both with original (optimized/fit) and substituted constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The new complexity of the simplified form may or may not be smaller 22 Background Knowledge in Symbolic Regression A PREPRINT than the original complexity so both attributes are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Finally, the thermodynamic constraints that may have been used to guide the search are checked for each expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='7 Identifying “interesting" expressions An important component of this work is identification of interesting or meaningful expressions generated by the SR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' An expression is most obviously meaningful if it is, or can be simplified to, the canonical form of an already known expression such as the Langmuir or BET adsorption isotherms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Unfortunately, determining if one expression is equivalent to another is a very difficult, and sometimes undecidable problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In fact, while not true for this work due to limits on operators and constants used, Richardson’s theorem shows that given the right set of operators and the transcendental numbers π and e, it may be impossible to show that one expression is equivalent to another [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' It is also important to note that there is no incentive for SR to generate expressions that are easily read or understood – if it even had a concept of what expressions fit those criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Even when ground truth expressions are found, they can often be in unrecognisable forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Furthermore, while SymPy does have a reasonably capable simplification function, it is of course not perfect and may encounter expressions it cannot handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of these facts, this work uses a more complex function built on top of SymPy’s simplification function and also accepts that much analysis will have to be done manually, at least for now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='8 Simplification Function While SymPy is capable of simplifying symbolic math expressions and equations, there are definitely expressions that it may not simplify to their shortest form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Furthermore, while constants may be optimized per expression within both PySR and BMS, the expressions themselves are the true object of the search so constants are often left symbolic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' c1, c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of this, in the interest of effectively simplifying without naively filling all symbolic constants with random numbers, symbolic constants are substituted with prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This avoids situations where one constant can completely remove another through multiplication or division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this method has some advantages, there are also reasons to simplify using fit constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Primarily, some expressions such as the BET isotherm have the same constant in multiple places, meaning it can be factored out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This should also be the case with well fit constants for newly generated expressions (especially those that turn out to be the same structure as previous isotherms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The conversion from the original BET isotherm to the form the simplification function finds is shown below: c1 ∗ (p/p0) (1 − p/p0) ∗ (1 − p/p0 + c2(p/p0)) → c1 ∗ p (c2 ∗ p + c3 + p2) (14) Once constants are decided (either fit or primes), ordinary SymPy simplification is used to compress the expression as much as is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Then, if the resulting expression is rational (has only integer exponents and no division by zero), it can be written as a fraction and possibly simplified further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Specifically, if there is a common factor between the numerator and denominator because they are both degree 1 or larger, it may be possible to remove up to the difference in their degrees (see example below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While this situation is not guaranteed, having the capability to simplify it is still useful, especially when examining many generated expressions and differences between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 3x 2x + 3x2 → 3 2 + 3x (15) 23 Background Knowledge in Symbolic Regression A PREPRINT Algorithm 3 Simplification Function Input: expr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' vars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' pars Output: can (Canonical form of expression) function SIMPLIFY(expr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' vars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' pars) expr ← expression vars ← variables pars ← parameters for p in pars do p ← prime number ▷ Substitute constants with unique primes end for Simplify using SymPy if expr is rational then num,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' denom ← expr as fraction ▷ Calculated using SymPy if degree(num) > degree(denom) then factor ← leading term of num else factor ← leading term of denom end if expr ← num/factor denom/factor ▷ Remove common factor end if return expr end function It is important to note that the simplification function will not always return the simplest possible form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While it does obviously reduce complexity, the original goal of the function was to have a canonical form for each expression such that whenever it is found, in whatever form, it will be simplified to that form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This effect is achieved by always writing the expression as a rational function if possible (and not simplifying further from that point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of this, when expressions are plotted on Pareto fronts (or their complexities are compared in general) the minimum complexity between the generated form and the simplified form is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='9 Fitting Constants in PySR By default, PySR uses Nelder-Mead optimization to fit constants for expressions as they are generated [31] [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Nelder-Mead is well suited to optimizing parameters for arbitrary generated expressions because it does not require any derivative or gradient information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Simply put, the algorithm evaluates the function at n + 1 points where n is the number of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Then the next point is selected by finding the point with the highest function evaluation and looking to the opposite side of the rest of the points (a simplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This iteratively moves the worst point to a likely better location, eventually moving towards a local optimum regardless of how the function is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' One downside to this method is that it is not guaranteed (or even expected) to find global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This issue is usually rectified by allowing for multiple starts from random locations, and selecting the best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In PySR, each expression gets 8 attempts by default, randomizing the parameters between each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This is typically enough but there are occasionally cases where an expression should be much more accurate than it is due to poorly fitted constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' To fit ground-truth expressions and check constants in post-processing, we also applied Nelder-Mead optimization using SciPy [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' In this case, many more iterations were allowed to ensure ground truths and interesting functional forms had the best chance to show their effectiveness (or lack thereof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='10 Runtime An important consideration when examining the effectiveness of the thermodynamic constraints in guiding SR is the impact on computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While not extreme, symbolic math (in SymPy) can be slow, especially compared to the otherwise efficient and optimized Julia code running behind the PySR front-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' The following plot shows the difference in iteration time (the time between one Pareto front and the next being printed by PySR) across different datasets, and more importantly, across different constraint penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' It shows that if SymPy can avoid being loaded, runtime is not affected but that in all other cases, it is increased by about an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 24 Background Knowledge in Symbolic Regression A PREPRINT Figure 10: Average runtimes across all datasets and combinations of thermodynamic constraint penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Runs with all penalties set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0 are highlighted in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Standard deviation is shown by error bars at the top of each bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='11 Environment Testing was done on both the batch and cpu2021 partitions of the UMBC High Performance Computing Facility (https://hpcf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='umbc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='edu/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' This allowed for the longer runtimes and larger parameter exploration necessitated by adding thermodynamic constraints with variable penalties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' It was also important to be able to run long term because BMS requires some build-up time to converge to the desired distribution, since it is based off of MCMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' While BMS is entirely based in Python, SymbolicRegression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl (the backend for PySR) is entirely in Julia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' Because of the need to use SymPy, the Julia package PyCall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='jl was used to allow Python code and libraries to be run by Julia (at the time this work was completed, symbolic math libraries in Julia could not evaluate the constraints considered in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' To allow for parallel use of Python / SymPy from within Julia, PySR was run in distributed mode, necessitating more overhead than threaded mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 25 Average Runtime Per Iteration Nitrogen [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3] Nitrogen [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3] Nitrogen [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0] Methane [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3] Methane [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3] Methane [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='01 Isobutane [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3] Isobutane [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3] Isobutane [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='0] 1 BET [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='31 BET[1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} +page_content='5 Iteration Time (in seconds)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFKT4oBgHgl3EQf1S6Z/content/2301.11919v1.pdf'} diff --git a/stE4T4oBgHgl3EQfww21/content/tmp_files/2301.05253v1.pdf.txt b/stE4T4oBgHgl3EQfww21/content/tmp_files/2301.05253v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5de718944c13175d95e2ed683117a01103412cfe --- /dev/null +++ b/stE4T4oBgHgl3EQfww21/content/tmp_files/2301.05253v1.pdf.txt @@ -0,0 +1,395 @@ +A Scalable Technique for Weak-Supervised Learning +with Domain Constraints +Sudhir Agarwal +Anu Sreepathy +Lalla Mouatadid +Intuit AI Research Center, Mountain View, CA, USA +{firstname_lastname@intuit.com} +Abstract +We propose a novel scalable end-to-end pipeline that uses symbolic domain knowl- +edge as constraints for learning a neural network for classifying unlabeled data in +a weak-supervised manner. Our approach is particularly well-suited for settings +where the data consists of distinct groups (classes) that lends itself to clustering- +friendly representation learning and the domain constraints can be reformulated +for use of efficient mathematical optimization techniques by considering multiple +training examples at once. We evaluate our approach on a variant of the MNIST +image classification problem where a training example consists of image sequences +and the sum of the numbers represented by the sequences – e.g., ( +, +, 9), and +show that our approach scales significantly better than previous approaches that rely +on computing all constraint satisfying combinations for each training example. +1 +Introduction +Integrating logical reasoning and machine learning is one of the main challenges of AI. Doing so can +lead to systems that can solve more complex problems, learn from less data, comply with domain +knowledge, or have better performance [1, 2]. +In particular, incorporating domain knowledge as symbolic constraints during ML training has +shown to improve the accuracy of ML predictions by learning an ML model that tries to enforce +the constraints as much as possible. These techniques are especially promising when training +data is limited and constraints are available or can be easily acquired. Some recent approaches +handle constraints by incorporating probabilities from neural networks into logic programs. For +instance, models that incorporate constraints expressed in ProbLog [3] and Answer Set Programs +have been proposed in [4] and [5] respectively. Some other approaches such as [6] and [7] use +stochastic grammar based logic programs and probabilistic deductive database with differentiable +reasoning respectively. However, due to the limitations arising from model grounding, all these +approaches don’t scale well when the complexity of the problem increases. The grounding of a +constraint is the computation of all satisfying assignments of the constrained variables, which can +lead to intractable combinatorial blow-up in cases with a large number of constrained variables. +This problem, known as the grounding bottleneck, arises in logical reasoning domains, such as +statistical relational AI (StarAI) [8] and Answer Set Programming (ASP) [9]. While the techniques +presented in [4, 5, 6] are highly expressive and accurate, they rely on naively iterating through all +possible output combinations for each training example in order to compute the satisfying output +combinations. For instance, for ( +, +, 5), they would iterate over 102 combinations in order to +compute the satisfying combinations (0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0). While expressiveness +can be traded for scalability – DeepStochLog [6] vs. DeepProbLog [4] –, the combinatorial blow-up +still poses a major hurdle for more complex problems; with n images per example, the number of +possible output combinations grows exponentially (10n). To the best of our knowledge, there is no +previous approach that can be used for a multitude of industry use cases, which often have complex +domain constraints and require high scalability. +Has it Trained Yet? Workshop at the Conference on Neural Information Processing Systems (NeurIPS 2022). +arXiv:2301.05253v1 [cs.LG] 12 Jan 2023 + +又73In this work, we develop a scalable weak-supervised learning technique to incorporate symbolic +domain constraints into neural networks. Our key contribution is an end-to-end pipeline that avoids +the grounding bottleneck and scales to state-of-the-art results on a challenging neuro-symbolic +problem. Our approach is based on the insight that in some cases (a) the data consists of distinct +groups (e.g., digit images) that are partially recoverable through clustering-friendly representation +learning, and (b) the constraint (e.g., addition) is the same across all training examples and the +properties of the constraint are known and can be exploited to our advantage. +Height +Width +Figure 1: An exam- +ple with h = 3, w = +2, and s = 93. +In this paper, we focus on a variant of the MNIST image classification problem +where each training example is, instead of an (image, label) pair, a sequence of +w × h MNIST images and an integer s where w (width) represents the number +of digits in a number and h (height) represents the number of numbers in the +example whose sum is equal to s. Unlike in the standard MNIST classification +problem, since we do not have the image labels, we first infer the labels, and +then use them to train the classifier. A practical example of such a use case is in +information extraction from hand-filled forms such as financial documents (e.g., +loan and tax forms) where certain "total" fields are pre-filled in typesetting, +while other fields leading up to the total are handwritten information from the +user. +For the MNIST classification variant, the state-of-the art solution [6] has been +shown to work for w ≤ 4, h = 2. We show that our approach scales in both +width and height as our model achieves between 92% and 97% accuracy for w ≤ 10, h ≤ 6 with the +training time being independent of w and h. +2 +Method +Overall, our solution can be broken down into the four main steps as presented in Figure 2. +Assign each image the +label of its cluster +Use constraints to detect +and where possible fix +wrong image labels +Train image classifier with +inferred image labels +Predict cluster labels using +Mathematical Optimization +Infer Image Labels +, +, 86 +, +, 98 +Substitute each image by the +variable corr. to its cluster +10c1 + c2 + 10c3 + c4 = 86 +10c4 + c3 + 10c5 + c1 = 98 + … +Use constraints to compute +variable assignment that +satisfies most examples +… +Training examples +c1=3, c2=4, +c3=5, c4=2, +c5=7, … +Cluster +c0 +c3 +c1 +c2 +c9 +c4 +c6 +c5 +c7 +c8 +Autoencoder +Figure 2: The main steps of our proposed method. +1. Autoencoder-based clustering: We first pre-train a fully connected symmetric autoencoder with +dense layer dimensions [500, 500, 2000, 10] for 300 epochs, similar to the one in [10], then use the +weights of the encoder to cluster with k-means using k-means++ initialization and 10 clusters. The +input for this step is the set of individual images across all training examples. +2. Cluster label prediction: A training example of width w, height h, and sum s can be represented +as a linear equation �h +i=1 +�w +j=1 v(imgi,j) × 10w−j = s, where v(imgi,j) represents the variable +assigned to the cluster that contains the jth image in the ith number of the training example. If the +clustering was 100% accurate, one could simply assign each cluster ci, i ∈ [0, 9] a variable vi, and +solve a system of linear equations to determine the values of each vi. However, since we don’t start +with a clustering of 100% purity, in this step, we formulate the problem as an integer linear program +using CVXPY [11, 12]. We split the training examples into batches of size 100, solve the optimization +problem for each batch using L1-norm as the objective function, and determine the overall result as +the batch result that satisfies the most training examples over all batches. As a result, this step assigns +each cluster an integer from [0, 9]. +2 + +又3632sInput layer +Hidden layer +Output layer +X1 +X2 +no +X3 +Out2 +X4 +X5 +Outm0 +1 +2 +3 +3 +4 +3 +力 +5 +S +6 +8 +8 +Network training +9 +7 +Data & Labels +8 +93273. Image label inference: The cluster labels computed in the previous step give the initial labeling +for each image in the training dataset. This labeling can be at most as accurate as the clustering purity. +The goal of this step is to improve image labeling based on rule-based inference. Currently, this step +consists of only one rule based on the fact that a variable in a linear equation can be resolved if all +other variables are known. We use an iterative process by which we increasingly resolve variables in +the system of equations with every iteration. In order to be able to do so, we need a way to detect +correctly clustered images. Since we do not know with certainty which images are correctly clustered, +we assume the images closer to their respective cluster’s centroid as correctly labeled as they have a +higher chance of being correctly clustered. +Algorithm 1 Image label inference +1: Correct ← {} +2: for 1 ≤ radius ≤ 5 do +3: +NewCorrect ← GetImagesWithinRadius(radius) +4: +Correct ← Correct ∪ NewCorrect +5: +Changed ← True +6: +while Changed = True do +7: +Changed ← InferCorrectLabels(Correct) +8: +end while +9: end for +Algorithm 2 InferCorrectLabels +1: Changed ← False +2: for all Ex ∈ TrainingExamples do +3: +UnresolvedImages ← GetUnresolvedImages(Correct, Ex) +4: +if |UnresolvedImages| = 1 then +5: +Let UnresolvedImages = {img} +6: +ResolveImageLabel(Correct, Ex, img) +7: +Correct ← Correct ∪ {img} +8: +Changed ← True +9: +end if +10: end for +11: return Changed +The image label inference (Algorithm 1) works as follows. It starts with an empty set of correctly +clustered images. At every iteration 1 ≤ radius ≤ 5, it gets the images that are at most radius away +from their resp. cluster centroids using the method GetImagesWithinRadius, and adds them to the set +of correctly labeled images. Based on this set of correctly labeled images, the algorithm then tries to +infer further (correct) image labels. It does so by using the method InferCorrectLabels (Algorithm 2) +in a while loop as long as it can infer new image labels as correct labels. InferCorrectLabels +iterates over all training examples. Ex denotes the training example considered in one such iteration. +InferCorrectLabels first computes the set of unresolved images in Ex, i.e., the images in Ex that +have not been identified as correct thus far (Line 3). It does so by calling the GetUnresolvedImages +method, which simply iterates over the images in the given training example, and returns the ones that +are not in the set of correctly labeled images thus far. If a training example has only one unresolved +image, say img, (Line 4), then InferCorrectLabels calls the method ResolveImageLabel to compute +its correct label (Line 6). ResolveImageLabel then computes the correct label by substituting all +images in Ex except img by their respective labels and solving a linear equation for img such that the +sum of the numbers represented by the sequences of images in Ex is equal to the sum given in Ex. +Finally, ResolveImageLabel updates the label of img to the inferred label. As last step of the iteration, +InferCorrectLabels adds img to the set of correct images (Line 7). +4. Classification: The final step of the pipeline is to train a CNN-based classifier [13] using the final +inferred image labels from Step 3. The network consists of a convolutional layer with 32 filters and a +kernel of size 3 × 3, followed by a max pooling layer, two more convolutional layers with 64 filters +and kernel size 3 × 3 each, another max pooling layer and a dense layer of 100 nodes before the +output layer. All layers use ReLu activation and He weight initialization. We use stochastic gradient +descent optimizer with a learning rate of 0.01 and momentum of 0.9. We train for 10 epochs. +3 + +3 +Results +Figures 3 and 4 show the classification accuracy and training time respectively for varying widths +and heights. For every w × h combination, the training dataset uses all 60K images in the MNIST +training dataset exactly once. This means that the higher the w × h combinations, the smaller the +dataset. Yet our accuracy remains above 90%. The reported accuracy is the accuracy of the MNIST +classifier (Step 4) using the inferred image labels (Step 3) as described in Section 2. The reported +time is the total time for all 4 steps of the pipeline 1. All our experiments are run on a MacBook Pro +2021 (Apple M1 Max, 64GB). +width +Classification accuracy (%) +92.00 +93.00 +94.00 +95.00 +96.00 +97.00 +2 +4 +6 +8 +10 +height = 2 +3 +4 +5 +6 +Figure 3: Classification accuracy % for varying w × h combinations. +width +Training time (s) +150 +175 +200 +225 +250 +2 +4 +6 +8 +10 +height = 2 +3 +4 +5 +6 +Figure 4: Total training time in seconds for varying w × h combinations. +As previously mentioned, DeepStochLog [6], the state-of-the-art solution so far, can solve for +examples with h = 2, w ≤ 4, and scales better than earlier approaches [4, 5], both of which can +handle h = 2, w ≤ 2. As shown by the results in Figure 3, our approach not only scales horizontally +(w > 4), but also vertically (h > 2). Furthermore, as shown in Figure 4 our approach doesn’t blow up +in the total training time as w and h grow. Our clustering component (Step 1), has about 91.3% purity. +As a consequence, after the optimization step (Step 2), assuming the optimizer finds the correct +cluster labels, the image label accuracy is also about 91.3%, which leads to the classification accuracy +between 92% and 93%. The reason for higher classification accuracy for smaller w, h combinations +is that in these cases the image label inference algorithm (Step 3) has a higher chance of inferring the +correct image label since it is more likely that an example has only one unresolved image that can be +resolved. This increases the accuracy of the final image labels used for training the classifier (Step 4). +In order to compare the addition accuracy with that of DeepStochLog [6], we augmented our pipeline +with a component for performing addition on the output of our classifier. With this architecture, we get +an addition accuracy of 95%, 87%, 78.5% and 72% respectively for w = 1 to w = 4 and h = 2. The +1The total training time we report doesn’t include the autoencoder training time. This step takes roughly 30 +minutes. We trained the autoencoder once and used the same encoder weights for all w × h runs since the input +to the autoencoder is just the set of individual images across all training examples and is independent of w and h. +4 + +reason for lower addition accuracy than the classification accuracy (Figure 3) is that with increasing +number of images per example it is more likely that a training example has at least one wrongly +classified image resulting in a wrong prediction of the sum. With over 99% classification accuracy, we +would get addition accuracy similar to that of DeepStochLog. Improving the classification accuracy +is a planned future work as discussed in Section 4. +One reason why we can avoid the combinatorial blow-up is attributed to the nature of the problem. +The addition constraint is linear and can be solved efficiently for a set of examples if they can be +formulated using the same set of variables. In our case, this is made possible by the clustering step +which essentially reduces the number of variables from 60k (one for each image) to 10. Furthermore, +our approach illustrates the trade-off between expressiveness and the scalability of the language. +Similar to how DeepStochLog scales better than DeepProbLog by using stochastic logic programs +that are less expressive than probabilistic logic programs used by DeepProbLog, our approach scales +even further but can only work for problems that fit our "cluster-then-optimize" paradigm. +4 +Discussion & Future Work +Our classification accuracy depends strongly on the clustering purity since the image label accuracy +after the optimization step is upper bounded by it. This is why we now use the autoencoder as a +pre-processing step which gives us approximately 91.3% clustering purity. Our future work is to +make the pipeline more robust w.r.t. the representation learning step. One way to achieve this may be +to keep track of the previous (wrong) labels and the new (correct) ones in the image label inference, +and use this information during the training of the classifier or as feedback to prior steps. +As of now, the batch size in the optimization step is fixed to 100. This may not be the best choice for +larger combinations, say (w = 12, h = 12), and the optimizer may not find the correct result. We +thus plan to dynamically compute the batches of examples (and their size) to increase the performance. +Furthermore, adding more inference rules or heuristics to the image label inference step — currently +it relies on one simple rule — would help achieve higher image label accuracy for more cases. +In our experiments, similar to [6, 5], we have used the minimum number of training examples for +every w × h combination such that the MNIST training dataset (60K images) is used entirely where +every image appears exactly once across all examples. However, we have preliminary results that +show that for small w × h combinations, increasing the number of training examples (i.e., allowing +images to appear more than once in the training dataset) leads to higher classification accuracy. For +instance, for h = 2, w = 5, with a minimum number of examples (n = 6000), we achieve 93.05% +classification accuracy, where as increasing n to 18000 achieves 98% accuracy. This is not surprising +since having an image appear more than once gives the image label inference algorithm a higher +chance of correctly resolving its label. +In this work, we have presented a scalable technique for weak-supervised learning using domain +constraints instead of labels, and evaluated it on the MNIST dataset with the addition constraint. We +have shown that our approach scales better than most recent approaches with respect to both width +and height of training examples while the total training time is independent of the width and height. +Our approach is applicable to problems with numerical and logical constraints where the input can +be clustered and multiple examples can be used together to efficiently evaluate the constraints. To +achieve a more general implementation, we plan to tackle the handwritten formula problem as defined +in [14] as our next use case. Furthermore, to illustrate the combination of both numerical and logical +constraints, we also plan to tackle the Sudoku problem as defined in [5]. +Acknowledgments and Disclosure of Funding +We thank our colleagues Kamalika Das and Jiaxin Zhang from Intuit AI Research Center for +discussions and comments. +5 + +References +[1] Don Monroe. Neurosymbolic ai. Communications of the ACM, 65(10):11–13, 2022. +[2] Pascal Hitzler and Md. Kamruzzaman Sarker, editors. Neuro-Symbolic Artificial Intelligence: +The State of the Art, volume 342 of Frontiers in Artificial Intelligence and Applications. IOS +Press, 2021. +[3] Luc De Raedt and Angelika Kimmig. Probabilistic (logic) programming concepts. Mach. +Learn., 100(1):5–47, 2015. +[4] Robin Manhaeve, Sebastijan Dumancic, Angelika Kimmig, Thomas Demeester, and Luc +De Raedt. +Deepproblog: Neural probabilistic logic programming. +Advances in Neural +Information Processing Systems, 31, 2018. +[5] Zhun Yang, Adam Ishay, and Joohyung Lee. Neurasp: Embracing neural networks into answer +set programming. In 29th International Joint Conference on Artificial Intelligence (IJCAI +2020), 2020. +[6] Thomas Winters, Giuseppe Marra, Robin Manhaeve, and Luc De Raedt. Deepstochlog: Neural +stochastic logic programming. In Thirty-Sixth AAAI Conference on Artificial Intelligence, +AAAI 2022, pages 10090–10100. AAAI Press, 2022. +[7] William W Cohen, Fan Yang, and Kathryn Rivard Mazaitis. Tensorlog: Deep learning meets +probabilistic databases. Journal of Artificial Intelligence Research, 1:1–15, 2018. +[8] Efthymia Tsamoura, Víctor Gutiérrez-Basulto, and Angelika Kimmig. Beyond the grounding +bottleneck: datalog techniques for inference in probabilistic logic programs. In Proceedings of +the AAAI Conference on Artificial Intelligence, pages 10284–10291. AAAI Press, 2020. +[9] Bernardo Cuteri, Carmine Dodaro, Francesco Ricca, and Peter Schüller. Overcoming the +grounding bottleneck due to constraints in asp solving: Constraints become propagators. In +IJCAI, pages 1688–1694, 2020. +[10] Chengwei. How to do unsupervised clustering with keras? +https://www.dlology.com/ +blog/how-to-do-unsupervised-clustering-with-keras/, 2018. +[11] Steven Diamond and Stephen Boyd. CVXPY: A Python-embedded modeling language for +convex optimization. Journal of Machine Learning Research, 17(83):1–5, 2016. +[12] Akshay Agrawal, Robin Verschueren, Steven Diamond, and Stephen Boyd. A rewriting system +for convex optimization problems. Journal of Control and Decision, 5(1):42–60, 2018. +[13] Jason Brownlee. How to develop a cnn for mnist handwritten digit classification. https: +//machinelearningmastery.com/how-to-develop-a-convolutional-neural- +network-from-scratch-for-mnist-handwritten-digit-classification/, 2019. +[14] Qing Li, Siyuan Huang, Yining Hong, Yixin Chen, Ying Nian Wu, and Song-Chun Zhu. Closed +loop neural-symbolic learning via integrating neural perception, grammar parsing, and symbolic +reasoning. In International Conference on Machine Learning, pages 5884–5894. PMLR, 2020. +6 + diff --git a/stE4T4oBgHgl3EQfww21/content/tmp_files/load_file.txt b/stE4T4oBgHgl3EQfww21/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..86147ce4e86b012a667dc31b30e05e1682a1b5cf --- /dev/null +++ b/stE4T4oBgHgl3EQfww21/content/tmp_files/load_file.txt @@ -0,0 +1,204 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf,len=203 +page_content='A Scalable Technique for Weak-Supervised Learning with Domain Constraints Sudhir Agarwal Anu Sreepathy Lalla Mouatadid Intuit AI Research Center, Mountain View, CA, USA {firstname_lastname@intuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='com} Abstract We propose a novel scalable end-to-end pipeline that uses symbolic domain knowl- edge as constraints for learning a neural network for classifying unlabeled data in a weak-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Our approach is particularly well-suited for settings where the data consists of distinct groups (classes) that lends itself to clustering- friendly representation learning and the domain constraints can be reformulated for use of efficient mathematical optimization techniques by considering multiple training examples at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We evaluate our approach on a variant of the MNIST image classification problem where a training example consists of image sequences and the sum of the numbers represented by the sequences – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=', ( , , 9), and show that our approach scales significantly better than previous approaches that rely on computing all constraint satisfying combinations for each training example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 1 Introduction Integrating logical reasoning and machine learning is one of the main challenges of AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Doing so can lead to systems that can solve more complex problems, learn from less data, comply with domain knowledge, or have better performance [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In particular, incorporating domain knowledge as symbolic constraints during ML training has shown to improve the accuracy of ML predictions by learning an ML model that tries to enforce the constraints as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' These techniques are especially promising when training data is limited and constraints are available or can be easily acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Some recent approaches handle constraints by incorporating probabilities from neural networks into logic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' For instance, models that incorporate constraints expressed in ProbLog [3] and Answer Set Programs have been proposed in [4] and [5] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Some other approaches such as [6] and [7] use stochastic grammar based logic programs and probabilistic deductive database with differentiable reasoning respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' However, due to the limitations arising from model grounding, all these approaches don’t scale well when the complexity of the problem increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The grounding of a constraint is the computation of all satisfying assignments of the constrained variables, which can lead to intractable combinatorial blow-up in cases with a large number of constrained variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This problem, known as the grounding bottleneck, arises in logical reasoning domains, such as statistical relational AI (StarAI) [8] and Answer Set Programming (ASP) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' While the techniques presented in [4, 5, 6] are highly expressive and accurate, they rely on naively iterating through all possible output combinations for each training example in order to compute the satisfying output combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' For instance, for ( , , 5), they would iterate over 102 combinations in order to compute the satisfying combinations (0, 5), (1, 4), (2, 3), (3, 2), (4, 1), (5, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' While expressiveness can be traded for scalability – DeepStochLog [6] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' DeepProbLog [4] –, the combinatorial blow-up still poses a major hurdle for more complex problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' with n images per example, the number of possible output combinations grows exponentially (10n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' To the best of our knowledge, there is no previous approach that can be used for a multitude of industry use cases, which often have complex domain constraints and require high scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Has it Trained Yet?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Workshop at the Conference on Neural Information Processing Systems (NeurIPS 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='05253v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='LG] 12 Jan 2023 又73In this work, we develop a scalable weak-supervised learning technique to incorporate symbolic domain constraints into neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Our key contribution is an end-to-end pipeline that avoids the grounding bottleneck and scales to state-of-the-art results on a challenging neuro-symbolic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Our approach is based on the insight that in some cases (a) the data consists of distinct groups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=', digit images) that are partially recoverable through clustering-friendly representation learning, and (b) the constraint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=', addition) is the same across all training examples and the properties of the constraint are known and can be exploited to our advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Height Width Figure 1: An exam- ple with h = 3, w = 2, and s = 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In this paper, we focus on a variant of the MNIST image classification problem where each training example is, instead of an (image, label) pair, a sequence of w × h MNIST images and an integer s where w (width) represents the number of digits in a number and h (height) represents the number of numbers in the example whose sum is equal to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Unlike in the standard MNIST classification problem, since we do not have the image labels, we first infer the labels, and then use them to train the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' A practical example of such a use case is in information extraction from hand-filled forms such as financial documents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=', loan and tax forms) where certain "total" fields are pre-filled in typesetting, while other fields leading up to the total are handwritten information from the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' For the MNIST classification variant, the state-of-the art solution [6] has been shown to work for w ≤ 4, h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We show that our approach scales in both width and height as our model achieves between 92% and 97% accuracy for w ≤ 10, h ≤ 6 with the training time being independent of w and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 2 Method Overall, our solution can be broken down into the four main steps as presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Assign each image the label of its cluster Use constraints to detect and where possible fix wrong image labels Train image classifier with inferred image labels Predict cluster labels using Mathematical Optimization Infer Image Labels , , 86 , , 98 Substitute each image by the variable corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' to its cluster 10c1 + c2 + 10c3 + c4 = 86 10c4 + c3 + 10c5 + c1 = 98 … Use constraints to compute variable assignment that satisfies most examples … Training examples c1=3, c2=4, c3=5, c4=2, c5=7, … Cluster c0 c3 c1 c2 c9 c4 c6 c5 c7 c8 Autoencoder Figure 2: The main steps of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Autoencoder-based clustering: We first pre-train a fully connected symmetric autoencoder with dense layer dimensions [500, 500, 2000, 10] for 300 epochs, similar to the one in [10], then use the weights of the encoder to cluster with k-means using k-means++ initialization and 10 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The input for this step is the set of individual images across all training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Cluster label prediction: A training example of width w, height h, and sum s can be represented as a linear equation �h i=1 �w j=1 v(imgi,j) × 10w−j = s, where v(imgi,j) represents the variable assigned to the cluster that contains the jth image in the ith number of the training example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' If the clustering was 100% accurate, one could simply assign each cluster ci, i ∈ [0, 9] a variable vi, and solve a system of linear equations to determine the values of each vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' However, since we don’t start with a clustering of 100% purity, in this step, we formulate the problem as an integer linear program using CVXPY [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We split the training examples into batches of size 100, solve the optimization problem for each batch using L1-norm as the objective function, and determine the overall result as the batch result that satisfies the most training examples over all batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' As a result, this step assigns each cluster an integer from [0, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 2 又3632sInput layer Hidden layer Output layer X1 X2 no X3 Out2 X4 X5 Outm0 1 2 3 3 4 3 力 5 S 6 8 8 Network training 9 7 Data & Labels 8 93273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Image label inference: The cluster labels computed in the previous step give the initial labeling for each image in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This labeling can be at most as accurate as the clustering purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The goal of this step is to improve image labeling based on rule-based inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Currently, this step consists of only one rule based on the fact that a variable in a linear equation can be resolved if all other variables are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We use an iterative process by which we increasingly resolve variables in the system of equations with every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In order to be able to do so, we need a way to detect correctly clustered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Since we do not know with certainty which images are correctly clustered, we assume the images closer to their respective cluster’s centroid as correctly labeled as they have a higher chance of being correctly clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Algorithm 1 Image label inference 1: Correct ← {} 2: for 1 ≤ radius ≤ 5 do 3: NewCorrect ← GetImagesWithinRadius(radius) 4: Correct ← Correct ∪ NewCorrect 5: Changed ← True 6: while Changed = True do 7: Changed ← InferCorrectLabels(Correct) 8: end while 9: end for Algorithm 2 InferCorrectLabels 1: Changed ← False 2: for all Ex ∈ TrainingExamples do 3: UnresolvedImages ← GetUnresolvedImages(Correct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Ex) 4: if |UnresolvedImages| = 1 then 5: Let UnresolvedImages = {img} 6: ResolveImageLabel(Correct,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' img) 7: Correct ← Correct ∪ {img} 8: Changed ← True 9: end if 10: end for 11: return Changed The image label inference (Algorithm 1) works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' It starts with an empty set of correctly clustered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' At every iteration 1 ≤ radius ≤ 5, it gets the images that are at most radius away from their resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' cluster centroids using the method GetImagesWithinRadius, and adds them to the set of correctly labeled images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Based on this set of correctly labeled images, the algorithm then tries to infer further (correct) image labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' It does so by using the method InferCorrectLabels (Algorithm 2) in a while loop as long as it can infer new image labels as correct labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' InferCorrectLabels iterates over all training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Ex denotes the training example considered in one such iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' InferCorrectLabels first computes the set of unresolved images in Ex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=', the images in Ex that have not been identified as correct thus far (Line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' It does so by calling the GetUnresolvedImages method, which simply iterates over the images in the given training example, and returns the ones that are not in the set of correctly labeled images thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' If a training example has only one unresolved image, say img, (Line 4), then InferCorrectLabels calls the method ResolveImageLabel to compute its correct label (Line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' ResolveImageLabel then computes the correct label by substituting all images in Ex except img by their respective labels and solving a linear equation for img such that the sum of the numbers represented by the sequences of images in Ex is equal to the sum given in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Finally, ResolveImageLabel updates the label of img to the inferred label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' As last step of the iteration, InferCorrectLabels adds img to the set of correct images (Line 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Classification: The final step of the pipeline is to train a CNN-based classifier [13] using the final inferred image labels from Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The network consists of a convolutional layer with 32 filters and a kernel of size 3 × 3, followed by a max pooling layer, two more convolutional layers with 64 filters and kernel size 3 × 3 each, another max pooling layer and a dense layer of 100 nodes before the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' All layers use ReLu activation and He weight initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We use stochastic gradient descent optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='01 and momentum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We train for 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 3 3 Results Figures 3 and 4 show the classification accuracy and training time respectively for varying widths and heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' For every w × h combination, the training dataset uses all 60K images in the MNIST training dataset exactly once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This means that the higher the w × h combinations, the smaller the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Yet our accuracy remains above 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The reported accuracy is the accuracy of the MNIST classifier (Step 4) using the inferred image labels (Step 3) as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The reported time is the total time for all 4 steps of the pipeline 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' All our experiments are run on a MacBook Pro 2021 (Apple M1 Max, 64GB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' width Classification accuracy (%) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='00 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='00 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='00 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='00 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='00 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='00 2 4 6 8 10 height = 2 3 4 5 6 Figure 3: Classification accuracy % for varying w × h combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' width Training time (s) 150 175 200 225 250 2 4 6 8 10 height = 2 3 4 5 6 Figure 4: Total training time in seconds for varying w × h combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' As previously mentioned, DeepStochLog [6], the state-of-the-art solution so far, can solve for examples with h = 2, w ≤ 4, and scales better than earlier approaches [4, 5], both of which can handle h = 2, w ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' As shown by the results in Figure 3, our approach not only scales horizontally (w > 4), but also vertically (h > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Furthermore, as shown in Figure 4 our approach doesn’t blow up in the total training time as w and h grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Our clustering component (Step 1), has about 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='3% purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' As a consequence, after the optimization step (Step 2), assuming the optimizer finds the correct cluster labels, the image label accuracy is also about 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='3%, which leads to the classification accuracy between 92% and 93%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The reason for higher classification accuracy for smaller w, h combinations is that in these cases the image label inference algorithm (Step 3) has a higher chance of inferring the correct image label since it is more likely that an example has only one unresolved image that can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This increases the accuracy of the final image labels used for training the classifier (Step 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In order to compare the addition accuracy with that of DeepStochLog [6], we augmented our pipeline with a component for performing addition on the output of our classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' With this architecture, we get an addition accuracy of 95%, 87%, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='5% and 72% respectively for w = 1 to w = 4 and h = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The 1The total training time we report doesn’t include the autoencoder training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This step takes roughly 30 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We trained the autoencoder once and used the same encoder weights for all w × h runs since the input to the autoencoder is just the set of individual images across all training examples and is independent of w and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 4 reason for lower addition accuracy than the classification accuracy (Figure 3) is that with increasing number of images per example it is more likely that a training example has at least one wrongly classified image resulting in a wrong prediction of the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' With over 99% classification accuracy, we would get addition accuracy similar to that of DeepStochLog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Improving the classification accuracy is a planned future work as discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' One reason why we can avoid the combinatorial blow-up is attributed to the nature of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' The addition constraint is linear and can be solved efficiently for a set of examples if they can be formulated using the same set of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In our case, this is made possible by the clustering step which essentially reduces the number of variables from 60k (one for each image) to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Furthermore, our approach illustrates the trade-off between expressiveness and the scalability of the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Similar to how DeepStochLog scales better than DeepProbLog by using stochastic logic programs that are less expressive than probabilistic logic programs used by DeepProbLog, our approach scales even further but can only work for problems that fit our "cluster-then-optimize" paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 4 Discussion & Future Work Our classification accuracy depends strongly on the clustering purity since the image label accuracy after the optimization step is upper bounded by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This is why we now use the autoencoder as a pre-processing step which gives us approximately 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='3% clustering purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Our future work is to make the pipeline more robust w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' the representation learning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' One way to achieve this may be to keep track of the previous (wrong) labels and the new (correct) ones in the image label inference, and use this information during the training of the classifier or as feedback to prior steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' As of now, the batch size in the optimization step is fixed to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This may not be the best choice for larger combinations, say (w = 12, h = 12), and the optimizer may not find the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We thus plan to dynamically compute the batches of examples (and their size) to increase the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Furthermore, adding more inference rules or heuristics to the image label inference step — currently it relies on one simple rule — would help achieve higher image label accuracy for more cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In our experiments, similar to [6, 5], we have used the minimum number of training examples for every w × h combination such that the MNIST training dataset (60K images) is used entirely where every image appears exactly once across all examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' However, we have preliminary results that show that for small w × h combinations, increasing the number of training examples (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=', allowing images to appear more than once in the training dataset) leads to higher classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' For instance, for h = 2, w = 5, with a minimum number of examples (n = 6000), we achieve 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content='05% classification accuracy, where as increasing n to 18000 achieves 98% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' This is not surprising since having an image appear more than once gives the image label inference algorithm a higher chance of correctly resolving its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' In this work, we have presented a scalable technique for weak-supervised learning using domain constraints instead of labels, and evaluated it on the MNIST dataset with the addition constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' We have shown that our approach scales better than most recent approaches with respect to both width and height of training examples while the total training time is independent of the width and height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Our approach is applicable to problems with numerical and logical constraints where the input can be clustered and multiple examples can be used together to efficiently evaluate the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' To achieve a more general implementation, we plan to tackle the handwritten formula problem as defined in [14] as our next use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Furthermore, to illustrate the combination of both numerical and logical constraints, we also plan to tackle the Sudoku problem as defined in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Acknowledgments and Disclosure of Funding We thank our colleagues Kamalika Das and Jiaxin Zhang from Intuit AI Research Center for discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 5 References [1] Don Monroe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' Neurosymbolic ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 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International Conference on Machine Learning, pages 5884–5894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfww21/content/2301.05253v1.pdf'} diff --git a/u9AyT4oBgHgl3EQfm_jD/content/tmp_files/2301.00482v1.pdf.txt b/u9AyT4oBgHgl3EQfm_jD/content/tmp_files/2301.00482v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3e5300206b027437e05b1ac21493e3fca782186 --- /dev/null +++ b/u9AyT4oBgHgl3EQfm_jD/content/tmp_files/2301.00482v1.pdf.txt @@ -0,0 +1,1832 @@ +FEVA: FAST EVENT VIDEO ANNOTATION TOOL +Snehesh Shrestha, William Sentosatio, Huiashu Peng, Cornelia Fermüller, Yiannis Aloimonos +The University of Maryland College Park +College Park, MD 20740 +snehesh@umd.edu +ABSTRACT +Video Annotation is a crucial process in computer science and social science alike. Many video +annotation tools (VAT) offer a wide range of features for making annotation possible. We conducted an +extensive survey of over 59 VAT and interviewed interdisciplinary researchers to evaluate the usability +of the VAT. Our findings suggest that most current VAT have overwhelming user interfaces, poor +interaction techniques, and difficult-to-understand features. These often lead to longer annotation time, +label inconsistencies, and user fatigue. We introduce FEVA, a video annotation tool with streamlined +interaction techniques and a dynamic interface that makes labeling tasks easy and fast. FEVA focuses +on speed, accuracy, and simplicity to make annotation quick, consistent, and straightforward. For +example, annotators can control the speed and direction of the video and mark the onset and the offset +of a label in real time with single key presses. In our user study, FEVA users, on average, require 36% +less interaction than the most popular annotation tools (Advene, ANVIL, ELAN, VIA, and VIAN). +The participants (N=32) rated FEVA as more intuitive and required less mental demand. The code +and demo are available at http://www.snehesh.com/feva. +Figure 1: “Speed Label” enables you to create annotations (red rectangle) in real-time by marking start (stp) and end +time (etp) with a single key press respectively. FEVA automatically adjusts these times of the event annotation based on +your reaction-time (∆r) in order to give you the most precise intended times (sti and eti). +Keywords Video Annotation Tools · User Interface Design · User Interaction +arXiv:2301.00482v1 [cs.HC] 1 Jan 2023 + +1 st Key Press +2nd Key Press +00:01:20.000 +0001:22.000 +00:01:24.000 +00:01:26.000 +00:01:28.000 +Ar +sti +stp +et;FEVA: Fast Event Video Annotation Tool +1 +Introduction +Social scientists need to code conversations and behaviors of videotaped interviews and experiments that quickly add +up to hours of footage [1, 2]. Computer scientists need datasets with appropriately labeled ground truth for machine +learning that contains video clips spanning hundreds of hours [3, 4, 5], or even thousands of hours [6]. Annotating +videos is thus time-consuming and tedious. +There are existing VATs [7, 8, 9, 10, 11, 12] that offer a range of different features for video annotation activities. +However, they often fail to meet the need of the researchers due to the steep learning curves with complicated features, +overwhelming interfaces, and poor interaction techniques leading to longer annotation time, inconsistencies, and user +fatigue. +For example, to analyze soccer games, researchers annotate player ball possessions, kicks, and assists. Annotating +with tools that pause for the user to name the annotation every time, needing to annotate with mouse context menu +options, and not allowing overlapping annotations make it an extremely tedious and time-consuming task. On the +contrary, coding by hand is more straightforward. An annotator can play the video slower rate, use a clicker count or a +stopwatch [13] to lap the timestamps in real-time, then enter them into a spreadsheet when completed. As a result, many +annotators still code by hand and rely on spreadsheets [14]. At scale, when you have hours of such footage, computer +scientists often outsource or crowdsource such annotation tasks [15, 16, 17, 18]. This can raise concerns about privacy +and reliability [19]. To get around this, researchers blur faces to obfuscate participant identity. However, this could +compromise the quality of the emotional judgment, causing inconsistencies in the results. +In this paper, we aim to design a video annotation tool that makes labeling tasks easy and fast. We interviewed +researchers from different disciplines to understand standard practices, workflows, and tools used for video annotation. +Specifically, we interviewed 13 researchers from 5 fields (neuroscience, behavior psychology, film studies, and computer +science). Researchers expressed reservations with existing VATs leading to avoiding them. We further surveyed 59 +VATs (the list is available in the appendix ). We categorized their main features and interface design from a firsthand +using experience and analyzing video tutorials for tools that are not accessible. According to the interviews and survey +results, we propose five design criteria that would benefit video annotation activities, which are detailed in section 4: +• D1. Organize space based on operational workflow +• D2. Streamline high-frequency actions +• D3. Use algorithmic support when possible +• D4. Adopt what works and redesign what doesn’t +• D5. Allow flexibility +Figure 2: The shift keys activate the fine-tuning feature. The left and right shift keys correspond to the start and end +time of the label, respectively. Pressing the arrow key while holding down the shift key adjusts the label’s start or end +time by a single frame. +These criteria inform the design of Fast Event Video Annotation (FEVA), a video annotation tool with streamlined +interaction techniques and a dynamic interface that makes labeling tasks easy and fast. Simplified UI and features such +as real-time labeling with reaction time adjustments (figure 1), and precise fine-tuning mechanisms (figure 2), help +annotators create a large number of accurate labels faster than any VAT. +To evaluate FEVA, we conducted two comparative studies. In the first study, we compared the number of inputs the +users need to provide to complete the same tasks with FEVA versus five other event-based VATs [7, 8, 9, 11, 12]. As +2 + +2450 +2451 +2452 +2453 +2454 +Apple Hit Butterfly +↑ shift +Shift ↑FEVA: Fast Event Video Annotation Tool +seen in table 3, on average, FEVA required 36.0% fewer inputs than competing VATs to do that same task. In the second +study, we asked 34 participants to perform annotations with one of the VAT in random order. We found that regardless +of the background, users thought FEVA was more intuitive at 88% and easier to use at 91% of the time. Users also rated +FEVA as requiring lower mental demand by 46% (p < 0.00003), caused fewer frustrations by 62% (p <0.00147), had +less physical demand by 41% (p < 0.00187), and required less effort by 34% (p < 0.00324). +In Summary, this paper contributes +• a comprehensive understanding of existing VATs with interviews, tool surveys, and pilot tests. +• a list of criteria for VAT tool design +• FEVA, an event-based video annotation tool that let’s you annotate faster and more accurately. +• a comparative study and a user study that confirms FEVA is more intuitive and requires less effort to get the +task done while creating more consistent and accurate annotations. +2 +Related Work +In this section, we introduce event-based annotation and tools we build upon, the workflow used in the annotation +process, and finally, the user interface and the related interaction techniques. To understand the annotation workflow and +goals, we interviewed researchers and surveyed the literature on the steps required to complete the desired annotations. +We found two primary groups of annotators: ones that annotate with the assistance of VAT and ones that rely on more +heuristic methods. +2.1 +Event Based Annotation +Video annotation is the process of marking regions of interest (ROI) either a) spacial (object annotation, OA) or b) +temporal (event annotation, EA). OA is marked on a single-time instance, referred to as frame-based annotation. EA +is the period when the event occurs. OA is prevalent in the CV community for object detection and recognition, and +bounding boxes [20, 21, 22, 23], dots [24, 11], or polygon a.k.a segmentation or masks [24, 25, 22] are used to mark the +annotation. While some VATs also track objects over time [11, 24, 21, 23], it is different from EA. EA focuses on what +is happening in the scene, what the objects are doing, or what is done to the objects rather than the objects themselves. +So the start and the end time marks are used. EA includes behaviors, interactions, emotional response, speech, and +movements [26, 27, 28]. Their application range across disciplines from computer science for activity recognition [11], +psychology for behavior analysis [10, 29, 27], to journalism to track and present stories over timelines [30]. However, +events annotation can be challenging due to the complex nature of temporal navigation, the ability to mark at the exact +desired time of sliding events, use of available features of the tools to facilitate the annotation. Therefore, the focus of +developing FEVA was to simplify the workflow and optimize the steps for the annotation of events to make annotation +easier and faster. +2.2 +Heuristic Workflows +Some researchers prefer not to use specialized video annotation tools. For example, some researchers thought it was +easier to use a clicker to count the time certain events occurred. This is error-prone, less reliable, and makes it difficult +to review the records in the future. Even though some VATs can provide a similar functionality [11] using keypresses, +researchers showed hesitation using VATs with the fear of the initial setup time and the repeated learning curve for +new coders. In another study [13], research assistants (RA) used a stopwatch to obtain the time taken "from when +the OK button was pressed to when the device beeped to signal completion of the RR count." These clicker counts or +stopwatch intervals are recorded either by hand with pen and paper or entered in a spreadsheet. In another workflow, an +interpersonal relationship researcher described how, from a video of two people interacting, they asked RAs to respond +to research questions set up by the researchers while watching a video. For example, multiple RAs focus on the entire +interaction or a specific individual in the video and respond to a Likert scale on how attentive one of the partners was +when the other spoke. These are either recorded on paper or online using Qualtrics or Google forms. +2.3 +Video Annotation Tools Workflows +The primary workflow using a VAT does not differ from the heuristic methods for simple coding. The main difference +is the higher learning curve and a longer setup process. However, as the coding gets more complex, heuristic methods +have a diminishing return on speed as you have more annotators. The annotation process is much slower, needing to do +3 + +FEVA: Fast Event Video Annotation Tool +a lot of steps typically tools might facilitate manually. However, not all VAT are created equal. The workflow design, +screen layout of features, interaction steps, and techniques differ significantly across VATs. +Computer scientists and engineers design most VATs for computer vision, machine learning, and robotics research +and applications [11, 27, 3, 15, 6]. However, there are a handful of tools specifically designed for and used by other +domains such as film studies [12, 31] where VAT allows for character analysis and scene analysis. Journalists use VAT +to annotate and synchronize events from multiple sources to present a cohesive story [30]. In sports, teams annotate +games and significant events and move to study for their teams [32, 33]. While most tools have different focuses, the +primary annotation is marking a binary exists or not and a temporal range of events of interest. To this end, events +annotation is the foundation that makes further work easier or, in some cases, possible. So it is essential to have accurate +annotations. Support for synchronized multi-camera views, micro and macro visualization of events in a timeline, and +convenient searching and reviewing video features make FEVA a favorable choice in these areas. +2.3.1 +Layout design: Balance of features and space usage +Some tools have a lot of features with complicated workflows and many permutations of features that can be customized +and used, so the UI is packed with small windows, buttons, and layers of menu items to get to them. These tools take +much longer to learn and get used to; however, they can be powerful once you learn them. Due to many features, such as +media player controls, annotation controls, and visualizations, the available screen real estate can be quite challenging. +So some tools group them into smaller windows [12, 8, 7], organize them through multiple levels of menus [8, 9], or +through different operational modes [11, 9, 12]. While this helps organize the functionality, it isn’t easy to access, and +one needs a lot of practice to remember the various steps required to use the software. While there are other much +simpler tools that serve a very niche purpose [33, 34, 31, 35, 36], and the layout is simple, and they make excellent use +of the screen real estate for the functionality they offer. These tools are easy to learn and start to use. However, the +features are limited. Our interviews found that most researchers’ and annotators’ needs are in between. Most features +are never used in the complex VAT, while simple VAT limits them from doing more than the niche they offer, and they +need to pair with other tools to augment the gap to complete their needs. We designed FEVA with the motivation of +creating a tool that is simple to navigate and use but comes with features that more complex tools offer without it being +overwhelming that you have to take a course to use a tool. +2.4 +Adoption +The keyboard shortcuts for VIA [11] are intuitive and practical, especially the hints that pop up based on the context +is something most tools lack that FEVA adopts as well. FEVA uses popular shortcuts that have become the standard +for media players, such as the spacebar to play or pause, ctrl+Z, and ctrl+Y to undo and redo, etc. VIAN [12] is the +only tool you don’t have to select before dragging the label with the mouse, which is the same in FEVA. Advene [8], +ELAN [9] and VIA [11] provide alternative ways to visualize or execute the playback option, such as continuous mode. +While useful in some instances, most annotators used the default way without changing, which could be confusing. +While some tools [11] allows one to create a label when the movie is playing, it only expects the start time with a fixed +length. Unfortunately, most users need to return to the label and readjust them. FEVA improves upon this interaction by +allowing a second key press to mark the endpoint. +3 +Understanding the State-Of-The-Art VATs and Their Comparisons +3.1 +Target Users +To understand if researchers from different disciplines annotate their data, what that entails, and what the workflow +looks like, we interviewed 3 neuroscience researchers, 3 behavior psychology researchers, 2 film study instructors, and +5 computer scientists. The interview was semi-structured to answer the following 3 questions: +• IQ1: The nature of their research involving human studies, the kinds of data collected, and if it entails video. +• IQ2: The workflow in the data collection process, post-processing of these data, and code generation. +• IQ3: The structure and workflow for annotating the videos, and how the annotations are used after. +Post-interview, the responses were tallied and coded for technical challenges. The interview insights (II) are as follows: +• II1. There were three primary temporal annotations. 1) A binary label to mark the presence or absence of +certain events. For example, researchers were interested in counting "how many times a person touched their +face as one indicator of how nervous the participants were." 2) A range label marks the beginning and end of +4 + +FEVA: Fast Event Video Annotation Tool +an event of interest. "Participants annotate videos for specific moments. For instance, a couple might interact, +then the researchers have each couple member watch the video and annotate whenever a specific thought or +feeling occurred to them. They do this with both participants to see convergence and dissonance of thoughts, +feelings, goals, etc., etc." 3) Certain kinds of labels, such as mood or scenario, lasted longer than an action +label. "For instance, we might want to compute a rating of how responsive or caring one individual is to +their relationship partner. So a Research Assistant might watch an entire interaction, focusing on a specific +individual in the video, and then answer a question about how attentive they are when their partner speaks." +Most VATs do a poor job of supporting these labels, so researchers had workarounds such as creating multiple +label tracks, each dedicated to a specific response. +• II2: The time, effort, and cost for data annotation are exponentially high. So any system that made some +improvements to make the annotation faster and more reliable was always a huge win. "We spend a lot of our +RA hours doing these annotations. If there were a tool that could cut the time by even an hour, I definitely +would be using that tool." +• II3: Even with very explicit codes, annotated data often had low temporal precision, so the agreement rules +were relaxed. "Many RAs review and annotate the videos. There will be slight variations in when each RA +thinks a certain event happened." +• II4: Collaboration and sharing of the annotation during the annotation process and analysis step was cumber- +some and required multiple steps to be in sync between the teams. "...the students need to refresh the page if +they are working on annotating the same clip simultaneously to see what their classmates are writing. That is +also a problem for me since I like to add my comments while they are working (to encourage them to elaborate +on points or explore a new point)." +• II5: Current VAT provided a poor interface for researchers to explore, analyze, and search the annotated data. +• II6: Crowd-sourced or AI-generated annotation often needed so much review that it was easier and faster +for researchers’ RAs to do the annotations. "So I used the [online automatic speech recognition tool]. With +this, with the premium, I still have to go in and edit everything. The labels for the actions and the labels for +the word-for-word speech need to match up in terms of start and end time. So creating the labels by hand is +actually easier for me." +This paper focuses on II2, II3, and II5, which help shape the design decisions made in section 4. +3.2 +Comparing Video Annotation Tools specifications +We created an extensive list of 59 VAT from the literature as listed in appendix B. We were able to find their websites, +download links or shared open source codes, or at the minimum online videos of either talk by the authors or how-to +videos. We cataloged typical features most software supported and unique features and techniques distinctive to each +tool. In this extensive survey, we share the table for the narrowed-down selected five tools. We detail the selection +method in section 6.1. Two types of tables were created: +• based on high-level taxonomy as shown in table 1 and +• based on features as shown in table 2 +As a contrasting difference, here is an example of the steps required to create a new annotation using 3 different VATs, +as shown in figure 3. We make a more thorough step-by-step comparison in the evaluation section in 6.2. +• In the upper left corner is ANVIL. It requires you to double-click the starting point and click on the endpoint +time mark to select the region. Then right-click the shaded area to select from the context menu to create and +edit the label. +• On the upper right corner is VIAN. In VIAN, you right-click in the space between the tracks and the ruler. +Then select create an annotation, then select text as the type for text annotation. +• In VIA, you create the track for the particular annotation and name it ("Bird" in this case), then play the video +till the movie’s starting point and hit the letter ’A’ to create a fixed-size annotation. You can go back and adjust +the annotation. +We surveyed the layout and the functionalities available in a number of events annotation software by using them to +annotate videos or watch videos posted by the authors. We noticed the following: +• Most VATs except for VIA [11] screen was filled with buttons, menus, and windows. While essential features +were buried in multiple levels of menu items, there needed to be more screen real estate utilized. The new VAT +requires to be simple to understand and easy to navigate. +5 + +FEVA: Fast Event Video Annotation Tool +SN +Features +Description +Advene +ANVIL +ELAN +VIA +VIAN +FEVA (Ours) +0 +Last Updated +Month Year +Jun 2020 +Mar 2019 +Mar 2020 +Jul 2020 +May 2020 +Sep 2020 +1 +SW Platform +Cloud vs Edge +Edge +Edge +Edge +Edge +Edge +Cloud or Edge +Native vs Web +Based +Native +Native +Native +Web Based +Native +Web Based +Modular +vs +Static +Static +Static +Static +Static +Static +Modular +2 +License +Open +Source +vs Proprietary +Open Source +Proprietary +Open Source +Open Source +Open Source +Open Source +Commercial vs +Open Access +Open Access +Open Access +Open Access +Open Access +Open Access +Open Access +Maintained vs +Outdated +Maintained +Maintained +Maintained +Maintained +Maintained +Maintained +3 +Cost +Free, low cost, +vs Expensive +Free +Free +Free +Free +Free/Low Cost +Free +One +time +vs +Subscription +N/A +N/A +N/A +N/A +N/A +N/A +4 +Collaboration +Single User +✓ +✓ +✓ +✓ +✓ +✓ +Multi-User (Si- +multaneous) + + + + + + +Crowd + + + + + + +5 +Target Users +Technical +vs +Non-Technical +Technical +Technical +Technical +Technical +Technical +Both +Academic +vs +Commercial +Academic +Academic +Academic +Academic +Academic +Academic +6 +Input Type +Image + + + +✓ + + +Video +✓ +✓ +✓ +✓ +✓ +✓ +Audio + + +✓ +✓ + +✓ +7 +Annotation +Type +Object + + + +✓ + +✓ +Action +✓ +✓ +✓ +✓ +✓ +✓ +Events +✓ +✓ +✓ +✓ +✓ +✓ +Hybrid + + + +✓ + +✓ +8 +Annotation Ap- +proach +Manual +✓ +✓ +✓ +✓ +✓ +✓ +Automatic + + + + + + +Hybrid + + + + + + +9 +Annotation +Format +JSON + +✓ + +✓ + +✓ +XML +✓ + +✓ + + + +SQL + + + + + + +Others? +✓ +✓ +✓ +✓ +✓ + +Table 1: VAT taxonomy comparison table. +6 + +FEVA: Fast Event Video Annotation Tool +SN +Features +Description +Advene +ANVIL +ELAN +VIA +VIAN +FEVA +1 +Annotation types +Object Bounding Box + + + +✓ + +✓ +Object Mask + + + +✓ + + +Object Dot + + + +✓ + +✓ +Temporal Events +✓ +✓ +✓ +✓ +✓ +✓ +2 +Playback controls +Play Pause FF RR +✓ +✓ +✓ +✓ +✓ +✓ +Speed +/- +✓ +✓ +✓ +✓ +✓ +✓ +Timeline Jump +✓ +✓ +✓ +✓ +✓ +✓ +3 +Preview +Thumbnail Previews +✓ + + + + +✓ +4 +Label +Multi-track +✓ +✓ +✓ +✓ +✓ +✓ +Group tracks + + + + + + +User-defined Label Types +✓ +✓ +✓ +✓ +✓ +✓ +Show/Hide/Collapse/Expand + + + + + + +5 +Speed Label +Sudo-Pedal + + + +✓ + +✓ +Transcribing Pedal Support + + + + + +✓ +6 +Resize +✓ +✓ +✓ +✓ +✓ +✓ +7 +Move +✓ +✓ +✓ +✓ +✓ +✓ +8 +Add +✓ +✓ +✓ +✓ +✓ +✓ +9 +Delete +✓ +✓ +✓ +✓ +✓ +✓ +10 +Edit +✓ +✓ +✓ +✓ +✓ +✓ +11 +Import +✓ +✓ +✓ + +✓ +✓ +12 +Import other formats +✓ +✓ +✓ + +✓ +✓ +13 +Video Support +MP4 +✓ +✓ +✓ +✓ +✓ +✓ +Others +✓ +✓ +✓ +✓ +✓ + +14 +Cameras +Multi-Cam +✓ + + + + +✓ +Switch View +✓ + + + + +✓ +Instant Switch View +✓ + + + + +✓ +15 +History +Undo/ Redo +✓ + +✓ + +✓ +✓ +16 +Search +Keyword +✓ + +✓ + +✓ +✓ +Filter by label type +✓ + +✓ + +✓ +✓ +17 +User Config +Remember/ Save + + +✓ + + +✓ +18 +Modular/ API +Add-In Support +✓ + +✓ + +✓ + +Full Open Source Support +✓ + +✓ +✓ + +✓ +Custom Layers + + + +✓ +✓ + +Custom Tracks +✓ +✓ +✓ +✓ +✓ +✓ +19 +Layers +Show/Hide Layers + + + + +✓ +✓ +Human Joints Keypoints Support + + + + + +✓ +Human Bounding Box Support + + + +✓ + +✓ +Human Mask Support + + + +✓ + + +20 +Export Support +Video Clips +✓ + +✓ + + +✓ +Image Frames + + +✓ + + +✓ +Closed Caption +✓ + +✓ + + +✓ +Table 2: VAT features comparison table. +7 + +FEVA: Fast Event Video Annotation Tool +Figure 3: Example of steps needed to annotate in 3 different VATs. The Upper left screenshot is ANVIL, the upper +right is VIAN, and the bottom screenshot is of VIA. +• Observations through video instructions and pilot testing, we noticed that only limited features and controls +were needed, primarily a) project and data management, b) annotation management, c) video navigation, d) +annotations space, and e) the tool configurations. The new VAT needs a layout such that these features are +upfront and eliminate unnecessary features or allocate them into rarely used spaces. +• The annotation visualization was limiting. An entire track was dedicated to a single label [12], and no +overlapping time labels were possible [7, 9]. Only some [11, 8] allow overlapping time labels. However, it is +difficult to distinguish and manipulate the labels. The new VAT needs to organize the annotations automatically +and better use annotation space. +• While some tools [11, 10] have good annotation UX controls to create and edit temporal placements, the use +of mouse and keyboard had to be used interchangeably between annotation steps. Some tools [7, 8, 9] have +a very cumbersome way to move or resize annotations. Some tools provided a history option to undo/ redo +[8, 9], while others provided no way to backtrack user mistakes or perform experimental steps. The new VAT +needed simple mouse control and default keyboard shortcuts while allowing users to redefine them. +• All tools required you to stop and annotate except VIA [11], which provided real-time annotation during video +playback. However, VIA only allowed fixed-length annotation flags requiring adjustment of the endpoint later. +Furthermore, VIA has poor visualization and lacks control for overlapping labels. The new VAT needs a fast +real-time way to continue annotating without stopping. +4 +Design Considerations +According to the interviews, literature survey, and the use of the tools, we propose five design criteria that would benefit +video annotation activities: +8 + +pframes) +THEPE +Start +Create +Create & Edit +Ae +Options +Edit +0:00 +00:00:10 +00:00:20 +00:00:30 +00:00:40 +00:00:50 +Extend +cification: +New Annotation Layer +ANVILispeclsa +Cut +New Segmentation +Delete +fied! +Create Annotation +Rectangle +jumprope +Play element +Ellipse +Play element slow +squirrel +Text +Play line-to-line +wandArrow +Image +nvil +FreeHand +Showhistograms +00:12 +00:13 +Showas table +0:16 +00:17 +00:18 +00:19 +Track analysis +00:00:17.453 +60:00:00.00 +00:00.28.681 +00:00:57.383 +00:01:28.045 +00:01:54.727 +00:02:23.408 +00:02:52.081 +00:03 20.773 +00:03:49.455 +00:04:18.137 +00:04:48.818 +00:05:15.500 +00.05:44.182 +00:08:12.884 +00.08:41.548 +00 +Events v +00:00:18 +00:00:19 +b000:.20 +00:00.21 +00:00-22 +50.00-23 +00:00:24 +CO:00-25 +00:00-26 +50.00:-27 +00:00:28 +00:00-29 +100:00:20 +00:00:31 +00:00:32 +00:00-33 +00:00:34 +0000.35 +00.0036 +00.0037 +00.00-33 +00:00:39 +CO:00:40 +00:00:4100:00 +Bird +add/del Events +Add Del +[Playback. Normal +Keybo +Paused. Press a to add a temporal segment. +Tab +to select and +to select another temporal segment timelineFEVA: Fast Event Video Annotation Tool +• D1. Organize space based on operational workflow: Features should be laid out in a logical flow of the +workflow while not veering too far away from standard software conventions. Features not needed in that +context should not be visible or active to free up valuable screen real estate. High-frequency controls and +features should be in the middle of the screen. +• D2. Streamline high-frequency actions: +The highly repeated actions, such as creating annotations and +fine-tuning them, should be optimized for easier and faster execution. Try to accomplish them with a single +key press when possible. Stick to a single device (i.e., do not require some steps to use the keyboard and the +other steps mouse movements, using up precious time in transitioning between the devices.) Leverage D1 +when possible to minimize user interaction required to accomplish the task. +• D3. Use algorithmic support when possible: Whenever possible, offload the user and rely on the algorithm to +take on the burden. For example, when the timing is concerned, consider user reaction time and adjust for +the lag in the user input from the intended time. Additionally, allowing external modules such as movement +detection, human detection, speech detection, and recognition can offload users’ need to find and annotate the +events manually. However, we should be cautious that no matter how good machine learning modules are, no +algorithm is 100% accurate. These should be considered as only additional assistance and not to be completely +relied on for ground truth generation. +• D4. Adopt what works and reinvent what doesn’t: Instead of reinventing the wheel, adopt from other tools +what works well based on user tests or pilot tests and not instincts and personal preferences. +• D5. Allow flexibility: While having D4, tested default input methods is great, adding flexibility by providing +redundancies with mouse context menus and keyboard shortcuts might be more intuitive for some users. +Additionally, let users redefine the key mapping as people have personal preferences. +5 +FEVA +Figure 4: A screenshot of FEVA, where a participant interacting with a robot, is being annotated. +The design considerations inform the minimalist design of FEVA, with most of the screen real estate allocated for the +video and the annotation. Inspired by the clicker/ stopwatch methods and transcribing pedal, we created the speed label. +This feature lets you create labels during video play, where a user can press a key to mark the starting and stopping +points and continue to watch, creating multiple labels with desired lengths. The system additionally accounts for the +user’s reaction time and adjusts the marked times as seen in figure 1. +9 + +器Localhost:5000 +C +@ localhost:5000 + Incognito +FEVA D40 +≤5 +p40_all +00:02:58.227 / 00:15:45.842 +5347/28375 +02:20.000 +00:02:24.000 +00:02:28.000 +00:02:32.000 +00:02:36.000 +00:02:40.000 +0:02:4.000 +00:02:48.00 +00:02:52.000 +00:02:56.0c 0 +00:03:00.000 +00:03:04.000 +00:03:08.000 +00:03:12.000 +00:03:16.000 +00:03:20.000 +00:03:24.000 +00:03:28.00 +00:03:32.000 +Event +LuttingApple +TurnToBacks ove +TurmToFrontStove +TurnToBackBasil + Nod +PotStoveOn +pON +PutLidon +Action +tun on burne +go to stove +turn on burner +put lid on water +pick up lid +pick uplid +put lid on water + Speech +ok +uh, so so he understands like my hand yup +uh Baxster tumn on +"pun +Baxster, go +tuni on +Baxster, put +Baxster, go +Bax... +now +Emotion + Training Completed + Status +Oz +Robotface +Instructions +Scene + PotTumOnGasFEVA: Fast Event Video Annotation Tool +The speed label enables users to annotate in real-time, the fine-tuning control lets the user make frame level adjustments, +and the label organizer arranges the label without ever overlapping them for direct access. The flat UI is responsive and +aesthetically pleasing. Using F-shape design [37], the initial workflow items are on the top left, while the most viewed +items are in the middle of the screen, and the most interacted components are at the bottom of the screen. Keeping most +used elements in the hot zones while less used elements such as camera selection and configuration buttons are on the +right-hand side in the less noticed area, making them easily tuned out unless needed. The UI has redundant user input +for flexible and fast interaction to cater to novice and expert users. And underlying context and configuration-dependent +UI model comes to life only when you need them. Expert users can annotate videos faster than in real-time by using +media speed control and speed labeler without touching the mouse. Researchers can also use the zoom feature to +visualize and analyze labels at a micro or macro level. FEVA UI can display the most number of labels on one screen +without losing their meaning. +5.1 +Workflow +To create any project, upon selecting a video, FEVA imports it and creates an empty annotation file, also referred to as a +label or the dataset file (D1, D2). Default tracks will be loaded, which can be customized from the configuration (D3). +Users can add or remove tracks. To play or pause the movie, users can use the media player overlay on the main video +or the keyboard shortcut ’spacebar.’ You can use your mouse scroll button to navigate the video in the timeline. You can +also use the arrow keys with or without the ’ctrl’ key to move the video by different amounts. You can click and drag +the filmstrip, roll over the global timeline and click at the desired time, or double-click the filmstrip or the local timeline +window (D1, D2, D5). You can also change the movie playback speed. You can zoom in and out at different time +intervals using the + and - icons around the white box on the global timeline or roll over the timeline and ctrl and scroll. +To annotate, users can right-click on the tracks and select the label type they want to create. Users can also hit the letter +’A’ key on the keyboard to mark the start and a second time to mark the stop. See figure 1. This is called the speed label, +as you can keep annotating without stopping the movie. Speed annotation requires a two-pass, but in our pilot tests, the +speed label is at least 1.5x faster than the traditional methods. +Following left-to-right and top-to-bottom conventions, workflow such as loading the project, dataset, and manipulating +the video and labels are organized in that order. The main video is at the center of the screen occupying the most space. +Below the video are the annotation tracks that follow conventions and eyes and the hand layout of users’ gaze and +action areas. See figure 5 and figure 4. +5.2 +View/ Layout +As seen in figure 5, ’b’ shows the project selector from which you create a new project and import videos. The ’d’ +shows your label file selector to create, load, save, merge, import, and export labels. You can search using the ’e’ and +filter labels by type. You can double-click the label from the ’2’ label list to find the corresponding label in the timeline +’j.’ You can see the thumbnail preview in ’h’ along with the current time window ’g’ and the global timeline ’f.’ +5.3 +Control: User input system +Every media control controlled by the mouse also has its associated keyboard shortcuts. For flexibility and efficiency, +GUI for novice users that are based on principles of recognition rather than recall [38], and shortcuts for expert users +for faster control. Users can press the play button overlayed on the video or use the spacebar key at any point to play or +pause the video. You can also choose to speed up or slow down. To create an annotation while the video is in play, +you can press the letter ’A’ twice to indicate the start of the label and end the label. This can be customized from the +configuration. A blank label is created after adjusting for your reaction time. You can also double-click an empty space +on a track or right-click the tracks and select a label type you wish to create. While these shortcuts were selected to be +consistent with existing standards from other VATs, all shortcuts can be user-defined easily in the user configuration. +To navigate, you can use your mouse to scroll or click and drag the filmstrips, double click the desired point in the +filmstrip or the global progress bar. You can also jump to a specific label from the label list by double-clicking it from +the label list. Users can choose what they prefer by providing multiple redundancies with both keyboard and mouse and +keyboard shortcuts that can be re-customized by the user. +5.4 +Model/ State: Underlying UI support system +The UI makes use of the limited screen space (x-y plain) by using the z-axis to layer components displayed and state +changes based on contexts such as if the video is playing, if labels are selected, if labels are being edited, or if your +10 + +FEVA: Fast Event Video Annotation Tool +Figure 5: FEVA Screen Default Layout black boxed areas are 1) Toolbar, 2) Label list, main video player, and multi- +camera selector, 3) Video navigation timeline, and 4) Label tracks. Key components are a) Logo, b) Project selection, +c) toolbar icons, d) Label data file selection, e) Search bar, f) Global progress bar timeline, g) Local timeline ruler, h) +Filmstrips, i) Label type, and j) tracks and labels. +mouse is hovering over a component or a specific feature is enabled in the configuration. Every area is compacted +with features that feel intuitive based on affordance users naturally would assign those components. An example is the +video player area. When the video is playing, one would only see the video. When a mouse pointer hovers over, media +controls, current time and frame number, and layer control are displayed. From the configuration, you can also enable +showing or hiding the video, human body keypoints [39], the bounding box of humans or objects, segmentation masks, +etc., that are extensible for researchers to customize. +6 +Evaluation +To evaluate FEVA, we compared FEVA with existing state-of-the-art (SOTA) VAT with two studies. +• Interaction Benchmark: To evaluate the theoretical limits of how fast users could annotate with each VAT, we +counted the number of user inputs required to perform various tasks. +• User Study: To evaluate user experience based on the user’s perceived workload with each VAT, we conducted +user studies where the users provided feedback based on their experience. +6.1 +The State-of-the-Art VAT Selection Method +We first created a master list of highly cited VAT that we could download and use. In this list, we only included software +that supported temporal annotation that could be downloaded, installed, and run without taking extreme measures for +practical reasons. Therefore, VCode [10], SVAT [40], and VACA [27] could not be included. Tools that were too +specific such as ToolScape [34], HistoryTracker [33], and CASAM [41] due to missing functionalities such as start +and end time, were removed from the list. Tools focused on crowd-sourcing such as Glance [42] and CoAT [29], were +excluded as we conducted a single-user study. Using these criteria, we could narrow down the tools to be compared in +the study. EagleView [28] was extremely unstable to run, so we could not test them. We narrowed down to Advene [8], +ANVIL [7], Elan [9], VIA [11], and VIAN [12]. +11 + +a +b +c +儿 +LogoProject Select +Toolbar icons +Label File Select +el +Search Labels +Multi-Camera +Main Video Player +Label List +Selector +fC +Global Timeline +g5 +Local Timeline +5 +h +Thumbnails (Filmstrips) +Label Type +4 +Tracks and Labels +Track Titles +儿FEVA: Fast Event Video Annotation Tool +6.2 +Interaction Benchmark +To compare the steps required to do a particular task with each VAT, we counted the number of clicks, double-clicks, +mouse movements, and keyboard key presses and took a cumulative sum as seen in table 3. If there were multiple ways +of completing a task, we included the fastest method for that tool. For example, if you can press Ctrl+N to create a new +project (keypresses count = 2) or can move your mouse to the main file menu, click the file, move the mouse to a new +project, and click on the menu item (mouse move = 2 and mouse clicks = 2, total = 4), then we took the lesser of the +two. +Table 3 shows the 15 tasks considered for the evaluation. These included basic setting, label creation, and manipulation +tasks. We selected tasks that the majority of the VATs could do. If a VAT missed a feature, we assigned the worst count +received by competing with the VATs. For example, [11] does not support "undo" or "redo" and received a count of 2. +The number of inputs required in FEVA is significantly less than the SOTA, as shown in table 3. On average, FEVA +requires 36% less input than the SOTA. Based on the T-test, FEVA required significantly less input than all tools except +VIA, which was not statistically significant. +SN +Tasks +Advene +ANVIL +ELAN +VIA +VIAN +FEVA +1 +Create a project + Import a video +7 +8 +8 +7 +14 +3 +2 +Create a single label +4 +6 +7 +2 +5 +2 +3 +Create multiple labels +8 +12 +14 +3 +9 +3 +4 +Create and name label +4 +9 +7 +7 +9 +7 +5 +Edit labels +4 +7 +3 +3 +3 +5 +6 +Resize labels +6 +5 +6 +4 +4 +4 +7 +Move labels +6 +12 +6 +5 +4 +4 +8 +Change label type +6 +6 +6 +6 +6 +4 +9 +Delete labels +3 +5 +4 +3 +3 +3 +10 +Find labels +3 +3 +5 +5 +3 +2 +11 +Save labels +2 +2 +2 +2 +2 +2 +12 +Load labels +4 +4 +4 +4 +8 +4 +13 +Navigate video +2 +6 +2 +1 +2 +1 +14 +Play/ Pause video +2 +1 +2 +1 +1 +1 +Play only label video +6 +5 +4 +3 +6 +2 +15 +Undo/ Redo +2 +2 +2 +2 +2 +2 +TOTAL SCORE +69 +93 +82 +58 +81 +49 +FEVA Faster by +29% +47% +40% +16% +40% +p-value +0.0255 +0.0013 +0.0149 +0.1321 +0.0223 +Table 3: The list of tasks done using the fastest possible methods in each software (shortcuts where applicable). Each +number reflects a cumulative sum of mouse clicks, double clicks, movement, and key presses. The last row shows how +much FEVA is faster than the SOTA in percent (%) and the T-test p-values. +6.3 +User Study +To evaluate user experience, we conducted a user study where participants used two VAT, FEVA, and one another +selected SOTA VATs (Advene, ANVIL, ELAN, VIA, and VIAN) in a round-robin fashion. We counterbalanced the +order of the two tools by alternating the order with the next participant. Due to the COVID-19 regulations, we conducted +our study via Zoom remote shared screen and control feature. This introduced some lag in the user experience. However, +since both tools were remote, we assumed that the effects of the lag on the outcome were not significantly discriminant. +Participants were given an approximate time range where an event occurs with clear descriptions of the events to +annotate, for instance, as seen in figure 6, "between 4 minutes and 20 seconds and 4 minutes 40 seconds, please annotate +bunny jump roping." Participants completed approximately 24 tasks until they gave up on the tool. After completing all +the tasks in the first software, they filled out the NASA Task Load Index [43] questionnaire with a 5-point Likert scale. +And this was repeated with the second tool. +12 + +FEVA: Fast Event Video Annotation Tool +Figure 6: An example of a task where a participant was asked to annotate an event of the bunny jump roping between +04:28 minutes mark and 04:31 minutes mark in FEVA. +6.3.1 +Participants +We recruited 34 participants in the University community via email and social media forums. In our study of N=32, the +participants were 53% male and 47% female, had a mean age of 30.4 +/- 5.9, with 84% not having video annotation +experience, and 66% had no experience with video editing. Two participants had to be dropped due to zoom connection +issues and are not counted in N=32. +6.3.2 +Procedure +For this study, we trained annotators for 3 minutes by watching a short training video that taught the basics of media +controls, video timeline navigation, and how to create and edit annotations, followed by practice trials for each item +with the research coordinator answering any questions. They then spent another 2 minutes exploring the tool on their +own. The participants spent the next 5 minutes practicing the tasks assigned individually by the researcher where they +were allowed to ask questions. Once they got comfortable, they were randomly given four categories of tasks shown in +the list below, with each type of task repeated at least three times. The tasks chosen were the most fundamental and +repeated tasks annotators must do during video annotation. The tasks ranged in complexity, with some tasks requiring +combinations of the fundamental steps. For instance, some tasks simply asked the participant to navigate the video to 2 +minutes and 20 seconds. In contrast, others asked participants to annotate three consecutive events between a specific +time and name them appropriately. They were no time limits to perform the tasks. They worked on the standard freely +available "Big Buck Bunny" video which is approximately 10 minutes long at 720p resolution. Figure 6 demonstrates +one example task. After completing all the tasks with the first tool, the participants filled out the NASA TLX workload +questionnaire and repeated the tasks with the second VAT. +We conducted the following four categories of basic tasks during the user study: +• Navigate the video a) play/ pause the video, b) jump to a specific point in the timeline, and c) jump to a precise +point where a particular label is. +13 + +FEVA + big_ buck_bunny +52 +big_buck_bunny_test +oirdhit +rawr +MOIIE PUE MOC +00:04:30.440/ 00:09:54.539 +8112/17836 +500 +00:04:26.000 +00:04:26.500 +00:04:27.000 +00:04:27.500 +00:04:28.000 +00:04:28.500 +00:04:29.000 +00:04:29.500 +00:04:30.000 +00:04:30.500 +00:04:31.000 +00:04:31.500 +Objects +Action +Speech +Emotion +Event +jump rope ++FEVA: Fast Event Video Annotation Tool +• Label Creation a) Create a new label at a specific time with a specific length, b) Create a label when a +participant shows a specific behavior (e.g., a character yawns, eats an apple, etc.), and c) create multiple labels +in a row. +• Label Content Manipulation a) Write text annotation for a label created and b) Modify annotation text. +• Label Temporal Manipulation a) Move the label by a specific number of seconds and b) Resize the label to +change its starting time or ending time to match a specific behavior by the person in the video +6.3.3 +Results +In this study, on average, the users felt less metal demand by 46% (p < 0.00003) with FEVA than the SOTA, less +physical demand by 41% (p < 0.00187), less effort was required by 34% (p < 0.00324), and felt less frustration by +62% (p <0.00147). The difference in the temporal demand and the performance level indexes was not significant. We +attributed this to there not being a time limit enforced during the study, and except for one user on VIAN, where the +user gave up, all other users completed all the tasks. +FEVA +Mental +Demand +Physical +Demand +Temporal +Demand +Performance +Level +Effort +Frustration +Level +MEAN (n=32) +1.8 +1.5 +1.8 +4.3 +2.3 +1.7 +Table 4: Shows FEVA’s average score on a NASA Task Load Index with a 5-point Likert scale. +FEVA vs. (%) +Mental +Demand +Physical +Demand +Temporal +Demand +Performance +Level +Effort +Frustration +Level +Advene (n=5) +75% +57% +0% +14% +67% +183% +ANVIL (n=8) +25% +36% +-8% +3% +17% +6% +ELAN (n=8) +29% +42% +13% +12% +38% +83% +VAI (n=8) +44% +36% +6% +11% +22% +36% +VIAN (n=3) +120% +40% +20% +17% +83% +140% +MEAN (n=32) +46% +41% +5% +10% +34% +62% +Table 5: Shows how FEVA compared to the other VAT. A positive number indicates how much people perceived FEVA +to be better than other VATs in the NASA TLX respective six dimensions, and a negative number indicates how much +worse. +6.4 +User Feedback +6.4.1 +The Good +On average, users expressed FEVA was more intuitive 88% of the time and that FEVA was easier to use 91% of the +time. The features users liked the most were the speed label, fine adjustments, "cooler feel," locating label and label +playback. A few users wished they could go back and change their feedback for the first tool once they used the second +tool. This was typical when they felt they gave the first tool too high scores after using FEVA. In contrast, this did not +happen when it was the other way around. One user said the user wanted to start a fan club and wanted to volunteer to +annotate because it was "so fun." +6.4.2 +The bad +One user mentioned that the user preferred traditional windowed UI for serious work. So the user thought FEVA looked +too mainstream tablet app-like. Another user stated, "while I think it was fine for me, I don’t think my mom will be able +to use either of the tools. So I gave low scores to both of them." A few users complained that they did not like pressing +the enter key to confirm the label after editing them. Clicking elsewhere, causing the loss of what was just typed as a +cancel feature, was not popular. +6.4.3 +The ugly +The majority of the confusion, however, was about the global and the local timeline due to needing a clear separation +and sharing the same preview component. Participants remotely controlling the UI of the VATs over a Zoom call on the +14 + +FEVA: Fast Event Video Annotation Tool +research coordinator’s computer noticed a lag in the effect of their actions. Some users complained that the UI did not +update fast enough due to the Zoom lags. "Maybe because I am controlling your computer through Zoom, but a huge +delay made it harder for me to resize the labels." +7 +Implementation +7.1 +Framework and Dependencies +We used ReactJS [44], an efficient component-based JavaScript library, and wrote the architecture to be lightweight and +responsive, so it works on most people’s computers. The installation has only two dependencies of Python and Flask. +The front end relies on standard HTML, Javascript, ReactJS, and CSS. We designed all the controls to optimize for +performance and flexibility to customize. We detail the UI layout breakdown in figure 5 and section 5. +7.2 +Architecture +FEVA uses a simple server-client architecture typical for many web-based applications. The server side runs on Python +3.5x or newer with Flask as the webserver. The server side primarily handles servicing data (web content, annotation +data, and video streaming) when requested by the client-side application. For FEVA, most of the modules and the +design are on the client side. We show more details of all the modules and their interaction with other modules in the +block diagram in appendix 8. +Figure 7: FEVA Client-Server Architecture Diagram. We include a more detailed block diagram of the different +modules on the client side in the appendix C. +8 +Discussion +This is the first version release of our tool FEVA, where we focused on building the fundamental tool while streamlining +the user interface and interactions to make annotating events faster, intuitive, robust, and more accurate. While these +early results look promising, there are more research questions that need to be further explored. +8.1 +User Study +In our pilot and user study, we conducted a short-duration study. In our user study, we assumed that 15 minutes +was sufficient time for participants to learn and practice annotating, which is how we designed the first evaluation +of comparing multiple VATs. However, we need to further our research by conducting a longitudinal user study to +understand the impact of our design on users as they get more comfortable with the software. +8.2 +User Input +In the input sector, we considered the keyboard and mouse/ trackpad as the interaction devices at this stage. Still, we +need to expand this to other kinds of inputs, such as touch, speech, and gesture, to explore the potential benefits of +multi-modal methods. +15 + +Data +Video +JSON +Server +Client +Flask (Python) +HTMLReactJSCSS +HTML TemplatesFEVA: Fast Event Video Annotation Tool +8.3 +Target Users +As more general public gets involved in the coding process, we focused this study so anyone can participate in the +coding process. Future studies to gather feedback from seasoned coders will be valuable in understanding how they use +VAT. +8.4 +VAT benchmarking study +In section 3 study, we counted the number of inputs as the pilot study showed the correlation between the number of +clicks and the time taken to complete a task. A more comprehensive study should be considered, including the mouse +movements and time taken that can reflect user confusion and a more accurate performance metric for completing the +task. +8.5 +Implicit +In section 6.3, we focused on the user’s perception to reflect their experience. Future studies should expand to more +quantitative measures implicit in evaluating user confusion, performance, and success. We would also like to conduct +more studies to understand labeling consistency. +8.6 +The layout +Based on user feedback, lessons learned from our observations, and the process of comparing with existing tools, we +could have done better. The multi-camera selector layout takes up a lot of screen real estate. Many users found the +global timeline being so close to the local timeline and the thumbnail view without any separation confusing and more +challenging to get used to. We have planned to redesign those experiences in the subsequent versions. We will focus on +several optimization opportunities in the next release to make FEVA even faster. +8.7 +Future Work +Beyond the incremental improvements, there were key features that we have planned for the future: +• Adaptive: All the input controls were linear. We plan to explore dynamic and adaptive control systems to +various new interaction techniques for faster annotation and a more intuitive experience. +• Extensible: An easier workflow for the open-source community to extend the features. +• AI assisted: While this had some algorithmic support, better integration with machine learning and deep +neural network models is needed. We will further research how AI can augment the annotation process while +exploring ways to inform the users of these models’ inaccuracies, uncertainties, and inherited bias. +• Remote videos: While our internal prototypes support YouTube, there are optimization opportunities that we +need to explore before they can be used seamlessly as an alternative to the local MPEG videos. We will also +explore other online streaming platforms. +• Case study: While we are working with some research labs in evaluating the FEVA for their video annotation, +we want to invite other interested research labs to try out FEVA, collaborate with us, and grow as a community +to address needs that may not have been realized by our research so far. +9 +Conclusion +We present a new event video annotation tool with streamlined interaction techniques and a dynamic UI, contextually +visible, and active features organized based on the workflow and usage frequency. With features like speed labeling, +users can accurately annotate videos in real-time. With simplified onboarding and workflow, researchers can set up and +start annotating videos using minimal time. We release FEVA source code in GitHub for everyone to try and further +extend its features. The community can also find project samples, tutorial videos, GitHub issues for support, and future +updates on the GitHub page. As we expand our case studies, we invite more researchers to use FEVA or contact us if +you wish to collaborate. +16 + +FEVA: Fast Event Video Annotation Tool +Acknowledgments +We thank Chethan Parameshwara, Levi Burner, Lindsay Little, and peers from UMD and the Perception and Robotics +Group for their valuable feedback and discussions. We extend special thanks to all our project contributors Johnny +Chiu, Rachelle Sims, John Gao, Leya Abraham, Vikram Sehgal, Swagata Chakroborty, Lucas Stuart, and Lin Chen. +References +[1] Pat Broadhead. 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Sage publications Sage CA: Los Angeles, CA, 2006. +[44] React – a javascript library for building user interfaces, May 2013. +19 + +FEVA: Fast Event Video Annotation Tool +A +Online Resources +Please visit FEVA website http://www.snehesh.com/feva for more information, code, instruction videos, samples, +and any updates. +B +List of Video Annotation Tools surveyed +• A Formative Study for Record-time Manual Annotation of First-person Videos +• A multi-level video annotation tool based on XML-dictionaries +• A Semi-Automatic Video Annotation Tool to Generate Ground Truth for Intelligent Video Surveillance +Systems +• Advene +• AIBU +• An innovative web-based collaborative platform for video annotation +• An Ontology Web Application-based Annotation Tool for Intangible Culture Heritage Dance Videos +• Anvil +• atlas.ti +• Augmented Studio: Projection Mapping on Moving Body for Physiotherapy Education +• Automatic tagging of video based on voice and localization +• Automatically Freezing Live Video for Annotation during Remote Collaboration +• AVISA: An annotation tool for video understanding +• BeaverDam: Video Annotation Tool for Computer Vision Training Labels +• BEDA: Visual Analysis of Relationships between Behavioral and Physiological Sensor Data +• CASAM: collaborative human-machine annotation of multimedia +• CLIPPER: Audiovisual Annotation in the Study of Physics +• CoAT: A Web-based, Collaborative Annotation Tool +• CoVidA: Pen-based collaborative video annotation +• Crowd-Guided Ensembles: How Can We Choreograph Crowd Workers for Video Segmentation? +• CrowdSport: Crowd-based Semantic Event Detection and Video Annotation for Sports Videos +• DarkLabel +• Demo: Semantic Human Activity Annotation Tool - Using Skeletonized Surveillance Videos +• EagleView: A Video Analysis Tool for Visualising and Querying Spatial Interactions of People and Devices +• Elan +• Generating annotations for how-to videos using crowdsourcing +• Glance: rapidly coding behavioral video with the crowd +• HistoryTracker: Minimizing Human Interactions in Baseball Game Annotation +• iSeg: Semi-automatic ground truth annotation in videos: An interactive tool for polygon-based object +annotation and segmentation +• iVAT: An interactive tool for manual, semi-automatic and automatic video annotation +• LabelMe +• Marquee: a tool for real-time video logging +• MediaDiver: viewing and annotating multi-view video +• MoViA: a mobile video annotation tool +• MRAS: Annotations for streaming video on the web +• Multimodal Video Annotation for Contemporary Dance Creation +• MuLVAT: A Video Annotation Tool Based on XML-Dictionaries and Shot Clustering +20 + +FEVA: Fast Event Video Annotation Tool +• NVivo +• Oudjat is dedicated to the manual annotation facial expressions of emotion(FEE) +• Redesigning video analysis: an interactive ink annotation tool +• Rethinking Engagement with Online News through Social and Visual Co-Annotation +• Sirio, orione and pan: an integrated web system for ontology-based video search and annotation +• Stabilized Annotations for Mobile Remote Assistance +• SVAT +• The MESH mobile video annotation tool +• Timelinely +• Tool Eval: Rapid Model-Driven Annotation and Evaluation for Object Detection in Videos +• ToolScape: Enhancing the Learning Experience of How-to Videos +• VACA: a tool for qualitative video analysis +• VAnnotator: Annotations as multiple perspectives of video content +• VATIC +• VCode and VData: Illustrating a new Framework for Supporting the Video Annotation Workflow +• VIA: The VIA Annotation Software for Images, Audio and Video +• ViBAT +• VideoAnt +• VideoJot: A Multifunctional Video Annotation Tool +• VidOR: Annotating Objects and Relations in User-Generated Videos +• ViTBAT: Video Tracking and Behavior Annotation Tool +• VoTT +21 + +FEVA: Fast Event Video Annotation Tool +C +FEVA Client Architecture +Figure 8: FEVA Client side block diagram of the various modules +22 + +Client +HTMLReactJSCSS +Configuration Manager +Modules +Project Manager +Data Manager +Context State Manager +Label List Manager +Main Video Player +Camera +Selector +User Input Manager +TimelineManager +Thumbnail Loader/Generator +Display Manager +Label Track Manager +UlComponents +Label Organizer +Label Organizer \ No newline at end of file diff --git a/u9AyT4oBgHgl3EQfm_jD/content/tmp_files/load_file.txt b/u9AyT4oBgHgl3EQfm_jD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fccbc2e7466404efbde0ef255b88141fb7d0db05 --- /dev/null +++ b/u9AyT4oBgHgl3EQfm_jD/content/tmp_files/load_file.txt @@ -0,0 +1,1494 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf,len=1493 +page_content='FEVA: FAST EVENT VIDEO ANNOTATION TOOL Snehesh Shrestha, William Sentosatio, Huiashu Peng, Cornelia Fermüller, Yiannis Aloimonos The University of Maryland College Park College Park, MD 20740 snehesh@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='edu ABSTRACT Video Annotation is a crucial process in computer science and social science alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Many video annotation tools (VAT) offer a wide range of features for making annotation possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We conducted an extensive survey of over 59 VAT and interviewed interdisciplinary researchers to evaluate the usability of the VAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Our findings suggest that most current VAT have overwhelming user interfaces, poor interaction techniques, and difficult-to-understand features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These often lead to longer annotation time, label inconsistencies, and user fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We introduce FEVA, a video annotation tool with streamlined interaction techniques and a dynamic interface that makes labeling tasks easy and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA focuses on speed, accuracy, and simplicity to make annotation quick, consistent, and straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, annotators can control the speed and direction of the video and mark the onset and the offset of a label in real time with single key presses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In our user study, FEVA users, on average, require 36% less interaction than the most popular annotation tools (Advene, ANVIL, ELAN, VIA, and VIAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The participants (N=32) rated FEVA as more intuitive and required less mental demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The code and demo are available at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='snehesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='com/feva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Figure 1: “Speed Label” enables you to create annotations (red rectangle) in real-time by marking start (stp) and end time (etp) with a single key press respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA automatically adjusts these times of the event annotation based on your reaction-time (∆r) in order to give you the most precise intended times (sti and eti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Keywords Video Annotation Tools · User Interface Design · User Interaction arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00482v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='HC] 1 Jan 2023 1 st Key Press 2nd Key Press 00:01:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 0001:22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:01:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:01:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:01:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 Ar sti stp et;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='FEVA: Fast Event Video Annotation Tool 1 Introduction Social scientists need to code conversations and behaviors of videotaped interviews and experiments that quickly add up to hours of footage [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Computer scientists need datasets with appropriately labeled ground truth for machine learning that contains video clips spanning hundreds of hours [3, 4, 5], or even thousands of hours [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Annotating videos is thus time-consuming and tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' There are existing VATs [7, 8, 9, 10, 11, 12] that offer a range of different features for video annotation activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, they often fail to meet the need of the researchers due to the steep learning curves with complicated features, overwhelming interfaces, and poor interaction techniques leading to longer annotation time, inconsistencies, and user fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, to analyze soccer games, researchers annotate player ball possessions, kicks, and assists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Annotating with tools that pause for the user to name the annotation every time, needing to annotate with mouse context menu options, and not allowing overlapping annotations make it an extremely tedious and time-consuming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' On the contrary, coding by hand is more straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' An annotator can play the video slower rate, use a clicker count or a stopwatch [13] to lap the timestamps in real-time, then enter them into a spreadsheet when completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' As a result, many annotators still code by hand and rely on spreadsheets [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' At scale, when you have hours of such footage, computer scientists often outsource or crowdsource such annotation tasks [15, 16, 17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This can raise concerns about privacy and reliability [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To get around this, researchers blur faces to obfuscate participant identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, this could compromise the quality of the emotional judgment, causing inconsistencies in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In this paper, we aim to design a video annotation tool that makes labeling tasks easy and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We interviewed researchers from different disciplines to understand standard practices, workflows, and tools used for video annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Specifically, we interviewed 13 researchers from 5 fields (neuroscience, behavior psychology, film studies, and computer science).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Researchers expressed reservations with existing VATs leading to avoiding them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We further surveyed 59 VATs (the list is available in the appendix ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We categorized their main features and interface design from a firsthand using experience and analyzing video tutorials for tools that are not accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' According to the interviews and survey results, we propose five design criteria that would benefit video annotation activities, which are detailed in section 4: D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Organize space based on operational workflow D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Streamline high-frequency actions D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Use algorithmic support when possible D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Adopt what works and redesign what doesn’t D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Allow flexibility Figure 2: The shift keys activate the fine-tuning feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The left and right shift keys correspond to the start and end time of the label, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Pressing the arrow key while holding down the shift key adjusts the label’s start or end time by a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These criteria inform the design of Fast Event Video Annotation (FEVA), a video annotation tool with streamlined interaction techniques and a dynamic interface that makes labeling tasks easy and fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Simplified UI and features such as real-time labeling with reaction time adjustments (figure 1), and precise fine-tuning mechanisms (figure 2), help annotators create a large number of accurate labels faster than any VAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To evaluate FEVA, we conducted two comparative studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In the first study, we compared the number of inputs the users need to provide to complete the same tasks with FEVA versus five other event-based VATs [7, 8, 9, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' As 2 2450 2451 2452 2453 2454 Apple Hit Butterfly ↑ shift Shift ↑FEVA: Fast Event Video Annotation Tool seen in table 3, on average, FEVA required 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0% fewer inputs than competing VATs to do that same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In the second study, we asked 34 participants to perform annotations with one of the VAT in random order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We found that regardless of the background, users thought FEVA was more intuitive at 88% and easier to use at 91% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Users also rated FEVA as requiring lower mental demand by 46% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00003), caused fewer frustrations by 62% (p <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00147), had less physical demand by 41% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00187), and required less effort by 34% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00324).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In Summary, this paper contributes a comprehensive understanding of existing VATs with interviews, tool surveys, and pilot tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' a list of criteria for VAT tool design FEVA, an event-based video annotation tool that let’s you annotate faster and more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' a comparative study and a user study that confirms FEVA is more intuitive and requires less effort to get the task done while creating more consistent and accurate annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2 Related Work In this section, we introduce event-based annotation and tools we build upon, the workflow used in the annotation process, and finally, the user interface and the related interaction techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To understand the annotation workflow and goals, we interviewed researchers and surveyed the literature on the steps required to complete the desired annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We found two primary groups of annotators: ones that annotate with the assistance of VAT and ones that rely on more heuristic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 Event Based Annotation Video annotation is the process of marking regions of interest (ROI) either a) spacial (object annotation, OA) or b) temporal (event annotation, EA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' OA is marked on a single-time instance, referred to as frame-based annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' EA is the period when the event occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' OA is prevalent in the CV community for object detection and recognition, and bounding boxes [20, 21, 22, 23], dots [24, 11], or polygon a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='a segmentation or masks [24, 25, 22] are used to mark the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While some VATs also track objects over time [11, 24, 21, 23], it is different from EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' EA focuses on what is happening in the scene, what the objects are doing, or what is done to the objects rather than the objects themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So the start and the end time marks are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' EA includes behaviors, interactions, emotional response, speech, and movements [26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Their application range across disciplines from computer science for activity recognition [11], psychology for behavior analysis [10, 29, 27], to journalism to track and present stories over timelines [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, events annotation can be challenging due to the complex nature of temporal navigation, the ability to mark at the exact desired time of sliding events, use of available features of the tools to facilitate the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Therefore, the focus of developing FEVA was to simplify the workflow and optimize the steps for the annotation of events to make annotation easier and faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 Heuristic Workflows Some researchers prefer not to use specialized video annotation tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, some researchers thought it was easier to use a clicker to count the time certain events occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This is error-prone, less reliable, and makes it difficult to review the records in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Even though some VATs can provide a similar functionality [11] using keypresses, researchers showed hesitation using VATs with the fear of the initial setup time and the repeated learning curve for new coders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In another study [13], research assistants (RA) used a stopwatch to obtain the time taken "from when the OK button was pressed to when the device beeped to signal completion of the RR count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" These clicker counts or stopwatch intervals are recorded either by hand with pen and paper or entered in a spreadsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In another workflow, an interpersonal relationship researcher described how, from a video of two people interacting, they asked RAs to respond to research questions set up by the researchers while watching a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, multiple RAs focus on the entire interaction or a specific individual in the video and respond to a Likert scale on how attentive one of the partners was when the other spoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These are either recorded on paper or online using Qualtrics or Google forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 Video Annotation Tools Workflows The primary workflow using a VAT does not differ from the heuristic methods for simple coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The main difference is the higher learning curve and a longer setup process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, as the coding gets more complex, heuristic methods have a diminishing return on speed as you have more annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The annotation process is much slower, needing to do 3 FEVA: Fast Event Video Annotation Tool a lot of steps typically tools might facilitate manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, not all VAT are created equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The workflow design, screen layout of features, interaction steps, and techniques differ significantly across VATs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Computer scientists and engineers design most VATs for computer vision, machine learning, and robotics research and applications [11, 27, 3, 15, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, there are a handful of tools specifically designed for and used by other domains such as film studies [12, 31] where VAT allows for character analysis and scene analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Journalists use VAT to annotate and synchronize events from multiple sources to present a cohesive story [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In sports, teams annotate games and significant events and move to study for their teams [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While most tools have different focuses, the primary annotation is marking a binary exists or not and a temporal range of events of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To this end, events annotation is the foundation that makes further work easier or, in some cases, possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So it is essential to have accurate annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Support for synchronized multi-camera views, micro and macro visualization of events in a timeline, and convenient searching and reviewing video features make FEVA a favorable choice in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 Layout design: Balance of features and space usage Some tools have a lot of features with complicated workflows and many permutations of features that can be customized and used, so the UI is packed with small windows, buttons, and layers of menu items to get to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These tools take much longer to learn and get used to;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' however, they can be powerful once you learn them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Due to many features, such as media player controls, annotation controls, and visualizations, the available screen real estate can be quite challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So some tools group them into smaller windows [12, 8, 7], organize them through multiple levels of menus [8, 9], or through different operational modes [11, 9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While this helps organize the functionality, it isn’t easy to access, and one needs a lot of practice to remember the various steps required to use the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While there are other much simpler tools that serve a very niche purpose [33, 34, 31, 35, 36], and the layout is simple, and they make excellent use of the screen real estate for the functionality they offer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These tools are easy to learn and start to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, the features are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Our interviews found that most researchers’ and annotators’ needs are in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Most features are never used in the complex VAT, while simple VAT limits them from doing more than the niche they offer, and they need to pair with other tools to augment the gap to complete their needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We designed FEVA with the motivation of creating a tool that is simple to navigate and use but comes with features that more complex tools offer without it being overwhelming that you have to take a course to use a tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 Adoption The keyboard shortcuts for VIA [11] are intuitive and practical, especially the hints that pop up based on the context is something most tools lack that FEVA adopts as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA uses popular shortcuts that have become the standard for media players, such as the spacebar to play or pause, ctrl+Z, and ctrl+Y to undo and redo, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' VIAN [12] is the only tool you don’t have to select before dragging the label with the mouse, which is the same in FEVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Advene [8], ELAN [9] and VIA [11] provide alternative ways to visualize or execute the playback option, such as continuous mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While useful in some instances, most annotators used the default way without changing, which could be confusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While some tools [11] allows one to create a label when the movie is playing, it only expects the start time with a fixed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Unfortunately, most users need to return to the label and readjust them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA improves upon this interaction by allowing a second key press to mark the endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 3 Understanding the State-Of-The-Art VATs and Their Comparisons 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 Target Users To understand if researchers from different disciplines annotate their data, what that entails, and what the workflow looks like, we interviewed 3 neuroscience researchers, 3 behavior psychology researchers, 2 film study instructors, and 5 computer scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The interview was semi-structured to answer the following 3 questions: IQ1: The nature of their research involving human studies, the kinds of data collected, and if it entails video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' IQ2: The workflow in the data collection process, post-processing of these data, and code generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' IQ3: The structure and workflow for annotating the videos, and how the annotations are used after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Post-interview, the responses were tallied and coded for technical challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The interview insights (II) are as follows: II1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' There were three primary temporal annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 1) A binary label to mark the presence or absence of certain events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, researchers were interested in counting "how many times a person touched their face as one indicator of how nervous the participants were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" 2) A range label marks the beginning and end of 4 FEVA: Fast Event Video Annotation Tool an event of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' "Participants annotate videos for specific moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For instance, a couple might interact, then the researchers have each couple member watch the video and annotate whenever a specific thought or feeling occurred to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' They do this with both participants to see convergence and dissonance of thoughts, feelings, goals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=', etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" 3) Certain kinds of labels, such as mood or scenario, lasted longer than an action label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' "For instance, we might want to compute a rating of how responsive or caring one individual is to their relationship partner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So a Research Assistant might watch an entire interaction, focusing on a specific individual in the video, and then answer a question about how attentive they are when their partner speaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" Most VATs do a poor job of supporting these labels, so researchers had workarounds such as creating multiple label tracks, each dedicated to a specific response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' II2: The time, effort, and cost for data annotation are exponentially high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So any system that made some improvements to make the annotation faster and more reliable was always a huge win.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' "We spend a lot of our RA hours doing these annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' If there were a tool that could cut the time by even an hour, I definitely would be using that tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" II3: Even with very explicit codes, annotated data often had low temporal precision, so the agreement rules were relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' "Many RAs review and annotate the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' There will be slight variations in when each RA thinks a certain event happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" II4: Collaboration and sharing of the annotation during the annotation process and analysis step was cumber- some and required multiple steps to be in sync between the teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='the students need to refresh the page if they are working on annotating the same clip simultaneously to see what their classmates are writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' That is also a problem for me since I like to add my comments while they are working (to encourage them to elaborate on points or explore a new point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" II5: Current VAT provided a poor interface for researchers to explore, analyze, and search the annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' II6: Crowd-sourced or AI-generated annotation often needed so much review that it was easier and faster for researchers’ RAs to do the annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' "So I used the [online automatic speech recognition tool].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' With this, with the premium, I still have to go in and edit everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The labels for the actions and the labels for the word-for-word speech need to match up in terms of start and end time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So creating the labels by hand is actually easier for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" This paper focuses on II2, II3, and II5, which help shape the design decisions made in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 Comparing Video Annotation Tools specifications We created an extensive list of 59 VAT from the literature as listed in appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We were able to find their websites, download links or shared open source codes, or at the minimum online videos of either talk by the authors or how-to videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We cataloged typical features most software supported and unique features and techniques distinctive to each tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In this extensive survey, we share the table for the narrowed-down selected five tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We detail the selection method in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Two types of tables were created: based on high-level taxonomy as shown in table 1 and based on features as shown in table 2 As a contrasting difference, here is an example of the steps required to create a new annotation using 3 different VATs, as shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We make a more thorough step-by-step comparison in the evaluation section in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In the upper left corner is ANVIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' It requires you to double-click the starting point and click on the endpoint time mark to select the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Then right-click the shaded area to select from the context menu to create and edit the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' On the upper right corner is VIAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In VIAN, you right-click in the space between the tracks and the ruler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Then select create an annotation, then select text as the type for text annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In VIA, you create the track for the particular annotation and name it ("Bird" in this case), then play the video till the movie’s starting point and hit the letter ’A’ to create a fixed-size annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can go back and adjust the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We surveyed the layout and the functionalities available in a number of events annotation software by using them to annotate videos or watch videos posted by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We noticed the following: Most VATs except for VIA [11] screen was filled with buttons, menus, and windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While essential features were buried in multiple levels of menu items, there needed to be more screen real estate utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The new VAT requires to be simple to understand and easy to navigate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='FEVA: Fast Event Video Annotation Tool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='SN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Features ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Advene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='ANVIL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='ELAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='VIA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='VIAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='FEVA (Ours) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Last Updated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Month Year ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Jun 2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Mar 2019 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Mar 2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Jul 2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='May 2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Sep 2020 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='SW Platform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Cloud vs Edge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Edge ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Table 2: VAT features comparison table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 7 FEVA: Fast Event Video Annotation Tool Figure 3: Example of steps needed to annotate in 3 different VATs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The Upper left screenshot is ANVIL, the upper right is VIAN, and the bottom screenshot is of VIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Observations through video instructions and pilot testing, we noticed that only limited features and controls were needed, primarily a) project and data management, b) annotation management, c) video navigation, d) annotations space, and e) the tool configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The new VAT needs a layout such that these features are upfront and eliminate unnecessary features or allocate them into rarely used spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The annotation visualization was limiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' An entire track was dedicated to a single label [12], and no overlapping time labels were possible [7, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Only some [11, 8] allow overlapping time labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, it is difficult to distinguish and manipulate the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The new VAT needs to organize the annotations automatically and better use annotation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While some tools [11, 10] have good annotation UX controls to create and edit temporal placements, the use of mouse and keyboard had to be used interchangeably between annotation steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Some tools [7, 8, 9] have a very cumbersome way to move or resize annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Some tools provided a history option to undo/ redo [8, 9], while others provided no way to backtrack user mistakes or perform experimental steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The new VAT needed simple mouse control and default keyboard shortcuts while allowing users to redefine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' All tools required you to stop and annotate except VIA [11], which provided real-time annotation during video playback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, VIA only allowed fixed-length annotation flags requiring adjustment of the endpoint later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Furthermore, VIA has poor visualization and lacks control for overlapping labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The new VAT needs a fast real-time way to continue annotating without stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 4 Design Considerations According to the interviews, literature survey, and the use of the tools, we propose five design criteria that would benefit video annotation activities: 8 pframes) THEPE Start Create Create & Edit Ae Options Edit 0:00 00:00:10 00:00:20 00:00:30 00:00:40 00:00:50 Extend cification: New Annotation Layer ANVILispeclsa Cut New Segmentation Delete fied!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Create Annotation Rectangle jumprope Play element Ellipse Play element slow squirrel Text Play line-to-line wandArrow Image nvil FreeHand Showhistograms 00:12 00:13 Showas table 0:16 00:17 00:18 00:19 Track analysis 00:00:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='453 60:00:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00 00:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='681 00:00:57.' metadata={'source': 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+page_content='08:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='548 00 Events v 00:00:18 00:00:19 b000:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='20 00:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='21 00:00-22 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00-23 00:00:24 CO:00-25 00:00-26 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00:-27 00:00:28 00:00-29 100:00:20 00:00:31 00:00:32 00:00-33 00:00:34 0000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='35 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0036 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0037 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00-33 00:00:39 CO:00:40 00:00:4100:00 Bird add/del Events Add Del [Playback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Normal Keybo Paused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Press a to add a temporal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Tab to select and to select another temporal segment timelineFEVA: Fast Event Video Annotation Tool D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Organize space based on operational workflow: Features should be laid out in a logical flow of the workflow while not veering too far away from standard software conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Features not needed in that context should not be visible or active to free up valuable screen real estate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' High-frequency controls and features should be in the middle of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Streamline high-frequency actions: The highly repeated actions, such as creating annotations and fine-tuning them, should be optimized for easier and faster execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Try to accomplish them with a single key press when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Stick to a single device (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=', do not require some steps to use the keyboard and the other steps mouse movements, using up precious time in transitioning between the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=') Leverage D1 when possible to minimize user interaction required to accomplish the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' D3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Use algorithmic support when possible: Whenever possible, offload the user and rely on the algorithm to take on the burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, when the timing is concerned, consider user reaction time and adjust for the lag in the user input from the intended time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Additionally, allowing external modules such as movement detection, human detection, speech detection, and recognition can offload users’ need to find and annotate the events manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, we should be cautious that no matter how good machine learning modules are, no algorithm is 100% accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These should be considered as only additional assistance and not to be completely relied on for ground truth generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Adopt what works and reinvent what doesn’t: Instead of reinventing the wheel, adopt from other tools what works well based on user tests or pilot tests and not instincts and personal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' D5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Allow flexibility: While having D4, tested default input methods is great, adding flexibility by providing redundancies with mouse context menus and keyboard shortcuts might be more intuitive for some users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Additionally, let users redefine the key mapping as people have personal preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 5 FEVA Figure 4: A screenshot of FEVA, where a participant interacting with a robot, is being annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The design considerations inform the minimalist design of FEVA, with most of the screen real estate allocated for the video and the annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Inspired by the clicker/ stopwatch methods and transcribing pedal, we created the speed label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This feature lets you create labels during video play, where a user can press a key to mark the starting and stopping points and continue to watch, creating multiple labels with desired lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The system additionally accounts for the user’s reaction time and adjusts the marked times as seen in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 9 器Localhost:5000 C @ localhost:5000 Incognito FEVA D40 ≤5 p40_all 00:02:58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='227 / 00:15:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='842 5347/28375 02:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 0:02:4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00 00:02:52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:02:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0c 0 00:03:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:03:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00 00:03:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 Event LuttingApple TurnToBacks ove TurmToFrontStove TurnToBackBasil Nod PotStoveOn pON PutLidon Action tun on burne go to stove turn on burner put lid on water pick up lid pick uplid put lid on water Speech ok uh, so so he understands like my hand yup uh Baxster tumn on "pun Baxster, go tuni on Baxster, put Baxster, go Bax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' now Emotion Training Completed Status Oz Robotface Instructions Scene PotTumOnGasFEVA: Fast Event Video Annotation Tool The speed label enables users to annotate in real-time, the fine-tuning control lets the user make frame level adjustments, and the label organizer arranges the label without ever overlapping them for direct access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The flat UI is responsive and aesthetically pleasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Using F-shape design [37], the initial workflow items are on the top left, while the most viewed items are in the middle of the screen, and the most interacted components are at the bottom of the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Keeping most used elements in the hot zones while less used elements such as camera selection and configuration buttons are on the right-hand side in the less noticed area, making them easily tuned out unless needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The UI has redundant user input for flexible and fast interaction to cater to novice and expert users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' And underlying context and configuration-dependent UI model comes to life only when you need them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Expert users can annotate videos faster than in real-time by using media speed control and speed labeler without touching the mouse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Researchers can also use the zoom feature to visualize and analyze labels at a micro or macro level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA UI can display the most number of labels on one screen without losing their meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 Workflow To create any project, upon selecting a video, FEVA imports it and creates an empty annotation file, also referred to as a label or the dataset file (D1, D2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Default tracks will be loaded, which can be customized from the configuration (D3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Users can add or remove tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To play or pause the movie, users can use the media player overlay on the main video or the keyboard shortcut ’spacebar.’ You can use your mouse scroll button to navigate the video in the timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can also use the arrow keys with or without the ’ctrl’ key to move the video by different amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can click and drag the filmstrip, roll over the global timeline and click at the desired time, or double-click the filmstrip or the local timeline window (D1, D2, D5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can also change the movie playback speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can zoom in and out at different time intervals using the + and - icons around the white box on the global timeline or roll over the timeline and ctrl and scroll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To annotate, users can right-click on the tracks and select the label type they want to create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Users can also hit the letter ’A’ key on the keyboard to mark the start and a second time to mark the stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' See figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This is called the speed label, as you can keep annotating without stopping the movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Speed annotation requires a two-pass, but in our pilot tests, the speed label is at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5x faster than the traditional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Following left-to-right and top-to-bottom conventions, workflow such as loading the project, dataset, and manipulating the video and labels are organized in that order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The main video is at the center of the screen occupying the most space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Below the video are the annotation tracks that follow conventions and eyes and the hand layout of users’ gaze and action areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' See figure 5 and figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 View/ Layout As seen in figure 5, ’b’ shows the project selector from which you create a new project and import videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The ’d’ shows your label file selector to create, load, save, merge, import, and export labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can search using the ’e’ and filter labels by type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can double-click the label from the ’2’ label list to find the corresponding label in the timeline ’j.’ You can see the thumbnail preview in ’h’ along with the current time window ’g’ and the global timeline ’f.’ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 Control: User input system Every media control controlled by the mouse also has its associated keyboard shortcuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For flexibility and efficiency, GUI for novice users that are based on principles of recognition rather than recall [38], and shortcuts for expert users for faster control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Users can press the play button overlayed on the video or use the spacebar key at any point to play or pause the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can also choose to speed up or slow down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To create an annotation while the video is in play, you can press the letter ’A’ twice to indicate the start of the label and end the label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This can be customized from the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' A blank label is created after adjusting for your reaction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can also double-click an empty space on a track or right-click the tracks and select a label type you wish to create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While these shortcuts were selected to be consistent with existing standards from other VATs, all shortcuts can be user-defined easily in the user configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' To navigate, you can use your mouse to scroll or click and drag the filmstrips, double click the desired point in the filmstrip or the global progress bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' You can also jump to a specific label from the label list by double-clicking it from the label list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Users can choose what they prefer by providing multiple redundancies with both keyboard and mouse and keyboard shortcuts that can be re-customized by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 Model/ State: Underlying UI support system The UI makes use of the limited screen space (x-y plain) by using the z-axis to layer components displayed and state changes based on contexts such as if the video is playing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' if labels are selected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' if labels are being edited,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' or if your 10 FEVA: Fast Event Video Annotation Tool Figure 5: FEVA Screen Default Layout black boxed areas are 1) Toolbar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 2) Label list,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' main video player,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' and multi- camera selector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 3) Video navigation timeline,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' and 4) Label tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Key components are a) Logo, b) Project selection, c) toolbar icons, d) Label data file selection, e) Search bar, f) Global progress bar timeline, g) Local timeline ruler, h) Filmstrips, i) Label type, and j) tracks and labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' mouse is hovering over a component or a specific feature is enabled in the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Every area is compacted with features that feel intuitive based on affordance users naturally would assign those components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' An example is the video player area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' When the video is playing, one would only see the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' When a mouse pointer hovers over, media controls, current time and frame number, and layer control are displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' From the configuration, you can also enable showing or hiding the video, human body keypoints [39], the bounding box of humans or objects, segmentation masks, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=', that are extensible for researchers to customize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6 Evaluation To evaluate FEVA, we compared FEVA with existing state-of-the-art (SOTA) VAT with two studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Interaction Benchmark: To evaluate the theoretical limits of how fast users could annotate with each VAT, we counted the number of user inputs required to perform various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' User Study: To evaluate user experience based on the user’s perceived workload with each VAT, we conducted user studies where the users provided feedback based on their experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 The State-of-the-Art VAT Selection Method We first created a master list of highly cited VAT that we could download and use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In this list, we only included software that supported temporal annotation that could be downloaded, installed, and run without taking extreme measures for practical reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Therefore, VCode [10], SVAT [40], and VACA [27] could not be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Tools that were too specific such as ToolScape [34], HistoryTracker [33], and CASAM [41] due to missing functionalities such as start and end time, were removed from the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Tools focused on crowd-sourcing such as Glance [42] and CoAT [29], were excluded as we conducted a single-user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Using these criteria, we could narrow down the tools to be compared in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' EagleView [28] was extremely unstable to run, so we could not test them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We narrowed down to Advene [8], ANVIL [7], Elan [9], VIA [11], and VIAN [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 11 a b c 儿 LogoProject Select Toolbar icons Label File Select el Search Labels Multi-Camera Main Video Player Label List Selector fC Global Timeline g5 Local Timeline 5 h Thumbnails (Filmstrips) Label Type 4 Tracks and Labels Track Titles 儿FEVA: Fast Event Video Annotation Tool 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 Interaction Benchmark To compare the steps required to do a particular task with each VAT, we counted the number of clicks, double-clicks, mouse movements, and keyboard key presses and took a cumulative sum as seen in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' If there were multiple ways of completing a task, we included the fastest method for that tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, if you can press Ctrl+N to create a new project (keypresses count = 2) or can move your mouse to the main file menu, click the file, move the mouse to a new project, and click on the menu item (mouse move = 2 and mouse clicks = 2, total = 4), then we took the lesser of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Table 3 shows the 15 tasks considered for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' These included basic setting, label creation, and manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We selected tasks that the majority of the VATs could do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' If a VAT missed a feature, we assigned the worst count received by competing with the VATs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For example, [11] does not support "undo" or "redo" and received a count of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The number of inputs required in FEVA is significantly less than the SOTA, as shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' On average, FEVA requires 36% less input than the SOTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Based on the T-test, FEVA required significantly less input than all tools except VIA, which was not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='SN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Tasks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Advene ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='ANVIL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='ELAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='VIA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='VIAN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='FEVA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Create a project + Import a video ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Create a single label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Create multiple labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Create and name label ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Edit labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Resize labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Play only label video ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Undo/ Redo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='TOTAL SCORE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='69 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='93 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='82 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='81 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='49 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='FEVA Faster by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='29% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='47% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='16% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='40% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='p-value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='0223 Table 3: The list of tasks done using the fastest possible methods in each software (shortcuts where applicable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Each number reflects a cumulative sum of mouse clicks, double clicks, movement, and key presses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The last row shows how much FEVA is faster than the SOTA in percent (%) and the T-test p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 User Study To evaluate user experience, we conducted a user study where participants used two VAT, FEVA, and one another selected SOTA VATs (Advene, ANVIL, ELAN, VIA, and VIAN) in a round-robin fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We counterbalanced the order of the two tools by alternating the order with the next participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Due to the COVID-19 regulations, we conducted our study via Zoom remote shared screen and control feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This introduced some lag in the user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, since both tools were remote, we assumed that the effects of the lag on the outcome were not significantly discriminant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Participants were given an approximate time range where an event occurs with clear descriptions of the events to annotate, for instance, as seen in figure 6, "between 4 minutes and 20 seconds and 4 minutes 40 seconds, please annotate bunny jump roping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" Participants completed approximately 24 tasks until they gave up on the tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' After completing all the tasks in the first software, they filled out the NASA Task Load Index [43] questionnaire with a 5-point Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' And this was repeated with the second tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 12 FEVA: Fast Event Video Annotation Tool Figure 6: An example of a task where a participant was asked to annotate an event of the bunny jump roping between 04:28 minutes mark and 04:31 minutes mark in FEVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 Participants We recruited 34 participants in the University community via email and social media forums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In our study of N=32, the participants were 53% male and 47% female, had a mean age of 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 +/- 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='9, with 84% not having video annotation experience, and 66% had no experience with video editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Two participants had to be dropped due to zoom connection issues and are not counted in N=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 Procedure For this study, we trained annotators for 3 minutes by watching a short training video that taught the basics of media controls, video timeline navigation, and how to create and edit annotations, followed by practice trials for each item with the research coordinator answering any questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' They then spent another 2 minutes exploring the tool on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The participants spent the next 5 minutes practicing the tasks assigned individually by the researcher where they were allowed to ask questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Once they got comfortable, they were randomly given four categories of tasks shown in the list below, with each type of task repeated at least three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The tasks chosen were the most fundamental and repeated tasks annotators must do during video annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The tasks ranged in complexity, with some tasks requiring combinations of the fundamental steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For instance, some tasks simply asked the participant to navigate the video to 2 minutes and 20 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In contrast, others asked participants to annotate three consecutive events between a specific time and name them appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' They were no time limits to perform the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' They worked on the standard freely available "Big Buck Bunny" video which is approximately 10 minutes long at 720p resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Figure 6 demonstrates one example task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' After completing all the tasks with the first tool, the participants filled out the NASA TLX workload questionnaire and repeated the tasks with the second VAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We conducted the following four categories of basic tasks during the user study: Navigate the video a) play/ pause the video, b) jump to a specific point in the timeline, and c) jump to a precise point where a particular label is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 13 FEVA big_ buck_bunny 52 big_buck_bunny_test oirdhit rawr MOIIE PUE MOC 00:04:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='440/ 00:09:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='539 8112/17836 500 00:04:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:04:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='500 00:04:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:04:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='500 00:04:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:04:28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='500 00:04:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:04:29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='500 00:04:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:04:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='500 00:04:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='000 00:04:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='500 Objects Action Speech Emotion Event jump rope +FEVA: Fast Event Video Annotation Tool Label Creation a) Create a new label at a specific time with a specific length, b) Create a label when a participant shows a specific behavior (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=', a character yawns, eats an apple, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' ), and c) create multiple labels in a row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Label Content Manipulation a) Write text annotation for a label created and b) Modify annotation text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Label Temporal Manipulation a) Move the label by a specific number of seconds and b) Resize the label to change its starting time or ending time to match a specific behavior by the person in the video 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 Results In this study, on average, the users felt less metal demand by 46% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00003) with FEVA than the SOTA, less physical demand by 41% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00187), less effort was required by 34% (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00324), and felt less frustration by 62% (p <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='00147).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The difference in the temporal demand and the performance level indexes was not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We attributed this to there not being a time limit enforced during the study, and except for one user on VIAN, where the user gave up, all other users completed all the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA Mental Demand Physical Demand Temporal Demand Performance Level Effort Frustration Level MEAN (n=32) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 Table 4: Shows FEVA’s average score on a NASA Task Load Index with a 5-point Likert scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' FEVA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' (%) Mental Demand Physical Demand Temporal Demand Performance Level Effort Frustration Level Advene (n=5) 75% 57% 0% 14% 67% 183% ANVIL (n=8) 25% 36% 8% 3% 17% 6% ELAN (n=8) 29% 42% 13% 12% 38% 83% VAI (n=8) 44% 36% 6% 11% 22% 36% VIAN (n=3) 120% 40% 20% 17% 83% 140% MEAN (n=32) 46% 41% 5% 10% 34% 62% Table 5: Shows how FEVA compared to the other VAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' A positive number indicates how much people perceived FEVA to be better than other VATs in the NASA TLX respective six dimensions, and a negative number indicates how much worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 User Feedback 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 The Good On average, users expressed FEVA was more intuitive 88% of the time and that FEVA was easier to use 91% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The features users liked the most were the speed label, fine adjustments, "cooler feel," locating label and label playback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' A few users wished they could go back and change their feedback for the first tool once they used the second tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' This was typical when they felt they gave the first tool too high scores after using FEVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In contrast, this did not happen when it was the other way around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' One user said the user wanted to start a fan club and wanted to volunteer to annotate because it was "so fun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 The bad One user mentioned that the user preferred traditional windowed UI for serious work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So the user thought FEVA looked too mainstream tablet app-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Another user stated, "while I think it was fine for me, I don’t think my mom will be able to use either of the tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' So I gave low scores to both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" A few users complained that they did not like pressing the enter key to confirm the label after editing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Clicking elsewhere, causing the loss of what was just typed as a cancel feature, was not popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 The ugly The majority of the confusion, however, was about the global and the local timeline due to needing a clear separation and sharing the same preview component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Participants remotely controlling the UI of the VATs over a Zoom call on the 14 FEVA: Fast Event Video Annotation Tool research coordinator’s computer noticed a lag in the effect of their actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Some users complained that the UI did not update fast enough due to the Zoom lags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' "Maybe because I am controlling your computer through Zoom, but a huge delay made it harder for me to resize the labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='" 7 Implementation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 Framework and Dependencies We used ReactJS [44], an efficient component-based JavaScript library, and wrote the architecture to be lightweight and responsive, so it works on most people’s computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The installation has only two dependencies of Python and Flask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The front end relies on standard HTML, Javascript, ReactJS, and CSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We designed all the controls to optimize for performance and flexibility to customize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We detail the UI layout breakdown in figure 5 and section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 Architecture FEVA uses a simple server-client architecture typical for many web-based applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The server side runs on Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5x or newer with Flask as the webserver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The server side primarily handles servicing data (web content, annotation data, and video streaming) when requested by the client-side application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' For FEVA, most of the modules and the design are on the client side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We show more details of all the modules and their interaction with other modules in the block diagram in appendix 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Figure 7: FEVA Client-Server Architecture Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We include a more detailed block diagram of the different modules on the client side in the appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8 Discussion This is the first version release of our tool FEVA, where we focused on building the fundamental tool while streamlining the user interface and interactions to make annotating events faster, intuitive, robust, and more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' While these early results look promising, there are more research questions that need to be further explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='1 User Study In our pilot and user study, we conducted a short-duration study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' In our user study, we assumed that 15 minutes was sufficient time for participants to learn and practice annotating, which is how we designed the first evaluation of comparing multiple VATs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' However, we need to further our research by conducting a longitudinal user study to understand the impact of our design on users as they get more comfortable with the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='2 User Input In the input sector, we considered the keyboard and mouse/ trackpad as the interaction devices at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Still, we need to expand this to other kinds of inputs, such as touch, speech, and gesture, to explore the potential benefits of multi-modal methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 15 Data Video JSON Server Client Flask (Python) HTMLReactJSCSS HTML TemplatesFEVA: Fast Event Video Annotation Tool 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3 Target Users As more general public gets involved in the coding process, we focused this study so anyone can participate in the coding process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Future studies to gather feedback from seasoned coders will be valuable in understanding how they use VAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='4 VAT benchmarking study In section 3 study, we counted the number of inputs as the pilot study showed the correlation between the number of clicks and the time taken to complete a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' A more comprehensive study should be considered, including the mouse movements and time taken that can reflect user confusion and a more accurate performance metric for completing the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='5 Implicit In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='3, we focused on the user’s perception to reflect their experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Future studies should expand to more quantitative measures implicit in evaluating user confusion, performance, and success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We would also like to conduct more studies to understand labeling consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='6 The layout Based on user feedback, lessons learned from our observations, and the process of comparing with existing tools, we could have done better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The multi-camera selector layout takes up a lot of screen real estate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Many users found the global timeline being so close to the local timeline and the thumbnail view without any separation confusing and more challenging to get used to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We have planned to redesign those experiences in the subsequent versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We will focus on several optimization opportunities in the next release to make FEVA even faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='7 Future Work Beyond the incremental improvements, there were key features that we have planned for the future: Adaptive: All the input controls were linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We plan to explore dynamic and adaptive control systems to various new interaction techniques for faster annotation and a more intuitive experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Extensible: An easier workflow for the open-source community to extend the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' AI assisted: While this had some algorithmic support, better integration with machine learning and deep neural network models is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We will further research how AI can augment the annotation process while exploring ways to inform the users of these models’ inaccuracies, uncertainties, and inherited bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Remote videos: While our internal prototypes support YouTube, there are optimization opportunities that we need to explore before they can be used seamlessly as an alternative to the local MPEG videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We will also explore other online streaming platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Case study: While we are working with some research labs in evaluating the FEVA for their video annotation, we want to invite other interested research labs to try out FEVA, collaborate with us, and grow as a community to address needs that may not have been realized by our research so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 9 Conclusion We present a new event video annotation tool with streamlined interaction techniques and a dynamic UI, contextually visible, and active features organized based on the workflow and usage frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' With features like speed labeling, users can accurately annotate videos in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' With simplified onboarding and workflow, researchers can set up and start annotating videos using minimal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We release FEVA source code in GitHub for everyone to try and further extend its features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' The community can also find project samples, tutorial videos, GitHub issues for support, and future updates on the GitHub page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' As we expand our case studies, we invite more researchers to use FEVA or contact us if you wish to collaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 16 FEVA: Fast Event Video Annotation Tool Acknowledgments We thank Chethan Parameshwara, Levi Burner, Lindsay Little, and peers from UMD and the Perception and Robotics Group for their valuable feedback and discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' We extend special thanks to all our project contributors Johnny Chiu, Rachelle Sims, John Gao, Leya Abraham, Vikram Sehgal, Swagata Chakroborty, Lucas Stuart, and Lin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' References [1] Pat Broadhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Cooperative play and learning from nursery to year one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' Play and learning in the early years, pages 43–60, 2010.' 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Angeles, CA, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' [44] React – a javascript library for building user interfaces, May 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' 19 FEVA: Fast Event Video Annotation Tool A Online Resources Please visit FEVA website http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='snehesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='com/feva for more information, code, instruction videos, samples, and any updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content=' B List of Video Annotation Tools surveyed A Formative Study for Record-time Manual Annotation of First-person Videos A multi-level video annotation tool based on XML-dictionaries A Semi-Automatic Video Annotation Tool to Generate Ground Truth for Intelligent Video Surveillance Systems Advene AIBU An innovative web-based collaborative platform for video annotation An Ontology Web Application-based Annotation Tool for Intangible Culture Heritage Dance Videos Anvil atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='ti ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Augmented Studio: Projection Mapping on Moving Body for Physiotherapy Education ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/u9AyT4oBgHgl3EQfm_jD/content/2301.00482v1.pdf'} +page_content='Automatic tagging of video based on voice and localization ' metadata={'source': 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Weil,1 Bhagya Subrayan,1 +Robert A. Fesen,5 Daniel J. Patnaude,6 Paul P. Plucinsky,6 Charles J. Law,6 William P. Blair,7 and +Jon A. Morse8, 9 +1Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA +2Brookhaven National Laboratory, Upton, New York, United States +3Integrative Data Science Initiative, Purdue University, West Lafayette, IN 47907, USA +4California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA +5Department of Physics and Astronomy, 6127 Wilder Laboratory, Dartmouth College, Hanover, NH 03755, USA +6Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA +7The William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD, +21218 +86 BoldlyGo Institute, 31 W 34 St, Floor 7 Suite 7159, New York, NY 10001, USA +9Visiting Associate in Astronomy, Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA +91125 +ABSTRACT +We present proper motion measurements of oxygen-rich ejecta of the LMC supernova remnant N132D +using two epochs of Hubble Space Telescope Advanced Camera for Surveys data spanning 16 years. +The proper motions of 120 individual knots of oxygen-rich gas were measured and used to calculate +a center of expansion (CoE) of α=5h25m01.71s and δ=−69◦38′41.′′64 (J2000) with a 1-σ uncertainty +of 2.′′90. This new CoE measurement is 9.′′2 and 10.′′8 from two previous CoE estimates based on the +geometry of the optically emitting ejecta. We also derive an explosion age of 2770 ± 500 yr, which +is consistent with recent age estimates of ≈ 2500 yr made from 3D ejecta reconstructions. We verify +our estimates of the CoE and age using a new automated procedure that detected and tracked the +proper motions of 137 knots, with 73 knots that overlap with the visually identified knots. We find +the proper motions of ejecta are still ballistic, despite the remnant’s age, and are consistent with the +notion that the ejecta are expanding into an ISM cavity. Evidence for explosion asymmetry from +the parent supernova is also observed. Using the visually measured proper motion measurements and +corresponding center of expansion and age, we compare N132D to other supernova remnants with +proper motion ejecta studies. +Keywords: ISM: individual(SNR N132D)– ISM: kinematics and dynamics – supernova remnants +1. INTRODUCTION +Supernova remnants (SNRs) provide valuable insights +into the explosion processes of supernovae that are oth- +erwise too distant to resolve (see Milisavljevic & Fesen +2017, for a review). They offer unique opportunities to +probe the elemental distribution of metal-rich ejecta and +investigate the progenitor star’s mass loss history at fine +scales (see Lopez & Fesen 2018, for a review). Young, +nearby oxygen-rich (O-rich) SNRs, created from the col- +Corresponding author: John Banovetz +jbanovet@purdue.edu +lapse of massive stars (ZAMS mass > 8M⊙; Smartt +2009), are especially informative to study core-collapse +dynamics because they are often associated with progen- +itor stars that were largely stripped of their hydrogen +envelopes (e.g., Blair et al. 2000; Chevalier 2005; Temim +et al. 2022). The kinematic and chemical properties of +their metal-rich ejecta retain information about the par- +ent supernova explosion that would otherwise be lost in +an H-rich explosion (Milisavljevic et al. 2010). +Tracking metal-rich ejecta over many years and mea- +suring their proper motion enables estimates of the cen- +ter of expansion (CoE) and explosion age, as well as +information about the progenitor system’s circumstellar +material (CSM) environment via ejecta interaction. The +arXiv:2301.02128v1 [astro-ph.HE] 5 Jan 2023 + +2 +Figure 1. +Left: Continuum subtracted image of N132D using images taken with HST/ACS and the F475W (green) and +F658N (red) filters (additional image information can be found in Table 1). The blue polygon indicates the field of view of the +1994 WFPC2/F502N image. The white square is the cropped image of N132D used for proper motion measurements. Right: +Enlarged view of the white square, highlighting in green the O-rich ejecta used for the proper motion measurements. +CoE and explosion age are important values for deter- +mining the kick velocity of compact objects (Vogt et al. +2018; Banovetz et al. 2021; Long et al. 2022), search- +ing for surviving companions (Kerzendorf et al. 2019; Li +et al. 2021), and measuring differences between optical +and X-ray centers (Katsuda et al. 2018). These values +can also serve as important tests for increasingly sophis- +ticated 2D and 3D supernova simulations (e.g., Wong- +wathanarat et al. 2015; Janka et al. 2016; Burrows et al. +2019; Ferrand et al. 2021; Orlando et al. 2021, 2022). +Only a handful of known O-rich SNRs are sufficiently +resolved to measure proper motion of high velocity +ejecta from multi-epoch observations. This small list in- +cludes Cassiopeia A (Cas A; Kamper & van den Bergh +1976; Thorstensen et al. 2001; Fesen et al. 2006; Ham- +mell & Fesen 2008), G292.0+1.8 (G292; Murdin & Clark +1979; Winkler et al. 2009), and 1E 0102.2-7219 (E0102; +Finkelstein et al. 2006; Banovetz et al. 2021). This paper +focuses on the O-rich SNR N132D, which to date has no +published proper motion measurements of its optically- +emitting ejecta. +N132D is located in the bar of the Large Magel- +lanic Cloud (LMC) and was first identified as a SNR +from radio emission (Westerlund & Mathewson 1966). +Later, it was found to contain high velocity O-rich ejecta +through optical spectra, classifying it as an O-rich SNR +(Danziger & Dennefeld 1976; Lasker 1980). +The par- +ent supernova may have been a Type Ib with a 10-35 +M⊙ ZAMS progenitor (Blair et al. 2000; Sharda et al. +2020). +Presently, the supernova continues to expand +into a cavity created by the pre-supernova mass loss of +the progenitor star (Hughes 1987; Sutherland & Dopita +1995; Blair et al. 2000; Chen et al. 2003; Sharda et al. +2020). +N132D is the brightest X-ray and gamma-ray SNR in +the LMC (Clark 1982; Favata et al. 1997; Borkowski +et al. 2007; H. E. S. S. Collaboration et al. 2015; +Ackermann et al. 2016). +X-ray images show a horse- +shoe shaped forward shock (e.g., Borkowski et al. 2007; +Bamba et al. 2018), the southern portion of which is as- +sociated with natal molecular clouds (Banas et al. 1997; +Dopita et al. 2018; Sano et al. 2020). X-ray and radio +observations indicate that N132D is transitioning from +a young to middle-aged SNR and is about to enter the +Sedov phase (Dickel & Milne 1995; Favata et al. 1997; +Bamba et al. 2018). +Previous estimates of N132D’s explosion age have +been made by dividing the radius of the SNR by the +maximum radial velocity of the ejecta, yielding age esti- +mates ranging from 1300-3440 yr (Danziger & Dennefeld +1976; Lasker 1980; Morse et al. 1995; Sutherland & Do- +pita 1995). Morse et al. (1995) gave two estimates for + +2004 +ACS/F658N +37:00.0 +ACS/F475W +38:30.0 +-69:40:00.0 +30" +30.0 +20.0 +10.0 5:25:00.0 50.0 +24:40.03 +Table 1. HST Observations of N132D +PI +Date +Exp. Time +Instrument +Filter +λcenter +Bandwidth +Pixel Scale +(s) +(˚A) +(˚A) +(′′ pixel−1) +Blair +1994/08/09 +3600 +WFPC2/PC +F502N +5012 +27 +0.0455 +Green +2004/01/22 +1440 +ACS/WFC +F658N +6584 +75 +0.049 +Green +2004/01/22 +1800 +ACS/WFC +F550M +5580 +389 +0.049 +Green +2004/01/21 +1440 +ACS/WFC +F775W +7702 +1300 +0.049 +Green* +2004/01/22 +1520 +ACS/WFC +F475W +4760 +1458 +0.049 +Milisavljevic* +2020/01/05 +2320 +ACS/WFC +F475W +4760 +1458 +0.049 +Milisavljevic +2020/01/05 +2480 +WFC3/UVIS +F502N +5013 +48 +0.040 +Note—* denotes images used in proper motion analysis +the CoE of N132D. The first estimate was made by fit- +ting an ellipse to the diffuse outer rim, and the second by +finding the geometric center of O-rich ejecta. Recent 3D +reconstructions of N132D use this geometrically-derived +center as the CoE and find that N132D’s optically emit- +ting oxygen-rich material is arranged in a torus distri- +bution, inclined at an angle of ≈ 25 − 28◦ in the plane +of the sky (Vogt & Dopita 2011; Law et al. 2020). They +also provide the most recent age estimates of ≈ 2500 yr. +This paper uses high resolution images obtained with +the Hubble Space Telescope (HST) to measure proper +motion of N132D’s oxygen-rich ejecta to estimate a CoE +and explosion age. Section 2 discusses observations of +N132D and the images used. +Section 3 describes our +proper motion measurements and analysis techniques. +Section A introduces an automated procedure to mea- +sure proper motions using computer vision. Section 5 +discusses the implications of the proper motion measure- +ments, CoE, and explosion age as it pertains to previous +estimates and other SNRs. We summarize and conclude +in Section 6. +2. OBSERVATIONS +Using the Mikulski Archive for Space Telescopes +(MAST) at the Space Telescope Science Institute, we ex- +amined three epochs of HST images that are sensitive to +[O III] λλ4959, 5007 emission tracing oxygen-rich ejecta +of N132D1. These consist of an image taken in 1994 us- +ing the Wide Field Planetary Camera 2 (WFPC2) with +the F502N filter (PI: Blair GO-5365), a 2004 image us- +ing the the Advanced Camera for Surveys (ACS) and +the F475W filter (PI: Green GO-12001), and a 2020 +image using the ACS/F475W setup (PI: Milisavljevic +1 The +specific +observations +analyzed +can +be +accessed +via +10.17909/4ppy-4e90 +Figure 2. +ACS/F475W images showing examples of the +expanding ejecta knots in 2004 (left) and 2020 (right) in +the area of the runaway knot (RK, see Section 5.1 for more +details). The 2004 knot centroids are shown as yellow crosses +while the 2020 centroids are shown as red crosses. The red +arrow points in the direction of the CoE. +GO-15818). The 1994 and 2020 F502N images use dif- +ferent instrument and filter configurations, whereas the +2004 and 2020 F475W images were both obtained with +ACS. Utilizing the same camera/filter setup greatly im- +proves the tracking confidence of the gas, as using differ- +ent camera/filters setups can cause ambiguity in precise +tracking due to brightening effects (see Banovetz et al. +2021). Thus, only the ACS images were used for proper +motion tracking. The 2020 F502N image was used to +confirm O-rich ejecta emission from possible continuum +emission. +All images were processed using Astrodriz- +zle (Gonzaga et al. 2012) and had a final image scale +of approximately 0.′′05 pixel−1. Table 1 contains more +information about the images used for analysis. + +2004 +2020 +X +X +CoE +CoE +X +N +1" +E+4 +To align the images, we use the geomap task in +PYRAF2 to create a transformation database using 30 +anchor stars between the two images (see Table A1 in +Appendix). +These anchors were chosen for their low +proper motions and small transformation residuals. The +transformation had resulting residuals of ≈ 0.3 pixels (≈ +0.′′015). We then used the PYRAF task geotran to ap- +ply this transformation, aligning the images. Once the +images were aligned, they were cropped to a 51.′′5×58.′′3 +field of view that contains only the O-rich portion of the +remnant (see Figure 1). The cropping extent was deter- +mined by visual examination of oxygen emission, and we +ensured that all high proper motion ejecta knots were +contained within the selected field of view. The World +Coordinate System (WCS) was calculated using a lo- +cally compiled version of the Astrometry.net3 (Lang +et al. 2010). This WCS solution is accurate to ≈ 0.′′17 +and was taken into account for the final CoE error. +3. PROPER MOTION MEASUREMENTS: +MANUAL ESTIMATION +Figure 3. The absolute proper motion vs radial distance +of the knots. The proper motions of the knots of ejecta are +shown as black points with their corresponding 1-σ error. +The red line indicates a linear fit to the data with the shaded +region indicating the 1-σ error. The location of the four knots +in the RK region are highlighted as blue points. +2 PYRAF is distributed by the National Optical Astronomy Ob- +servatory, which is operated by the AURA, Inc., under coopera- +tive agreement with the National Science Foundation. The Space +Telescope Science Data Analysis System (STSDAS) is distributed +by STScI. +3 Astrometry is distributed as open source under the GNU General +Public License and was developed on Linux. +Using the aligned images, we identified knots with +high proper motions. Knots were chosen by how well +they could be tracked visually, with most knots be- +ing optically bright and circular to have confidence in +the measurements. The shifts of the knots were calcu- +lated by blinking between the two images in SAOImage +DS9 and visually locating the centers of the knots or +other conspicuous features (see Figure 2). The centers +were measured multiple times to estimate positional er- +rors of each knot. During the 16 yrs, knots can possi- +bly brighten/dim or change morphology as they inter- +act with the surrounding medium (Fesen et al. 2011; +Banovetz et al. 2021). This interaction can skew results +if using the astrometric approach of fitting a Gaussian +for the knots. +We did apply a Gaussian based cen- +troid fitting procedure (see Appendix and Section 4) and +found more accurate results through manual inspection. +We applied our methodology to 120 knots (see Ta- +ble A2 in Appendix4), which resulted in proper motions +ranging from 3–14.5 miliarcseconds (mas) per year, with +a median proper motion of 7.54 mas yr−1 and average +relative error of 13% (Figure 3). This translates to a +median velocity of 1784 km s−1 assuming a distance to +the LMC of 50 kpc (Panagia et al. 1991). Our median +velocity is consistent within uncertainties to the average +expansion velocity of 1745 km s−1 calculated using a fit- +ted projected radius of N132D (Law et al. 2020). The +linear fit also gives a higher scaling factor S of 0.′′014 +per km s−1 compared to 0.′′010 per km s−1 of Law et al. +(2020). Figure 4 shows the locations and proper motions +of the 120 knots, as well as the O-rich regions discussed +in Morse et al. (1995). +3.1. Center of Expansion +Our approach to determine the CoE of N132D uses +the trajectories of the ejecta augmented with a likeli- +hood function. This method is similar to that used by +Banovetz et al. (2021) and Thorstensen et al. (2001) for +the calculation of E0102’s and Cas A’s CoE, respectively. +We favor this method because it only depends on the di- +rection of the knots, and is not sensitive to deceleration +over time. +We assume that the likelihood of the CoE in the plane +of sky coordinates (X,Y) is given by: +L(X, Y ) = Πi +wi +2σi +exp(−d2 +i⊥/(2σ2 +i )) +(1) +wi = +1 +PyiPxi +, +(2) +4 Also available in a machine readable format + +16 +Proper Motion of Knot (mas/yr) +14 +12 +10 +8 +10 +20 +30 +40 +50 +Distance away from Center of Expansion (")5 +Figure 4. 2004 ACS/F475W continuum- and hydrogen-subtracted image with vectors representing the measured shifts (mul- +tiplied by a factor of 20 for visual clarity) shown in blue. The regions identified in Morse et al. (1995) are also labeled. +where di⊥ is the perpendicular distance between (X,Y) +and the knot’s line of position, and σi is the uncer- +tainty associated with the point common to the knot’s +extended line of position and di⊥ (Banovetz et al. 2021). +We also define w, the probability of finding an individual +knot in a given X and Y position, denoted by Pxi and +Pyi, respectively, which was calculated using a kernel +density estimate (KDE) to fit knots in the (X,Y) plane. +The (X,Y) combination that maximizes this function +gives the CoE. The uncertainty of the CoE is derived +from 100,000 artificial data sets generated from position +and direction distributions of individual knots. +A notable difference from Banovetz et al. (2021) is the +addition of a weight, w. We used this weight to minimize +the effects of selection bias in our sample. As seen in Fig- +ure 4, N132D is unique compared to other O-rich SNRs +in that the knot distribution is skewed, with a larger +number of knots displaying proper motions in the north- +ern region of the remnant as compared to the southern +region. Without the weight, the CoE will skew in the di- +rection of the more populated region. This added weight +term compensates for sparse regions by giving propor- +tionally more weight to knots in these regions. +Applying this procedure to our proper motion mea- +surements yields a CoE of α=5h25m01.71s and δ=- +69◦38′41.′′64 (J2000) with a 1-σ uncertainty of 2.′′90. Fig- +ure 5 shows the trajectories of the knots as compared to +the derived CoE. +3.2. Explosion Age +Using the manually tracked knots and the associated +center of expansion estimate, we calculated the explo- +sion age of N132D by dividing the knot’s distance from +the CoE by their proper motion measurements. Figure +6 shows the calculated explosion age of all 120 knots. +Combining these ages resulted in an age of 2770 ± 500 +yr. + +38:10.0 +ACS/F475W +B4 +2004 +R1 +B3 +30.0 +R2 +40.0 +B1 +B2 +50.0 +-69:39:00.0 +N +5 +RK +1 pc +E +08.0 +06.0 +04.0 +02.0 +5:25:00.06 +Figure 5. Left: The visually measured proper motions of the 120 knots traced back 25′′ (≈3000 yr assuming an average proper +motion of 8.1 mas yr−1 ). The CoE is shown in yellow. Right: The trajectories of the visually measured knots if forced to +originate from our calculated CoE. Strong spatial asymmetry in the knot distribution is observed with respect to the CoE. +We also calculated the explosion age using only the +knots with the fastest proper motions. +A similar ap- +proach was used by Fesen et al. (2006) for the explo- +sion age of Cassiopeia A and Banovetz et al. (2021) for +E0102. This method assumes that knots with the fastest +proper motions are least decelerated, resulting in a more +accurate explosion age. Forty-nine of the 120 knots with +proper motions greater than the average (8.1 mas yr−1 ) +were selected. Almost all these knots correspond to the +region B4 from Morse et al. (1995). Using these knots +resulted in an explosion age of 2745 ± 404 yr, consistent +with the age using all of the knots. For further discus- +sion, we adopt the age of 2770 ± 500 yr, as this age is +representative of all the knots. +4. PROPER MOTION MEASUREMENTS: +AUTOMATED VIA COMPUTER VISION +We also implemented a novel computer vision based +approach to measure the proper motions of the ejecta. +This approach utilizes hydrogen and continuum sub- +tracted images between the epochs to insure that only +the O-rich material is being tracked. Then, regions of +high emission and/or high ejecta proper motions are +specified. These regions, or stamps, are passed through +an automated detection procedure to identify knots us- +ing image segmentation and deblending. To track the +knots, a kernel density estimate (KDE) estimates the +peaks within these segments, and we use these peaks to +measure the proper motion between the epochs. A more +Figure 6. +A comparison of the explosions age measure- +ments, assuming our CoE. The black line represents the av- +erage age of the data set, while the red dashed line is the 1-σ +uncertainty. This results in an explosion age of 2770 ± 500 +yr. +detailed explanation of this procedure can be found in +the Appendix. + +30 +20 +Y Location +10 +-10 +-20 +-30 +20 +10 +0 +10 +X location (30 +20 +Y Location +10 +-10 +-20 +-30 +20 +0 +10 +X location (120 +100 +80 +Knot ID +60 +40 +20 +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)7 +The automated procedure identified and measured the +proper motions of 137 knots of ejecta5. Seventy-three of +these 137 knots matched visually identified knots used +in the manual procedure. The average difference in in- +ferred values between the manual and automated pro- +cedures for the shift between epochs is 0.′′03 (or ≈ 1.8 +mas yr−1 ) and the vector angle is ≈ 12◦. +The error +of the proper motions was set to 0.4 pixels (≈ 0.′′2), +from the sub-pixel ratio of the KDE. While the auto- +mated proper-motion measurements generally followed +the same ballistic v ∝ r relationship found with the vi- +sually tracked knots, the measurements exhibited more +scatter and higher uncertainty. This discrepancy most +likely arises from tracking fainter, less dense knots that +are more susceptible to deceleration compared to the +bright, denser knots that were found visually. +While our automated procedure generally produces +similar results to the visually measured proper motions, +the sample is contaminated by the less dense knots, pos- +sibly skewing the results. Hence, we adopt the visual +measurement results for this paper. We note that, the +results above used conservative metrics in the measure- +ment of the proper motions in order to not be biased too +heavily by any one parameter. Attempts to improve the +results by fine-tuning these parameters can be found in +the Appendix. +5. DISCUSSION +5.1. Proper Motion Measurements +Our work presents the first proper motion measure- +ments of the O-rich ejecta of N132D. We visually iden- +tified and tracked 120 knots of ejecta across 16 years. +While the baseline is large and could be on the order +of shock cooling times, we are confident in our track- +ing ability. This is because the emission mechanism is +most likely a combination of shock excitation and pho- +toionization producing a high amount of [O III] emission +(Sutherland & Dopita 1995), the low densities of the +shock will increase the cooling times (Blair et al. 2000), +and we find similar morphology in the knots between +epochs. With these 120 knots, we find that the ejecta +follow homologous expansion with an average proper +motion of 8.1 mas yr−1 (median proper motion of 7.54 +mas yr−1 ) despite N132D’s advanced age approaching +the Sedov phase. +Figure 3 shows the proper motions of the knots versus +their distance away from our calculated CoE (see Section +3.1). Comparing this with spectroscopic measurements, +5 Proper motion measurements and locations can be found in a +machine readable format, a subset of which can be found in Table +A2 +the highest proper motion measurements are seen in the +B4 region, as first reported in Morse et al. (1995), and +shown in the left panel of Figure 4. This is to be ex- +pected, as B4 corresponds to a region of small Doppler +velocities in the O-rich ejecta (Morse et al. 1995; Vogt +& Dopita 2011; Law et al. 2020), as seen in the up- +per left of the plot in the right panel of Figure 7. We +find this inverse relationship between the proper mo- +tion measurements and Doppler velocities to hold true +except for the region B1. +B1 also corresponds to an +area of small Doppler velocities, but the proper motion +measurements are much smaller compared to B4. This +difference in proper motions could be a result of B1 pos- +sibly being outside the reverse shock, as proposed by +Vogt & Dopita (2011) (see Figure 7 for X-ray emission +tracing the shocks). However, as this region is consis- +tent with the ballistic trend, it is more likely associated +with an explosion asymmetry (see Section 5.3 for more +discussion). +Notably, the fit shown in Figure 3 does not pass +through the origin and has an offset of ≈ +1.6 mas yr−1 . +Forcing the line through the origin results in S ≈ 0.′′012 +per km s−1 , which is closer to the value reported in Law +et al. (2020). This offset in the original fit could be in- +dicative of deceleration experienced by the ejecta over +time. As the ejecta expands in the surrounding envi- +ronment, the fastest ejecta will interact and decelerate +at a different rate compared to the slower ejecta. This +will disrupt the v ∝ r relation between ejecta velocity +and distance from the CoE, introducing a positive offset +term to the linear fit. +A unique feature of N132D is the runaway knot (RK), +which is an isolated small clump of ejecta located in the +southwestern portion of the remnant and is unique in +that it is enhanced in Si and S but not O (Law et al. +2020). The RK was first reported in Morse et al. (1995) +and recent 3D reconstructions show that the knot is per- +pendicular to the main torus (Vogt & Dopita 2011; Law +et al. 2020). Explanations for the origin of the RK in- +clude evidence of a polar jet (Vogt & Dopita 2011) and +high velocity ejecta, similar to Cas A (Fesen & Gun- +derson 1996; Law et al. 2020). +We were able to find +and measure the proper motions of four O-rich knots +in the same region as the RK (see Figure 2 and Figure +A2). The proper motions are faster than the global aver- +age but are still consistent with proper motions of other +knots (≈ 9.5 mas yr−1 or ≈ 2250 km s−1 ). Combining +this with a Doppler velocity of 820 km s−1 (Law et al. +2020), the RK has a 3D velocity of ≈ 2395 km s−1 . This +is much lower than the total spatial velocity of ≈ 3650 +km s−1 calculated by Law et al. (2020), which is a con- + +8 +Figure 7. Left: Chandra image of N132D (PI: Borkowski). The proper motions from the visual measurements are in blue and +our calculated CoE in yellow (see Section 3.1). Right: Vectors of proper motions (black) with Doppler velocities (Law et al. +2020), centered on our CoE (see Section 3.1). +Table 2. Characteristics of Young O-rich SNRs +Parameter +G292 +E0102 +Cas A +N132D +Proper motion derived CoE (J2000) +11:24:34.4 [1] +1:04:02.48 [2] +23:23:27.77 [3] +5:25:01.71a +-59:15:51 +-72:01:53.92 ++58:48:49.4 +-69:38:41.64 +CoE 1-σ Error (′′) +5 +1.77 +0.4 +2.90 +Center of X-ray emission (J2000) +11:24:33.1 [4] +01:04:1.964 [5] +23:23:27.9 [4] +5:25:03.08b +-59:15:51.1 +-72:01:53.47 +58:48:56.2 +-69:38:32.6 +Current Age (yr) +∼2990 [1] +∼1740 [2] +∼350 [3] +∼2770a +Distance to Remnant (kpc) +6.2 ± 0.9 [7] +62.1 ± 1.9 [8,9] +3.4 +0.3 +−0.1 [10] +50.1 ± 3.1 [11] +Size of Remnant (arcmin) +8.4–9.6 [12] +0.7 [5] +5.6 [13] +1.8 [14] +Ejecta transverse velocity (km/s) +∼1500-3600 [1] +∼1300–2700 [2] +∼5500–14500 [15] +∼700–3400a +Progenitor ZAMS Mass (M⊙) +13–30 [16] +25–50 [17,18] +15–20 [19] +10–35 [17,20] +aThis work. +b Estimated using archival Chandra observations (Xi et al., private communication). +References—[1] Winkler et al. (2009), [2] Banovetz et al. (2021), [3] Thorstensen et al. (2001), [4] Katsuda et al. (2018), [5] Xi +et al. (2019), [6] Dickel & Milne (1995), [7] Gaensler & Wallace (2003), [8] Graczyk et al. (2014), [9] Scowcroft et al. (2016), +[10] Reed et al. (1995), [11] Panagia et al. (1991), [12] Park et al. (2007), [13] Vink et al. (2022), [14] Law et al. (2020), [15] +Fesen et al. (2006), [16] Bhalerao et al. (2019), [17] Blair et al. (2000), [18] Finkelstein et al. (2006), [19] Lee et al. (2014), [20] +Sharda et al. (2020) + +37:30. +Chandra +0*00:88:69- +39:00.0 +N +30.0 +20" +E. +20.0 +15.0 +10.0 +05.0 +5:25:00.0 +55.0 +24:50.0 +45.09 +Figure 8. Center of expansion estimates for N132D. This paper’s CoE calculations and associated 1-σ uncertainty is shown in +yellow. This CoE is centered at α=5h25m01.71s and δ=−69◦38′41.′′64 (J2000) with a 1-σ uncertainty of 2.′′90 and is 9.′′2 and +10.′′8 away from the estimates of Morse et al. (1995) found by fitting an ellipse to the diffuse outer rim (blue) and the O-rich +geometric center (red). + +WFC3/F502N +2020- +Diffuse.Rim +OCenter +COE +N +.: +T'pc +E10 +Figure 9. +Locations of O-rich knots (blue) from N132D (this work), E0102 (Banovetz et al. 2021), Cas A (Hammell & +Fesen 2008), and G292 (Winkler et al. 2009), with trajectories (black) forced to originate on their proper-motion derived CoEs +(Thorstensen et al. 2001; Winkler et al. 2009; Banovetz et al. 2021). Cas A also includes nitrogen rich knots (green) and fast +moving knots (yellow) (Hammell & Fesen 2008). The circle in the bottom right is 1 pc in radius. The physical distances were +calculated using distance estimates and CoEs from Table 2. +sequence of using a scaling relation found from the in- +nermost ejecta that was then extended to the RK. +Although we conclude that the RK does not have +kinematic features that distinguish it from the bulk of +N132D’s ejecta, there remains a conspicuous gap in the +proper motion measurements in the direction of the RK +and our analysis was unable to uncover any new rapidly +moving ejecta in this location. The isolated nature of the +RK may imply that it passed through the reverse shock +and only recently became optically bright again due to +interaction with the CSM/ISM. This interpretation is +supported by X-ray enhancement in close proximity to +the RK (Borkowski et al. 2007; Law et al. 2020), which +can be seen in the left panel of Figure 7, where the RK is +located in the southeast, very close to the rim of X-ray +emission. Further observations of this area may reveal +more knots interacting with the CSM/ISM and fill the +conspicuous gap. +5.2. CoE and Age Results +In Figure 10, we compare our resulting visually in- +spected CoE to the estimates from Morse et al. (1995) +and the automated procedure (see Section 4 for more +details) in Figure 8. Our estimate is located 9.′′2 to the +southwest of the oxygen geometric center and 10.′′8 to +the south and slightly east from the geometric center +of the diffuse rim. Notably, these geometric center es- +timates are ≈ 3.2σ and 3.7σ away from our estimate, + +N132D +Cas A +8 +8 +2770 yr +340 yr +6 +(od) +(od) +CoE +CoE +from +from +away +away +0 +Distance +Distance +Dec. +Dec. +6 +RA Distance awa +RA Distance awa +F +E0102 +G292 +8 +1740 yr +2990 yr +6 +6 +(od) +(od) +CoE +CoE +4 +from +from +away +away +Distance +Distance +Dec. +Dec. +6 +-4 +RA Distance away from CoE (pc) +RA Distance away from CoE (pc)11 +respectively. Our new age estimate of 2770 ± 500 yr +is consistent with the value of ≈ 2500 yr estimated by +Vogt & Dopita (2011) and Law et al. (2020), as well as +the estimate of 2350 ± 520 yr made by Sutherland & +Dopita (1995). +5.3. Comparison to other O-rich SNRs +With +our +new +proper +motion +measurements +of +optically-emitting ejecta of N132D, we expand the num- +ber of young (< 3000 yr) O-rich SNRs with proper mo- +tion studies from three to four.6 Table 2 shows proper- +ties of these remnants (E0102, Cas A, and G292) com- +pared to those of N132D. Figure 9 displays the positions +of O-rich knots with respect to their respective CoEs in +physical space. +Compared to the three other SNRs, N132D shows +the highest degree of spatial asymmetry in the distri- +bution of high velocity ejecta knots. The majority of +the knots are seen to the north of the CoE. Given that +CSM/ISM in the northwest is associated with higher +densities than in the south (Williams et al. 2006), this +unique morphology is likely strongly influenced by ex- +plosion asymmetry. Further supporting the notion that +N132D was an asymmetric explosion in a uniform envi- +ronment is the ballistic proper motions we measure, the +overall blueshift in the Doppler velocities of the ejecta +(Lasker 1980; Sutherland & Dopita 1995; Morse et al. +1995; Vogt & Dopita 2011; Law et al. 2020), asymme- +try in the elemental abundances in X-ray observations +(Sharda et al. 2020), and evidence of a bipolar explosion +from 3D reconstructions (Vogt & Dopita 2011). +In contrast, the distribution of high velocity ejecta +knots in E0102 is fairly uniform, with only the north- +ern part of the remnant lacking any high proper motion +knots (Vogt et al. 2017). There is also a notable asym- +metry in the proper motions, showing evidence that +E0102 is now undergoing non-homologous expansion of +optical ejecta (Banovetz et al. 2021). +Proper motion +measurements show non-ballisitic motion, with slower +material preferentially in the east (Banovetz et al. 2021), +suggesting that E0102 is likely interacting with an inho- +mogenous surrounding environment. X-ray studies also +show varying densities across the remnant, as well as a +non-spherical forward shock (Sasaki et al. 2006; Xi et al. +2019). While some level of explosion asymmetry may be +present, E0102’s morphology is most likely dominated +by effects from an inhomogenous surrounding environ- +ment. +6 Work on Puppis A reported by Winkler et al. (2010) remains +unpublished. +G292, which is elongated along the north-south di- +rection, shows very little evidence of an inhomogenous +environment and its shape comes mostly from explosion +asymmetry. Both proper motion studies and Doppler +measurements show an asymmetric nature to the explo- +sion (Ghavamian et al. 2005; Winkler et al. 2009). The +proper motion-derived CoE coinciding with the X-ray +and radio centers of emission (Winkler et al. 2009) also +support the notion of minimal CSM interaction. +De- +spite interaction with an equatorial bar of CSM mate- +rial, overall G292 appears to be expanding into a low +density environment (Ghavamian et al. 2005). However, +CSM interactions cannot be ruled out. +Recent simu- +lations show that inhomogenous surrounding environ- +ments are reflected in the forward and reverse shock for +only ≈ 2000 yr (Orlando et al. 2022). +There is also +evidence that G292’s morphology is influenced by the +motion of its surviving pulsar (Temim et al. 2022). +Cas A shows an asymmetry in the distribution of +its highest velocity oxygen knots, although not to the +extent of N132D. This remnant shows a main shell +of material moving at ≈4000 to 6000 km s−1 that is +broadly symmetric in the plane of the sky, but there +is also an extended component of sulfur-rich material +to the northeast that extends to velocities upwards of +≈ 1.5 × 104 km s−1 (Hammell & Fesen 2008; Fesen & +Milisavljevic 2016). A complementary high velocity out- +flow also exists in the southwest (Fesen 2001). Knots +of other chemical abundances of Cas A are more sym- +metrical (see Fesen et al. 2006; Hammell & Fesen 2008; +Milisavljevic & Fesen 2013), and although there is a gap +of O-rich material in the south of Cas A as seen in Fig- +ure 9, sulfur-rich main shell ejecta at slower velocities +is present. Simulations and observations of clumpy, fil- +amentary nebulosities have shown that Cas A likely in- +teracted with an inhomogenous CSM environment (Weil +et al. 2020; Orlando et al. 2022). Overall, Cas A is a +mixture of explosion asymmetry and an inhomogenous +surrounding environment. +6. CONCLUSION +We present the first proper motion measurements of +optical emitting ejecta of SNR N132D in the LMC. +The proper motions were measured using manual and +automated procedures applied to two epochs of high +resolution HST data taken 16 yr apart with the same +ACS/F475W instrument+filter combination sensitive to +[O III] λλ4959, 5007 emission. With these proper mo- +tion measurements, we have increased the number of +young, O-rich SNRs with proper motion derived CoEs +from three to four. + +12 +Our measurement of the CoE made via visual in- +spection converged on coordinates α=5h25m01.71s δ=- +69◦38′41.′′64 (J2000) with 1-σ uncertainty of 2.′′90. Our +new CoE estimate is approximately 9.′′2 and 10.′′8 from +previous estimates using geometric centers of emission +(Morse et al. 1995). Combining this CoE estimate with +the proper motion measurements leads to an age of 2770 +± 500 yr, consistent with recent age estimates of ≈ 2500 +yr by 3D reconstructions of N132D (Vogt & Dopita 2011; +Law et al. 2020). +Our new CoE and explosion age serves as a useful +guide for searches to possibly locate the associated neu- +tron star of the original core collapse explosion of N132D +(e.g., Holland-Ashford et al. 2017; Katsuda et al. 2018). +To date, no neutron star has been identified in N132D, +and our CoE identifies a region where a targeted search +can be performed with new 1 Msec Chandra observa- +tions (PI: Plucinsky). Our CoE and age estimates of +N132D can also effectively guide searches for a surviv- +ing binary companion to the progenitor system. To date, +there have been no surviving stellar companions found +for the population of nearby stripped-envelope SN rem- +nants (see, e.g., Kerzendorf et al. 2019). +The nearby +distance of N132D makes it possible to probe individual +stars in the remnant’s stellar neighborhood and avoid +distance uncertainties and source confusion encountered +in studies at extragalactic distances (Fox et al. 2022). +An attempt was recently made to identify the surviving +companion of E0102 (Li et al. 2021) using an updated +CoE; thus our new CoE for N132D makes the remnant +an excellent opportunity for a similar analysis. +ACKNOWLEDGMENTS +We thank the anonymous referees and data editor +for helping improving this paper. D. M. acknowledges +NSF support from grants PHY-1914448, PHY- 2209451, +AST-2037297, and AST-2206532. C.J.L. acknowledges +funding from the National Science Foundation Grad- +uate Research Fellowship under Grant DGE1745303. +This research is based on observations made with the +NASA/ESA Hubble Space Telescope obtained from the +Space Telescope Science Institute, which is operated by +the Association of Universities for Research in Astron- +omy, Inc., under NASA contract NAS 5-26555. These +observations are associated with HST programs 6052, +12001, 12858, and 13378. Support for program #13378 +was provided by NASA through a grant from the Space +Telescope Science Institute, which is operated by the +Association of Universities for Research in Astronomy, +Inc., under NASA contract NAS 5-26555. +Software: PYRAF (Green 2012), ds9 (Smithsonian +Astrophysical Observatory 2000), astrometry.net (Lang +et al. 2010), Astropy (Astropy Collaboration et al. 2013, +2018) +REFERENCES +Ackermann, M., Albert, A., Atwood, W. 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Previous Automation Techniques +In this paper, we used a new automated procedure for measuring the proper motions of supernova remnant ejecta. +Although manual inspection is a reliable method, using computer vision measurement techniques can allow for rapidly +reproducible results, testing various quantitative thresholds, and scale to a large number of proper motions more +efficiently. One common technique is using a cross-correlation method (e.g., Currie et al. 1996), that was been used +for the proper motion measurements of SNRs (e.g., Finkelstein et al. 2006; Winkler et al. 2009), stellar ejecta (e.g., +Morse et al. 2001; Kiminki et al. 2016), and protostellar jets (e.g., Hartigan et al. 2001; Bally et al. 2002). This uses +predefined box regions to outline a ‘clump’ in one epoch that is then translated to a new position. The translated image +is then subtracted by the second epoch and the translation that produces the minimum sum of the differences is the +translation that is used. Other computer vision techniques for measuring proper motions in SNRs include measuring +the optical flow, projection methods, and maximum likelihood functions (e.g., Borkowski et al. 2020). +The success of these procedures depends on the knots being bright and are best suited for large regions of gas. +However, with the high spatial resolution of HST, we are able to resolve individual knots and cover more of the +periphery of the SNR where faint, high proper motion ejecta knots are expected to be. Still, these knots can change +in illumination and shape between epochs (e.g. Fesen et al. 2011; Patnaude & Fesen 2014), which can lead to errors +in tracking. One of the goals of our new procedure is to be able to track fainter, individual knots for an older O-rich +remnant such as N132D. +A.2. Automated Proper Motion Measurement Procedure +A.2.1. Image Preparation +The first step in image preparation was to ensure that emission unrelated to N132D’s oxygen-rich material was +corrected for and removed. Doing so ensured that our computer vision procedure only tracked [O III] λλ4959, 5007 +emission and was not confused by CSM/ISM or stellar emission that was easily recognized in and avoided by the +manually inspected proper motion measurements. +We first scaled the F550M (continuum) and F658N (Hα) images taken in 2004 to the ACS/F475W images by +matching the flux of common stars and then subtracting the F550M and F658N emission from the ACS/F475W +image. This is to remove the stellar continuum and Hβ emission (using Hα as a tracer) from our images in order to +only track the oxygen-rich ejecta. We then manually removed any subtraction residuals outside of emission regions, +and performed additional cleaning of the images using cosmicray lacosmic task from the astropy package ccdproc. +A.2.2. Identifying Knots +Next, small regions were identified for further individual processing. +We chose regions that had strong oxygen +emission or showed evidence of proper motion measurements when blinking between epochs. We divide these regions +into 2′′ by 2′′ boxes, or stamps. Figure A1 shows all of the stamps while the first row of Figure A2 shows an enlarged +example of a stamp. The stamp’s size was chosen for its ability to encapsulate knot motion between epochs and to +sample the local background. This is important since N132D’s non-uniform emission properties makes using a global +background value impractical. We allowed large overlap between the stamps to ensure that no knots were missed. +We then used the detect sources and deblend sources tasks from the photutils package in astropy to identify +knots, shown in the second and third row of Figure A2, respectively. Detect sources creates a segmented image +which identifies sources of emission above a certain threshold. In our case, this was 2σ above the median background +of the 2′′ by 2′′ stamp. We further restricted our analysis to those knots containing emission that spanned at least +four adjacent pixels. This pixellimit was selected to account for smaller, fainter knots than those found through visual +inspection. +Deblend sources takes the segmented regions and finds the local maxima to separate potential overlapping knots. +For this process, we assumed a Gaussian kernel and a low contrast between the knots. We then apply three initial +filters to remove any embedded residuals or hot pixels that were missed with cosmicray lacosmic. The first filter +removes segments too close to the edges of the region, effectively reducing the region from 2′′ by 2′′ to 1.′′9 by 1.′′9. The + +16 +Figure A1. 2004 ACS/F475W continuum- and hydrogen-subtracted image with vectors representing the automated procedure’s +measured shifts (multiplied by a factor of 20 for visual clarity) shown in red. The stamps used in the automated procedure are +shown as blue boxes. Empty boxes represent areas where the automated procedure could not detect any motion and/or identify +knots. +next filter removes segments that do not move more than 1.5 mas yr−1 , to remove any remaining residuals. This lower +limit was chosen based on the manually inspected lowest proper motions around 3 mas yr−1 . The last filter removes +any segment that does not have a corresponding segment within 0.′′3 around it in the other epoch, to remove residuals +or hot pixels that appear in one epoch but not the other. +A.2.3. Matching Knots and Calculating Shifts +The changing brightness and shape of individual knots over time, in addition to any residual hot pixels, can affect +the local peak position and the geometric center determined for each knot. To mitigate these effects, the local peaks +in the identified knots are calculated using a KDE fit of the stamp (fourth row of Figure A2). To achieve sub-pixel +accuracy, we found the best results by resampling the stamp from 40x40 pixels to 100x100 before applying the KDE, +resulting in an error of 0.4 pixels (≈ 0.′′2) for the calculated centers. We also found the best bandwidth for the KDE +was the the Silverman kernel (Silverman 1982) through manual inspection of the resulting KDE centers. +The next step is to match the knots between the epochs. Each pair of centroids is found using an initial guess of +the center of expansion to find the position angle between the knot and the guess. We use the CoE calculated by +the visually tracked knots for the initial guess. Each individual pair of knots between epochs is given a score, which +is the shift between epochs multiplied by the difference between their trajectory and the position angle. The pair +with the lowest score is selected as the correct combination. The combinations are then passed through two filters to + +- +8422 +中 +V +5 +1 pc17 +Figure A2. Steps needed for the computer vision technique to identify and measure the proper motion of ejecta knots. The +2004 and 2020 images of the same region as Figure 2 are shown on the left and right panels, respectively. The top row shows +the original cutout of the continuum and hydrogen subtracted image in gray scale. The second and third rows show the results +of the detect sources and deblend sources tasks, respectively. The fourth row shows the remaining segments after applying +the KDE to the total region, identifying the localized peaks as shown in red. The final row shows the matched centroids. The +centers are shown in red (2004) and blue (2020), with a white vector showing the motion between the two epochs. +mitigate outliers. Knot combinations are discarded if the difference between the trajectory and the position angle is +greater than 90 degrees and if the shift is larger than 30 mas yr−1 , double the largest proper motion found using visual +measurements. The effect of changing these parameters are explored in Section 4. If multiple combinations use the +same endpoint (e.g. the deblending process splits a knot into two between epochs), the combination with the lowest +score is chosen and the other is discarded. The final row of Figure A2 shows an example final combinations. +Finally, we clip the top and bottom 5% of the proper motion vectors to remove outliers, average duplicate mea- +surements due to overlapping stamps, and remove measurements that are within 3 pixels of the mask for subtraction +residuals. The remaining measurements are used for the proper motion analysis as was done for the manual inspection +in Section 3. We ensured that the procedure is robust using a toy model simulation, detailed in the following section. + +Cleaned Stamp +Detect Sources +Deblend Sources +Filtered KDE Mask +2004 +202018 +Figure A3. Similar to Figure 8 with the addition of the automated procedure’s result of α=5h25m02.771s and δ=-69◦38′38.′′985 +(J2000) with 1-σ uncertainty of 4.′′47 in purple. +A.2.4. Simulated Proper Motion +We tested our procedure using a simulated, idealized SNR with ejecta moving ballisitically. We generated two 300 +by 300 pixel images at a scale similar to HST with randomized background emission using the make noise image +task in astropy’s photutils package. The first was an image of 45 knots randomly placed to concentrically surround +a test CoE located at the center of the image. The knots were drawn from a sample of the visually inspected HST +knots, the majority of which were in the top third of knot brightness, and were placed using the geometric center of +each knot. The second image was then created using a different randomized background and placing these knots a +certain distance, radially away from the simulated CoE with a proper motion with v ∝ r. We implemented a scanning +procedure to find regions containing the knots. This procedure scanned the image using subdivided regions to 40x40 +pixel boxes with overlap between them.7 +Our automated procedure recovers all 45 knots. There was an average positional difference of 2.12 pixels (≈ 0.′′1 +with HST resolution), an average shift difference of 1.53 pixels (≈3.5 mas yr−1 ), and an average angle difference of +0.12 radians between the inferred and true (simulated) values. We calculated a CoE of (154.77,155.97) with a 1σ error +radius of 10.88 pixels (≈ 0.′′58) as compared to the simulated CoE of (150,150). Overall, we were able to find all of the +7 We did not pass regions through the filters for hot pixels and star +removal, as neither of these features were present in the simulated +images. + +WFC3/F502N +2020 +Diffuse.Rim +.O.Center +Computer +Vision +Visual +Inspection +TpC +E19 +knots, match them correctly, and calculate their speed and trajectory to return a CoE that is within 1-σ of the true +value. +A.3. Automated Procedure Results +This procedure was able to identify and track 137 knots of ejecta with the error of the proper motions was set to 0.4 +pixels (≈ 0.′′2), from the sub-pixel ratio of the KDE. Figures A4–A7 show the knot locations, trajectory, and proper +motion trends using this procedure. The procedure measured proper motions ranging from 2 to 17 mas yr−1 and an S +of 0.′′015 per km s−1 . +Using the same CoE method as outlined in Section 3.1, the 137 proper motions yields a CoE of α=5h25m02.771s +and δ=-69◦38′38.′′985 (J2000) with 1-σ uncertainty of 4.′′47. Figure A3 in the Appendix shows this result as compared +to the other center of expansion estimates. +Utilizing this CoE and proper motions, we calculate an age of 3377 ± 2241 yr using all 137 knots, as shown in the +left panel of Figure A5. This large discrepancy is most likely due to the procedure measuring the proper motions +of artifacts or heavily decelerated knots, skewing the age estimates to higher values. To account for the decelerated +knots, we also calculated the age using knots above the median proper motion. These 70 knots yield an age of 2497 +± 638 yr (right panel of Figure A5), much closer to that derived from visually measured proper motions. +A.4. Effect of Tuning Parameters +Figures A4, A3, and A5 show the results of using the conservative metrics to measure the proper motions, their +trajectories, and subsequent CoE and explosion ages, respectively. These results are highlighted in Section 4. The +conservative parameter constraints used in the selection of knots adopted in our automated procedure were used to +incorporate many degrees of freedom. However, the associated proper motion measurement uncertainties were much +larger than those associated with our manual procedure. There are many ways that the parameters of knot selection +can be further constrained to reduce the uncertainties and better match the manually measured CoE and age. We +explored limiting the difference angle between the trajectory and position angle of the input CoE to 45 degrees (from +90), increasing the minimum proper motion to 3 mas yr−1 (from 1.5 mas yr−1 ), and using brighter knots by increasing +the signal to noise of selected knots from 2σ to 3σ. We found that by tightening the boundaries of these parameters, +the CoE of the automated procedure and the age calculation were both within 1-σ of the visual inspection results. +The following section contains a detailed discussion about the effect of each of these parameters. +This procedure is ideal for ejecta proper motion analysis of other SNRs. Two parameters that must be changed +between SNRs are the KDE bandwidth and the arbitrary CoE. The KDE bandwidth is very sensitive and needs to be +fine-tuned depending on the SNR. A bandwidth that is too small will identify many flux peaks within a knot while a +bandwidth that is too large can miss fainter knots. +Another parameter that we explored was the influence of the choice of the arbitrary CoE for the trajectory versus +position angle cutoff. +For our procedure outlined in Section A.2.3, we chose the CoE calculated using the visual +inspection method. Assuming this CoE was not available, we could have chosen the [O III] geometric center for our +arbitrary CoE (Morse et al. 1995). We experimented with the 45 and 90 degree cutoff with this arbitrary CoE and +found that the resulting CoE did not match that found with the visually inspected CoE using an arbitrary initial CoE. +However, we found that by iterating the arbitrary initial CoE calculation procedure, the CoE calculation converges to +the same CoE as if using the visually inspected CoE for the arbitrary CoE. Running through 10 initial guess estimates +of the CoE, we found that using a 45 degree cutoff would take 3–4 iterations, whereas the 90 degree cutoff would take +2 iterations before converging on the the same CoE as found by using the visual inspection CoE as the arbitrary CoE. +As such, we would recommend anyone using our procedure to use the iteration method to verify results, especially +if using a geometric center for the arbitrary CoE, as they are often offset from proper motion derived centers (see +Thorstensen et al. 2001; Katsuda et al. 2018; Banovetz et al. 2021). +A.4.1. Further Fine-tuning of Parameters +Here we discuss how measurement and calculation of the CoE and age are affected by knot selection parameters +in our automated procedure. Figure A4 shows the results of using the conservative parameters for the automated +procedure. Figure A5 show the results of the age calculation when using all 137 knots, or only the fastest (70). +Figures A6 and A7 show the results of changing the parameters of the automated procedure. The left, middle left, +middle right, and right panels show the effect of changing the parameters to incorporate a higher minimum speed + +20 +(3 mas yr−1 ), only bright knots (3σ), difference of trajectory and position angle of 45 degrees, and all three of the +changes, respectively. +Figure A4. Similar to Figure 3 (left) and 5 (middle and right) but using the proper motions from the conservative parameters +of the automated procedure. +Figure A5. Similar to Figure 6 but using the proper motions from the automated procedure and including the results of only +using the fastest ejecta. This results in an age of 3377 ± 2241 yr using all the knots and 2497 ± 638 yr using the fastest. The +red shaded points correspond to the matching visually measured knots when applicable. + +30 +30 +17.5 +(mas/yr) +20 +20 +15.0 +12 +Knot ( +Location (") +10 +10 +Location +10.0 +Proper Motion of i +0 +-10 +-10 +2 +-20 +-20 +0.0E +30 +20 +30 +40 +10 +10 +50 +-20 +0 +X location (") +X location (") +Distance away from Center of Expansion (")140 +120 +100 +80 +D +Knot I +60 +40 +20 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)140 +120 +100 +80 +D +Knot I +60 +40 +20 +Fast Eiecta +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)21 +Figure A6. Similar to Figure A4 for the 3 mas yr−1 lower limit (left), increase to 3-σ of brightness (middle left), and reducing +the trajectory and position angle difference of the knots to less than 45 degrees (middle right), and applying all three variations +(right). +Figure A7. Similar to Figure 6 for the 3 mas yr−1 lower limit (top left), increase to 3-σ of brightness (top right),the trajectory +and position angle difference of the knots to less than 45 degrees (bottom left), and using all the parameters (bottom right) +that match closest to the visual inspection result. + +18 +Proper Motion of Knot (mas/yr) +16 +14 +12 +10 +0 +10 +20 +30 +40 +50 +Distance away from Center of Expansion (")18 +Proper Motion of Knot (mas/yr) +16 +14 +12 +10 +8 +0 +10 +20 +30 +40 +50 +Distance away from Center of Expansion (")18 +16 +Proper Motion of Knot (mas/yr) +14 +12 +10 +0 +10 +20 +30 +40 +50 +Distance away from Center of Expansion (")18 +Proper Motion of Knot (mas/yr) +16 +14 +12 +10 +0 +10 +20 +30 +40 +50 +Distance away from Center of Expansion (")20 +Location (") +10 +-10 +-20 +30 +20 +10 +20 +X location ("30 +20 +10 +Y Location ( +-10 +-20 +20 +OT +20 +X location ()30 +20 +Location (") +10 +10 +-20 +30 +20 +10 +20 +X location ()30 +20 +Y Location (") +10 +-10 +-20 +20 +10 +OT +20 +X location ()140 +120 +100 +80 +D +Knot I +60 +40 +20 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)140 +120 +100 +80 +Knot ID +60 +40 +20 +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)140 +120 +100 +80 +Knot ID +60 +40 +20 +Fast Eiecta +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)140 +120 +100 +80 +Knot ID +60 +40 +20 +Fast Ejecta +0 +1000 +2000 +3000 +4000 +5000 +6000 +Explosion Age (years)22 +B. ADDITIONAL TABLES +Table A1. Anchor Star Coordinates +Star +RA (J2000) +Dec. (J2000) +1 +5h24m53.7308s +69d38m15.310s +2 +5h24m58.3686s +69d38m35.298s +3 +5h24m57.8822s +69d38m46.447s +4 +5h24m56.5386s +69d39m38.340s +5 +5h24m58.2838s +69d39m39.118s +6 +5h24m42.9113s +69d39m13.996s +7 +5h24m47.0373s +69d38m43.292s +8 +5h24m45.0608s +69d38m41.959s +9 +5h24m49.4502s +69d38m29.224s +10 +5h25m04.9414s +69d39m39.441s +11 +5h25m00.6230s +69d38m00.893s +12 +5h24m51.6282s +69d39m05.757s +13 +5h24m49.5039s +69d38m47.073s +14 +5h24m50.4438s +69d39m48.192s +15 +5h24m52.2404s +69d39m22.696s +16 +5h25m01.3169s +69d39m48.778s +17 +5h24m44.4366s +69d39m09.882s +18 +5h24m56.5344s +69d38m47.148s +19 +5h25m04.2141s +69d40m09.653s +20 +5h24m58.9991s +69d40m16.083s +21 +5h25m07.3918s +69d40m02.308s +22 +5h25m06.7103s +69d39m40.337s +23 +5h25m11.2145s +69d38m20.227s +24 +5h24m45.6713s +69d38m35.164s +25 +5h24m47.2420s +69d38m17.491s +26 +5h24m59.4994s +69d40m08.583s +27 +5h25m08.2993s +69d39m18.289s +28 +5h24m59.8352s +69d40m17.268s +29 +5h24m49.7550s +69d38m06.447s +30 +5h24m52.6476s +69d38m01.440s +Table A2. Manually Inspected Knot Measurements and Corresponding Automated Measurements +Knot +RA +Dec +Visual +Procedure +Automated +Procedure +µα +σµα +µδ +σµδ +µα +µδ +(J2000) +(J2000) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +1 +5h25m06.759s +69d39m04.815s +-8.095 +0.452 +-7.722 +0.431 +-7.386 +-7.386 +2 +5h25m06.817s +69d39m04.523s +-5.075 +0.472 +-4.399 +0.409 +-3.693 +-3.693 +Table A2 continued + +23 +Table A2 (continued) +Knot +RA +Dec +Visual +Procedure +Automated +Procedure +µα +σµα +µδ +σµδ +µα +µδ +(J2000) +(J2000) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +3 +5h25m06.791s +69d39m04.138s +-7.498 +0.469 +-6.602 +0.413 +-7.386 +-6.155 +4 +5h25m06.870s +69d39m04.341s +-7.866 +0.487 +-6.336 +0.392 +-7.386 +-6.155 +5 +5h24m58.937s +69d38m48.660s +5.251 +0.571 +-2.335 +0.254 +NA +NA +6 +5h24m59.498s +69d38m49.597s +2.766 +0.412 +-3.161 +0.470 +2.462 +-3.693 +7 +5h24m59.334s +69d38m50.789s +3.786 +0.403 +-4.496 +0.478 +NA +NA +8 +5h24m59.263s +69d38m49.396s +5.096 +0.520 +-3.409 +0.347 +NA +NA +9 +5h24m59.365s +69d38m49.148s +5.557 +0.543 +-3.162 +0.309 +NA +NA +10 +5h24m59.317s +69d38m49.096s +6.207 +0.550 +-3.349 +0.297 +NA +NA +11 +5h24m59.068s +69d38m47.891s +5.977 +0.555 +-3.097 +0.288 +4.924 +-3.693 +12 +5h24m58.914s +69d38m47.797s +7.006 +0.578 +-2.873 +0.237 +7.386 +-2.462 +13 +5h24m59.357s +69d38m46.738s +6.259 +0.589 +-2.238 +0.210 +NA +NA +14 +5h24m59.052s +69d38m43.732s +4.547 +0.578 +-1.877 +0.238 +3.693 +-1.231 +15 +5h24m58.979s +69d38m35.721s +3.913 +0.571 +1.736 +0.253 +NA +NA +16 +5h24m58.891s +69d38m34.856s +4.518 +0.513 +3.136 +0.356 +4.924 +2.462 +17 +5h24m59.026s +69d38m35.230s +4.951 +0.505 +3.605 +0.368 +NA +NA +18 +5h24m59.091s +69d38m33.413s +4.812 +0.543 +2.746 +0.310 +2.462 +2.462 +19 +5h24m59.355s +69d38m35.500s +4.221 +0.427 +4.511 +0.456 +NA +NA +20 +5h24m59.444s +69d38m35.482s +3.312 +0.478 +2.791 +0.403 +NA +NA +21 +5h24m59.387s +69d38m35.145s +5.867 +0.534 +3.571 +0.325 +NA +NA +22 +5h24m59.471s +69d38m35.149s +3.199 +0.407 +3.728 +0.474 +NA +NA +23 +5h24m59.061s +69d38m32.842s +3.358 +0.462 +3.061 +0.421 +2.462 +2.462 +24 +5h24m59.165s +69d38m32.899s +6.936 +0.545 +3.905 +0.307 +4.924 +3.693 +25 +5h24m59.336s +69d38m33.396s +5.870 +0.548 +3.218 +0.300 +NA +NA +26 +5h25m00.580s +69d38m33.323s +1.729 +0.248 +4.002 +0.574 +1.231 +1.231 +27 +5h25m00.657s +69d38m33.912s +1.930 +0.281 +3.832 +0.558 +NA +NA +28 +5h25m00.707s +69d38m35.283s +2.645 +0.433 +2.750 +0.451 +NA +NA +29 +5h25m00.614s +69d38m30.296s +2.558 +0.324 +4.227 +0.535 +NA +NA +30 +5h25m00.476s +69d38m33.102s +1.871 +0.244 +4.407 +0.575 +NA +NA +31 +5h25m00.700s +69d38m28.663s +0.142 +0.015 +5.776 +0.625 +0.000 +3.693 +32 +5h25m00.845s +69d38m27.202s +2.276 +0.252 +5.171 +0.572 +3.693 +2.462 +33 +5h25m00.594s +69d38m26.570s +1.869 +0.145 +7.859 +0.608 +NA +NA +34 +5h25m00.948s +69d38m26.000s +2.112 +0.195 +6.414 +0.594 +3.693 0 +2.462 +35 +5h25m00.447s +69d38m23.793s +1.778 +0.177 +6.031 +0.599 +1.231 +6.155 +36 +5h25m00.407s +69d38m22.524s +1.459 +0.119 +7.525 +0.614 +2.462 +7.386 +37 +5h25m00.231s +69d38m22.238s +2.345 +0.183 +7.649 +0.598 +NA +NA +38 +5h25m00.473s +69d38m23.509s +3.180 +0.264 +6.810 +0.566 +4.924 +7.386 +39 +5h24m59.998s +69d38m22.286s +2.610 +0.237 +6.370 +0.578 +NA +NA +40 +5h25m00.851s +69d38m18.795s +1.378 +0.099 +8.562 +0.617 +1.231 +6.155 +41 +5h25m00.920s +69d38m18.666s +2.360 +0.233 +5.875 +0.580 +0.000 +6.155 +Table A2 continued + +24 +Table A2 (continued) +Knot +RA +Dec +Visual +Procedure +Automated +Procedure +µα +σµα +µδ +σµδ +µα +µδ +(J2000) +(J2000) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +42 +5h25m00.956s +69d38m17.995s +0.808 +0.056 +8.964 +0.622 +NA +NA +43 +5h25m00.964s +69d38m18.212s +1.658 +0.138 +7.308 +0.610 +0.000 +6.155 +44 +5h25m00.764s +69d38m18.370s +1.616 +0.099 +10.030 +0.617 +2.462 +12.311 +45 +5h25m00.892s +69d38m18.986s +1.431 +0.113 +7.782 +0.615 +1.231 +6.155 +46 +5h25m01.869s +69d38m20.792s +-1.263 +0.074 +10.628 +0.621 +NA +NA +47 +5h25m02.200s +69d38m20.383s +-1.787 +0.150 +7.249 +0.607 +-3.693 +4.924 +48 +5h25m02.373s +69d38m20.365s +-2.316 +0.191 +7.216 +0.595 +-3.693 +4.924 +49 +5h25m02.258s +69d38m20.524s +-2.631 +0.187 +8.404 +0.596 +-3.693 +4.924 +50 +5h25m02.309s +69d38m21.170s +-1.858 +0.105 +10.877 +0.616 +-2.462 +11.080 +51 +5h25m02.559s +69d38m26.825s +-3.684 +0.396 +4.500 +0.484 +-3.693 +3.693 +52 +5h25m02.397s +69d38m30.352s +-1.014 +0.141 +4.365 +0.609 +0.000 +4.924 +53 +5h25m02.455s +69d38m30.719s +-1.352 +0.141 +5.836 +0.609 +-1.231 +6.155 +54 +5h25m02.538s +69d38m30.703s +-4.625 +0.405 +5.428 +0.476 +-3.693 +4.924 +55 +5h25m03.081s +69d38m30.761s +-2.899 +0.377 +3.830 +0.498 +-1.231 +3.693 +56 +5h25m03.079s +69d38m29.633s +-2.074 +0.300 +3.791 +0.548 +-2.462 +4.924 +57 +5h25m03.225s +69d38m29.853s +-4.040 +0.461 +3.700 +0.422 +-1.231 +2.462 +58 +5h25m03.387s +69d38m28.857s +-2.748 +0.259 +6.035 +0.569 +-3.693 +4.924 +59 +5h25m03.553s +69d38m31.191s +-4.421 +0.466 +3.954 +0.417 +-3.693 +3.693 +60 +5h25m03.644s +69d38m31.047s +-4.108 +0.393 +5.077 +0.486 +-4.924 +2.462 +61 +5h25m03.738s +69d38m30.971s +-4.446 +0.443 +4.420 +0.441 +-2.462 +3.693 +62 +5h25m03.792s +69d38m30.488s +-3.843 +0.400 +4.604 +0.480 +-2.462 +4.924 +63 +5h25m04.737s +69d38m37.017s +-4.387 +0.599 +1.320 +0.180 +NA +NA +64 +5h25m04.584s +69d38m38.620s +-7.044 +0.540 +4.111 +0.315 +NA +NA +65 +5h25m04.800s +69d38m36.432s +-6.623 +0.483 +5.433 +0.396 +NA +NA +66 +5h25m03.965s +69d38m41.964s +-2.947 +0.602 +0.823 +0.168 +-2.462 +1.231 +67 +5h25m04.343s +69d38m40.948s +-4.563 +0.553 +2.407 +0.292 +NA +NA +68 +5h25m03.941s +69d38m42.226s +-5.002 +0.622 +0.481 +0.060 +-2.462 +1.231 +69 +5h25m04.387s +69d38m30.020s +-5.281 +0.410 +6.064 +0.471 +-6.155 +2.462 +70 +5h25m04.664s +69d38m29.489s +-5.792 +0.441 +5.809 +0.443 +NA +NA +71 +5h25m04.712s +69d38m29.118s +-5.238 +0.465 +4.714 +0.418 +NA +NA +72 +5h25m05.020s +69d38m28.243s +-5.438 +0.469 +4.799 +0.414 +-4.924 +6.155 +73 +5h25m05.009s +69d38m28.027s +-4.951 +0.417 +5.517 +0.465 +-4.924 +6.155 +74 +5h25m04.751s +69d38m27.468s +-4.284 +0.342 +6.548 +0.523 +-3.693 +6.155 +75 +5h25m05.117s +69d38m28.383s +-3.508 +0.360 +4.986 +0.511 +-3.693 +7.386 +76 +5h25m04.831s +69d38m26.683s +-4.528 +0.413 +5.136 +0.469 +-4.924 +7.386 +77 +5h25m04.797s +69d38m26.002s +-5.496 +0.402 +6.537 +0.478 +-6.155 +7.386 +78 +5h25m02.870s +69d38m21.258s +-3.230 +0.273 +6.654 +0.562 +-2.462 +4.924 +79 +5h25m02.811s +69d38m21.684s +-5.987 +0.404 +7.064 +0.477 +-2.462 +4.924 +80 +5h25m03.575s +69d38m22.023s +-3.531 +0.270 +7.370 +0.564 +0.000 +1.231 +Table A2 continued + +25 +Table A2 (continued) +Knot +RA +Dec +Visual +Procedure +Automated +Procedure +µα +σµα +µδ +σµδ +µα +µδ +(J2000) +(J2000) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +81 +5h25m03.874s +69d38m20.501s +-3.165 +0.291 +6.015 +0.553 +-3.693 +7.386 +82 +5h25m03.436s +69d38m23.201s +-2.373 +0.198 +7.098 +0.593 +NA +NA +83 +5h25m04.893s +69d38m22.848s +-4.533 +0.306 +8.081 +0.545 +-3.693 +7.386 +84 +5h25m04.797s +69d38m21.717s +-7.334 +0.410 +8.452 +0.472 +NA +NA +85 +5h25m04.663s +69d38m20.967s +-7.508 +0.428 +7.979 +0.455 +NA +NA +86 +5h25m04.954s +69d38m18.989s +-5.288 +0.281 +10.497 +0.558 +NA +NA +87 +5h25m05.116s +69d38m18.782s +-6.096 +0.363 +8.557 +0.509 +NA +NA +88 +5h25m05.411s +69d38m19.262s +-7.661 +0.502 +5.670 +0.372 +NA +NA +89 +5h25m08.036s +69d38m20.676s +-11.101 +0.512 +7.791 +0.359 +-12.311 +4.924 +90 +5h25m07.979s +69d38m20.386s +-10.437 +0.505 +7.603 +0.368 +-12.311 +4.924 +91 +5h25m07.921s +69d38m20.738s +-8.091 +0.394 +9.951 +0.485 +-9.848 +6.155 +92 +5h25m07.220s +69d38m16.762s +-9.750 +0.454 +9.215 +0.429 +-8.617 +8.617 +93 +5h25m07.046s +69d38m17.499s +-10.700 +0.463 +9.720 +0.420 +NA +NA +94 +5h25m06.250s +69d38m19.801s +-8.183 +0.501 +6.116 +0.374 +-8.617 +6.155 +95 +5h25m06.290s +69d38m19.507s +-8.475 +0.492 +6.644 +0.386 +-8.617 +6.155 +96 +5h25m06.317s +69d38m17.245s +-6.808 +0.433 +7.101 +0.451 +-7.386 +7.386 +97 +5h25m05.259s +69d38m16.604s +-6.608 +0.401 +7.889 +0.479 +-7.386 +7.386 +98 +5h25m05.423s +69d38m15.945s +-5.608 +0.298 +10.350 +0.550 +NA +NA +99 +5h25m05.352s +69d38m15.983s +-6.149 +0.357 +8.852 +0.513 +NA +NA +100 +5h25m05.648s +69d38m15.853s +-6.412 +0.367 +8.849 +0.506 +-6.155 +8.617 +101 +5h25m05.767s +69d38m16.143s +-6.894 +0.343 +10.506 +0.523 +-6.155 +8.617 +102 +5h25m05.794s +69d38m15.290s +-6.897 +0.366 +9.563 +0.507 +NA +NA +103 +5h25m06.181s +69d38m13.869s +-7.882 +0.410 +9.053 +0.471 +-9.848 +8.617 +104 +5h25m05.608s +69d38m13.780s +-6.183 +0.314 +10.619 +0.540 +-6.155 +8.617 +105 +5h25m06.015s +69d38m13.013s +-8.498 +0.391 +10.610 +0.488 +NA +NA +106 +5h25m05.661s +69d38m13.407s +-6.182 +0.323 +10.241 +0.535 +-6.155 +8.617 +107 +5h25m05.593s +69d38m12.558s +-6.322 +0.313 +10.925 +0.541 +NA +NA +108 +5h25m07.099s +69d38m11.216s +-9.856 +0.435 +10.153 +0.448 +-9.848 +9.848 +109 +5h25m06.241s +69d38m10.852s +-7.146 +0.375 +9.535 +0.500 +-7.386 +8.617 +110 +5h25m05.609s +69d38m10.424s +-7.850 +0.387 +9.943 +0.491 +-6.155 +9.848 +111 +5h25m05.343s +69d38m08.948s +-5.298 +0.301 +9.658 +0.548 +NA +NA +112 +5h25m05.381s +69d38m11.992s +-6.841 +0.346 +10.297 +0.521 +NA +NA +113 +5h25m05.438s +69d38m10.191s +-7.160 +0.344 +10.850 +0.522 +-6.155 +9.848 +114 +5h25m04.640s +69d38m10.040s +-3.924 +0.241 +9.388 +0.577 +-7.386 +9.848 +115 +5h25m04.454s +69d38m09.844s +-4.261 +0.246 +9.951 +0.575 +NA +NA +116 +5h24m58.646s +69d38m25.981s +6.982 +0.425 +7.526 +0.458 +NA +NA +117 +5h25m01.395s +69d38m49.800s +1.921 +0.302 +-3.487 +0.547 +NA +NA +118 +5h25m01.442s +69d38m50.223s +0.035 +0.005 +-4.109 +0.625 +NA +NA +119 +5h24m59.732s +69d38m20.293s +3.725 +0.301 +6.767 +0.548 +0.000 +3.693 +Table A2 continued + +26 +Table A2 (continued) +Knot +RA +Dec +Visual +Procedure +Automated +Procedure +µα +σµα +µδ +σµδ +µα +µδ +(J2000) +(J2000) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +(mas yr−1) +120 +5h25m05.432s +69d38m08.588s +-6.583 +0.334 +10.405 +0.528 +NA +NA +Note—Positive values indicate direction to the north and east for RA and Dec. respectively. Results of the automated procedure +are matched when applicable. + diff --git a/vtA0T4oBgHgl3EQfMP8I/content/tmp_files/load_file.txt b/vtA0T4oBgHgl3EQfMP8I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ca010f5c7c4442d9e2fe5554d232babcd7da04f --- /dev/null +++ b/vtA0T4oBgHgl3EQfMP8I/content/tmp_files/load_file.txt @@ -0,0 +1,2410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf,len=2409 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX63 HST Proper Motion Measurements of Supernova Remnant N132D: Center of Expansion and Age John Banovetz,1, 2 Dan Milisavljevic,1, 3 Niharika Sravan,4 Kathryn E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Weil,1 Bhagya Subrayan,1 Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Fesen,5 Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Patnaude,6 Paul P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Plucinsky,6 Charles J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law,6 William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Blair,7 and Jon A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Morse8, 9 1Department of Physics and Astronomy, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA 2Brookhaven National Laboratory, Upton, New York, United States 3Integrative Data Science Initiative, Purdue University, West Lafayette, IN 47907, USA 4California Institute of Technology, 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' California Blvd, Pasadena, CA 91125, USA 5Department of Physics and Astronomy, 6127 Wilder Laboratory, Dartmouth College, Hanover, NH 03755, USA 6Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 7The William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Charles Street, Baltimore, MD, 21218 86 BoldlyGo Institute, 31 W 34 St, Floor 7 Suite 7159, New York, NY 10001, USA 9Visiting Associate in Astronomy, Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125 ABSTRACT We present proper motion measurements of oxygen-rich ejecta of the LMC supernova remnant N132D using two epochs of Hubble Space Telescope Advanced Camera for Surveys data spanning 16 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The proper motions of 120 individual knots of oxygen-rich gas were measured and used to calculate a center of expansion (CoE) of α=5h25m01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='71s and δ=−69◦38′41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′64 (J2000) with a 1-σ uncertainty of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This new CoE measurement is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′8 from two previous CoE estimates based on the geometry of the optically emitting ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We also derive an explosion age of 2770 ± 500 yr, which is consistent with recent age estimates of ≈ 2500 yr made from 3D ejecta reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We verify our estimates of the CoE and age using a new automated procedure that detected and tracked the proper motions of 137 knots, with 73 knots that overlap with the visually identified knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We find the proper motions of ejecta are still ballistic, despite the remnant’s age, and are consistent with the notion that the ejecta are expanding into an ISM cavity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Evidence for explosion asymmetry from the parent supernova is also observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Using the visually measured proper motion measurements and corresponding center of expansion and age, we compare N132D to other supernova remnants with proper motion ejecta studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Keywords: ISM: individual(SNR N132D)– ISM: kinematics and dynamics – supernova remnants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' INTRODUCTION Supernova remnants (SNRs) provide valuable insights into the explosion processes of supernovae that are oth- erwise too distant to resolve (see Milisavljevic & Fesen 2017, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' They offer unique opportunities to probe the elemental distribution of metal-rich ejecta and investigate the progenitor star’s mass loss history at fine scales (see Lopez & Fesen 2018, for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Young, nearby oxygen-rich (O-rich) SNRs, created from the col- Corresponding author: John Banovetz jbanovet@purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='edu lapse of massive stars (ZAMS mass > 8M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Smartt 2009), are especially informative to study core-collapse dynamics because they are often associated with progen- itor stars that were largely stripped of their hydrogen envelopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Blair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Chevalier 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Temim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The kinematic and chemical properties of their metal-rich ejecta retain information about the par- ent supernova explosion that would otherwise be lost in an H-rich explosion (Milisavljevic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Tracking metal-rich ejecta over many years and mea- suring their proper motion enables estimates of the cen- ter of expansion (CoE) and explosion age, as well as information about the progenitor system’s circumstellar material (CSM) environment via ejecta interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='02128v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='HE] 5 Jan 2023 2 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Left: Continuum subtracted image of N132D using images taken with HST/ACS and the F475W (green) and F658N (red) filters (additional image information can be found in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The blue polygon indicates the field of view of the 1994 WFPC2/F502N image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The white square is the cropped image of N132D used for proper motion measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Right: Enlarged view of the white square, highlighting in green the O-rich ejecta used for the proper motion measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' CoE and explosion age are important values for deter- mining the kick velocity of compact objects (Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2022), search- ing for surviving companions (Kerzendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021), and measuring differences between optical and X-ray centers (Katsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These values can also serve as important tests for increasingly sophis- ticated 2D and 3D supernova simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Wong- wathanarat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Janka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Burrows et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Ferrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Only a handful of known O-rich SNRs are sufficiently resolved to measure proper motion of high velocity ejecta from multi-epoch observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This small list in- cludes Cassiopeia A (Cas A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Kamper & van den Bergh 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Thorstensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Ham- mell & Fesen 2008), G292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='8 (G292;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Murdin & Clark 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2009), and 1E 0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2-7219 (E0102;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This paper focuses on the O-rich SNR N132D, which to date has no published proper motion measurements of its optically- emitting ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' N132D is located in the bar of the Large Magel- lanic Cloud (LMC) and was first identified as a SNR from radio emission (Westerlund & Mathewson 1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Later, it was found to contain high velocity O-rich ejecta through optical spectra, classifying it as an O-rich SNR (Danziger & Dennefeld 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Lasker 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The par- ent supernova may have been a Type Ib with a 10-35 M⊙ ZAMS progenitor (Blair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Sharda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Presently, the supernova continues to expand into a cavity created by the pre-supernova mass loss of the progenitor star (Hughes 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Sutherland & Dopita 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Blair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Sharda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' N132D is the brightest X-ray and gamma-ray SNR in the LMC (Clark 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Favata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Borkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Ackermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' X-ray images show a horse- shoe shaped forward shock (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Borkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Bamba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018), the southern portion of which is as- sociated with natal molecular clouds (Banas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Dopita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Sano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' X-ray and radio observations indicate that N132D is transitioning from a young to middle-aged SNR and is about to enter the Sedov phase (Dickel & Milne 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Favata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Bamba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Previous estimates of N132D’s explosion age have been made by dividing the radius of the SNR by the maximum radial velocity of the ejecta, yielding age esti- mates ranging from 1300-3440 yr (Danziger & Dennefeld 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Lasker 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Sutherland & Do- pita 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995) gave two estimates for 2004 ACS/F658N 37:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ACS/F475W 38:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 69:40:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 30" 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 5:25:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 24:40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='03 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' HST Observations of N132D PI Date Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Time Instrument Filter λcenter Bandwidth Pixel Scale (s) (˚A) (˚A) (′′ pixel−1) Blair 1994/08/09 3600 WFPC2/PC F502N 5012 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0455 Green 2004/01/22 1440 ACS/WFC F658N 6584 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='049 Green 2004/01/22 1800 ACS/WFC F550M 5580 389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='049 Green 2004/01/21 1440 ACS/WFC F775W 7702 1300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='049 Green* 2004/01/22 1520 ACS/WFC F475W 4760 1458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='049 Milisavljevic* 2020/01/05 2320 ACS/WFC F475W 4760 1458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='049 Milisavljevic 2020/01/05 2480 WFC3/UVIS F502N 5013 48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='040 Note—* denotes images used in proper motion analysis the CoE of N132D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The first estimate was made by fit- ting an ellipse to the diffuse outer rim, and the second by finding the geometric center of O-rich ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Recent 3D reconstructions of N132D use this geometrically-derived center as the CoE and find that N132D’s optically emit- ting oxygen-rich material is arranged in a torus distri- bution, inclined at an angle of ≈ 25 − 28◦ in the plane of the sky (Vogt & Dopita 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' They also provide the most recent age estimates of ≈ 2500 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This paper uses high resolution images obtained with the Hubble Space Telescope (HST) to measure proper motion of N132D’s oxygen-rich ejecta to estimate a CoE and explosion age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Section 2 discusses observations of N132D and the images used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Section 3 describes our proper motion measurements and analysis techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Section A introduces an automated procedure to mea- sure proper motions using computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Section 5 discusses the implications of the proper motion measure- ments, CoE, and explosion age as it pertains to previous estimates and other SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We summarize and conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' OBSERVATIONS Using the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope Science Institute, we ex- amined three epochs of HST images that are sensitive to [O III] λλ4959, 5007 emission tracing oxygen-rich ejecta of N132D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These consist of an image taken in 1994 us- ing the Wide Field Planetary Camera 2 (WFPC2) with the F502N filter (PI: Blair GO-5365), a 2004 image us- ing the the Advanced Camera for Surveys (ACS) and the F475W filter (PI: Green GO-12001), and a 2020 image using the ACS/F475W setup (PI: Milisavljevic 1 The specific observations analyzed can be accessed via 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='17909/4ppy-4e90 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' ACS/F475W images showing examples of the expanding ejecta knots in 2004 (left) and 2020 (right) in the area of the runaway knot (RK, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The 2004 knot centroids are shown as yellow crosses while the 2020 centroids are shown as red crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The red arrow points in the direction of the CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' GO-15818).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The 1994 and 2020 F502N images use dif- ferent instrument and filter configurations, whereas the 2004 and 2020 F475W images were both obtained with ACS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Utilizing the same camera/filter setup greatly im- proves the tracking confidence of the gas, as using differ- ent camera/filters setups can cause ambiguity in precise tracking due to brightening effects (see Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Thus, only the ACS images were used for proper motion tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The 2020 F502N image was used to confirm O-rich ejecta emission from possible continuum emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' All images were processed using Astrodriz- zle (Gonzaga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2012) and had a final image scale of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′05 pixel−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Table 1 contains more information about the images used for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2004 2020 X X CoE CoE X N 1" E+4 To align the images, we use the geomap task in PYRAF2 to create a transformation database using 30 anchor stars between the two images (see Table A1 in Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These anchors were chosen for their low proper motions and small transformation residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The transformation had resulting residuals of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3 pixels (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We then used the PYRAF task geotran to ap- ply this transformation, aligning the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Once the images were aligned, they were cropped to a 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′5×58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′3 field of view that contains only the O-rich portion of the remnant (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The cropping extent was deter- mined by visual examination of oxygen emission, and we ensured that all high proper motion ejecta knots were contained within the selected field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The World Coordinate System (WCS) was calculated using a lo- cally compiled version of the Astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='net3 (Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This WCS solution is accurate to ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′17 and was taken into account for the final CoE error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' PROPER MOTION MEASUREMENTS: MANUAL ESTIMATION Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The absolute proper motion vs radial distance of the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The proper motions of the knots of ejecta are shown as black points with their corresponding 1-σ error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The red line indicates a linear fit to the data with the shaded region indicating the 1-σ error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The location of the four knots in the RK region are highlighted as blue points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2 PYRAF is distributed by the National Optical Astronomy Ob- servatory, which is operated by the AURA, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', under coopera- tive agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The Space Telescope Science Data Analysis System (STSDAS) is distributed by STScI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 3 Astrometry is distributed as open source under the GNU General Public License and was developed on Linux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Using the aligned images, we identified knots with high proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Knots were chosen by how well they could be tracked visually, with most knots be- ing optically bright and circular to have confidence in the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The shifts of the knots were calcu- lated by blinking between the two images in SAOImage DS9 and visually locating the centers of the knots or other conspicuous features (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The centers were measured multiple times to estimate positional er- rors of each knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' During the 16 yrs, knots can possi- bly brighten/dim or change morphology as they inter- act with the surrounding medium (Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This interaction can skew results if using the astrometric approach of fitting a Gaussian for the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We did apply a Gaussian based cen- troid fitting procedure (see Appendix and Section 4) and found more accurate results through manual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We applied our methodology to 120 knots (see Ta- ble A2 in Appendix4), which resulted in proper motions ranging from 3–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 miliarcseconds (mas) per year, with a median proper motion of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='54 mas yr−1 and average relative error of 13% (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This translates to a median velocity of 1784 km s−1 assuming a distance to the LMC of 50 kpc (Panagia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Our median velocity is consistent within uncertainties to the average expansion velocity of 1745 km s−1 calculated using a fit- ted projected radius of N132D (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The linear fit also gives a higher scaling factor S of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′014 per km s−1 compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′010 per km s−1 of Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure 4 shows the locations and proper motions of the 120 knots, as well as the O-rich regions discussed in Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Center of Expansion Our approach to determine the CoE of N132D uses the trajectories of the ejecta augmented with a likeli- hood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This method is similar to that used by Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2021) and Thorstensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2001) for the calculation of E0102’s and Cas A’s CoE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We favor this method because it only depends on the di- rection of the knots, and is not sensitive to deceleration over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We assume that the likelihood of the CoE in the plane of sky coordinates (X,Y) is given by: L(X, Y ) = Πi wi 2σi exp(−d2 i⊥/(2σ2 i )) (1) wi = 1 PyiPxi , (2) 4 Also available in a machine readable format 16 Proper Motion of Knot (mas/yr) 14 12 10 8 10 20 30 40 50 Distance away from Center of Expansion (")5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2004 ACS/F475W continuum- and hydrogen-subtracted image with vectors representing the measured shifts (mul- tiplied by a factor of 20 for visual clarity) shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The regions identified in Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995) are also labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' where di⊥ is the perpendicular distance between (X,Y) and the knot’s line of position, and σi is the uncer- tainty associated with the point common to the knot’s extended line of position and di⊥ (Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We also define w, the probability of finding an individual knot in a given X and Y position, denoted by Pxi and Pyi, respectively, which was calculated using a kernel density estimate (KDE) to fit knots in the (X,Y) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The (X,Y) combination that maximizes this function gives the CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The uncertainty of the CoE is derived from 100,000 artificial data sets generated from position and direction distributions of individual knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A notable difference from Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2021) is the addition of a weight, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We used this weight to minimize the effects of selection bias in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' As seen in Fig- ure 4, N132D is unique compared to other O-rich SNRs in that the knot distribution is skewed, with a larger number of knots displaying proper motions in the north- ern region of the remnant as compared to the southern region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Without the weight, the CoE will skew in the di- rection of the more populated region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This added weight term compensates for sparse regions by giving propor- tionally more weight to knots in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Applying this procedure to our proper motion mea- surements yields a CoE of α=5h25m01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='71s and δ=- 69◦38′41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′64 (J2000) with a 1-σ uncertainty of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Fig- ure 5 shows the trajectories of the knots as compared to the derived CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Explosion Age Using the manually tracked knots and the associated center of expansion estimate, we calculated the explo- sion age of N132D by dividing the knot’s distance from the CoE by their proper motion measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure 6 shows the calculated explosion age of all 120 knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Combining these ages resulted in an age of 2770 ± 500 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 38:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ACS/F475W B4 2004 R1 B3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 R2 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 B1 B2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 69:39:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 N 5 RK 1 pc E 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 5:25:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='06 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Left: The visually measured proper motions of the 120 knots traced back 25′′ (≈3000 yr assuming an average proper motion of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 mas yr−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The CoE is shown in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Right: The trajectories of the visually measured knots if forced to originate from our calculated CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Strong spatial asymmetry in the knot distribution is observed with respect to the CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We also calculated the explosion age using only the knots with the fastest proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A similar ap- proach was used by Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2006) for the explo- sion age of Cassiopeia A and Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2021) for E0102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This method assumes that knots with the fastest proper motions are least decelerated, resulting in a more accurate explosion age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Forty-nine of the 120 knots with proper motions greater than the average (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 mas yr−1 ) were selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Almost all these knots correspond to the region B4 from Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Using these knots resulted in an explosion age of 2745 ± 404 yr, consistent with the age using all of the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' For further discus- sion, we adopt the age of 2770 ± 500 yr, as this age is representative of all the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' PROPER MOTION MEASUREMENTS: AUTOMATED VIA COMPUTER VISION We also implemented a novel computer vision based approach to measure the proper motions of the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This approach utilizes hydrogen and continuum sub- tracted images between the epochs to insure that only the O-rich material is being tracked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Then, regions of high emission and/or high ejecta proper motions are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These regions, or stamps, are passed through an automated detection procedure to identify knots us- ing image segmentation and deblending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' To track the knots, a kernel density estimate (KDE) estimates the peaks within these segments, and we use these peaks to measure the proper motion between the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A more Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A comparison of the explosions age measure- ments, assuming our CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The black line represents the av- erage age of the data set, while the red dashed line is the 1-σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This results in an explosion age of 2770 ± 500 yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' detailed explanation of this procedure can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 30 20 Y Location 10 10 20 30 20 10 0 10 X location (30 20 Y Location 10 10 20 30 20 0 10 X location (120 100 80 Knot ID 60 40 20 0 0 1000 2000 3000 4000 5000 6000 Explosion Age (years)7 The automated procedure identified and measured the proper motions of 137 knots of ejecta5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Seventy-three of these 137 knots matched visually identified knots used in the manual procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The average difference in in- ferred values between the manual and automated pro- cedures for the shift between epochs is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′03 (or ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='8 mas yr−1 ) and the vector angle is ≈ 12◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The error of the proper motions was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 pixels (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2), from the sub-pixel ratio of the KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' While the auto- mated proper-motion measurements generally followed the same ballistic v ∝ r relationship found with the vi- sually tracked knots, the measurements exhibited more scatter and higher uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This discrepancy most likely arises from tracking fainter, less dense knots that are more susceptible to deceleration compared to the bright, denser knots that were found visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' While our automated procedure generally produces similar results to the visually measured proper motions, the sample is contaminated by the less dense knots, pos- sibly skewing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Hence, we adopt the visual measurement results for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We note that, the results above used conservative metrics in the measure- ment of the proper motions in order to not be biased too heavily by any one parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Attempts to improve the results by fine-tuning these parameters can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' DISCUSSION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Proper Motion Measurements Our work presents the first proper motion measure- ments of the O-rich ejecta of N132D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We visually iden- tified and tracked 120 knots of ejecta across 16 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' While the baseline is large and could be on the order of shock cooling times, we are confident in our track- ing ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This is because the emission mechanism is most likely a combination of shock excitation and pho- toionization producing a high amount of [O III] emission (Sutherland & Dopita 1995), the low densities of the shock will increase the cooling times (Blair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2000), and we find similar morphology in the knots between epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' With these 120 knots, we find that the ejecta follow homologous expansion with an average proper motion of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 mas yr−1 (median proper motion of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='54 mas yr−1 ) despite N132D’s advanced age approaching the Sedov phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure 3 shows the proper motions of the knots versus their distance away from our calculated CoE (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Comparing this with spectroscopic measurements, 5 Proper motion measurements and locations can be found in a machine readable format, a subset of which can be found in Table A2 the highest proper motion measurements are seen in the B4 region, as first reported in Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995), and shown in the left panel of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This is to be ex- pected, as B4 corresponds to a region of small Doppler velocities in the O-rich ejecta (Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Vogt & Dopita 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020), as seen in the up- per left of the plot in the right panel of Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We find this inverse relationship between the proper mo- tion measurements and Doppler velocities to hold true except for the region B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' B1 also corresponds to an area of small Doppler velocities, but the proper motion measurements are much smaller compared to B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This difference in proper motions could be a result of B1 pos- sibly being outside the reverse shock, as proposed by Vogt & Dopita (2011) (see Figure 7 for X-ray emission tracing the shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' However, as this region is consis- tent with the ballistic trend, it is more likely associated with an explosion asymmetry (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3 for more discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Notably, the fit shown in Figure 3 does not pass through the origin and has an offset of ≈ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6 mas yr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Forcing the line through the origin results in S ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′012 per km s−1 , which is closer to the value reported in Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This offset in the original fit could be in- dicative of deceleration experienced by the ejecta over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' As the ejecta expands in the surrounding envi- ronment, the fastest ejecta will interact and decelerate at a different rate compared to the slower ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This will disrupt the v ∝ r relation between ejecta velocity and distance from the CoE, introducing a positive offset term to the linear fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A unique feature of N132D is the runaway knot (RK), which is an isolated small clump of ejecta located in the southwestern portion of the remnant and is unique in that it is enhanced in Si and S but not O (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The RK was first reported in Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995) and recent 3D reconstructions show that the knot is per- pendicular to the main torus (Vogt & Dopita 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Explanations for the origin of the RK in- clude evidence of a polar jet (Vogt & Dopita 2011) and high velocity ejecta, similar to Cas A (Fesen & Gun- derson 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We were able to find and measure the proper motions of four O-rich knots in the same region as the RK (see Figure 2 and Figure A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The proper motions are faster than the global aver- age but are still consistent with proper motions of other knots (≈ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 mas yr−1 or ≈ 2250 km s−1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Combining this with a Doppler velocity of 820 km s−1 (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020), the RK has a 3D velocity of ≈ 2395 km s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This is much lower than the total spatial velocity of ≈ 3650 km s−1 calculated by Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2020), which is a con- 8 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Left: Chandra image of N132D (PI: Borkowski).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The proper motions from the visual measurements are in blue and our calculated CoE in yellow (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Right: Vectors of proper motions (black) with Doppler velocities (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020), centered on our CoE (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Characteristics of Young O-rich SNRs Parameter G292 E0102 Cas A N132D Proper motion derived CoE (J2000) 11:24:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 [1] 1:04:02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='48 [2] 23:23:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='77 [3] 5:25:01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='71a 59:15:51 72:01:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='92 +58:48:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 69:38:41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='64 CoE 1-σ Error (′′) 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='90 Center of X-ray emission (J2000) 11:24:33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 [4] 01:04:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='964 [5] 23:23:27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='9 [4] 5:25:03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='08b 59:15:51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 72:01:53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='47 58:48:56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2 69:38:32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6 Current Age (yr) ∼2990 [1] ∼1740 [2] ∼350 [3] ∼2770a Distance to Remnant (kpc) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='9 [7] 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='9 [8,9] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 [10] 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1 [11] Size of Remnant (arcmin) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6 [12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='7 [5] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6 [13] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='8 [14] Ejecta transverse velocity (km/s) ∼1500-3600 [1] ∼1300–2700 [2] ∼5500–14500 [15] ∼700–3400a Progenitor ZAMS Mass (M⊙) 13–30 [16] 25–50 [17,18] 15–20 [19] 10–35 [17,20] aThis work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' b Estimated using archival Chandra observations (Xi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', private communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' References—[1] Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2009), [2] Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2021), [3] Thorstensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2001), [4] Katsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2018), [5] Xi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2019), [6] Dickel & Milne (1995), [7] Gaensler & Wallace (2003), [8] Graczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2014), [9] Scowcroft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2016), [10] Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995), [11] Panagia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1991), [12] Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2007), [13] Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2022), [14] Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2020), [15] Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2006), [16] Bhalerao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2019), [17] Blair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2000), [18] Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2006), [19] Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2014), [20] Sharda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2020) 37:30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Chandra 0*00:88:69- 39:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 N 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 20" E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 5:25:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 24:50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='09 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Center of expansion estimates for N132D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This paper’s CoE calculations and associated 1-σ uncertainty is shown in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This CoE is centered at α=5h25m01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='71s and δ=−69◦38′41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′64 (J2000) with a 1-σ uncertainty of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′90 and is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′8 away from the estimates of Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995) found by fitting an ellipse to the diffuse outer rim (blue) and the O-rich geometric center (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' WFC3/F502N 2020- Diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Rim OCenter COE N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=" : T'pc E10 Figure 9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Locations of O-rich knots (blue) from N132D (this work), E0102 (Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021), Cas A (Hammell & Fesen 2008), and G292 (Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2009), with trajectories (black) forced to originate on their proper-motion derived CoEs (Thorstensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Cas A also includes nitrogen rich knots (green) and fast moving knots (yellow) (Hammell & Fesen 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The circle in the bottom right is 1 pc in radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The physical distances were calculated using distance estimates and CoEs from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' sequence of using a scaling relation found from the in- nermost ejecta that was then extended to the RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Although we conclude that the RK does not have kinematic features that distinguish it from the bulk of N132D’s ejecta, there remains a conspicuous gap in the proper motion measurements in the direction of the RK and our analysis was unable to uncover any new rapidly moving ejecta in this location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The isolated nature of the RK may imply that it passed through the reverse shock and only recently became optically bright again due to interaction with the CSM/ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This interpretation is supported by X-ray enhancement in close proximity to the RK (Borkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020), which can be seen in the left panel of Figure 7, where the RK is located in the southeast, very close to the rim of X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Further observations of this area may reveal more knots interacting with the CSM/ISM and fill the conspicuous gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' CoE and Age Results In Figure 10, we compare our resulting visually in- spected CoE to the estimates from Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (1995) and the automated procedure (see Section 4 for more details) in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Our estimate is located 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2 to the southwest of the oxygen geometric center and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′8 to the south and slightly east from the geometric center of the diffuse rim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Notably, these geometric center es- timates are ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2σ and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='7σ away from our estimate, N132D Cas A 8 8 2770 yr 340 yr 6 (od) (od) CoE CoE from from away away 0 Distance Distance Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 6 RA Distance awa RA Distance awa F E0102 G292 8 1740 yr 2990 yr 6 6 (od) (od) CoE CoE 4 from from away away Distance Distance Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 6 4 RA Distance away from CoE (pc) RA Distance away from CoE (pc)11 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Our new age estimate of 2770 ± 500 yr is consistent with the value of ≈ 2500 yr estimated by Vogt & Dopita (2011) and Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2020), as well as the estimate of 2350 ± 520 yr made by Sutherland & Dopita (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Comparison to other O-rich SNRs With our new proper motion measurements of optically-emitting ejecta of N132D, we expand the num- ber of young (< 3000 yr) O-rich SNRs with proper mo- tion studies from three to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6 Table 2 shows proper- ties of these remnants (E0102, Cas A, and G292) com- pared to those of N132D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure 9 displays the positions of O-rich knots with respect to their respective CoEs in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Compared to the three other SNRs, N132D shows the highest degree of spatial asymmetry in the distri- bution of high velocity ejecta knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The majority of the knots are seen to the north of the CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Given that CSM/ISM in the northwest is associated with higher densities than in the south (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2006), this unique morphology is likely strongly influenced by ex- plosion asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Further supporting the notion that N132D was an asymmetric explosion in a uniform envi- ronment is the ballistic proper motions we measure, the overall blueshift in the Doppler velocities of the ejecta (Lasker 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Sutherland & Dopita 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Vogt & Dopita 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020), asymme- try in the elemental abundances in X-ray observations (Sharda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020), and evidence of a bipolar explosion from 3D reconstructions (Vogt & Dopita 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' In contrast, the distribution of high velocity ejecta knots in E0102 is fairly uniform, with only the north- ern part of the remnant lacking any high proper motion knots (Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' There is also a notable asym- metry in the proper motions, showing evidence that E0102 is now undergoing non-homologous expansion of optical ejecta (Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Proper motion measurements show non-ballisitic motion, with slower material preferentially in the east (Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021), suggesting that E0102 is likely interacting with an inho- mogenous surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' X-ray studies also show varying densities across the remnant, as well as a non-spherical forward shock (Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Xi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' While some level of explosion asymmetry may be present, E0102’s morphology is most likely dominated by effects from an inhomogenous surrounding environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 6 Work on Puppis A reported by Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (2010) remains unpublished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' G292, which is elongated along the north-south di- rection, shows very little evidence of an inhomogenous environment and its shape comes mostly from explosion asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Both proper motion studies and Doppler measurements show an asymmetric nature to the explo- sion (Ghavamian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The proper motion-derived CoE coinciding with the X-ray and radio centers of emission (Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2009) also support the notion of minimal CSM interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' De- spite interaction with an equatorial bar of CSM mate- rial, overall G292 appears to be expanding into a low density environment (Ghavamian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' However, CSM interactions cannot be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Recent simu- lations show that inhomogenous surrounding environ- ments are reflected in the forward and reverse shock for only ≈ 2000 yr (Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' There is also evidence that G292’s morphology is influenced by the motion of its surviving pulsar (Temim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Cas A shows an asymmetry in the distribution of its highest velocity oxygen knots, although not to the extent of N132D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This remnant shows a main shell of material moving at ≈4000 to 6000 km s−1 that is broadly symmetric in the plane of the sky, but there is also an extended component of sulfur-rich material to the northeast that extends to velocities upwards of ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 × 104 km s−1 (Hammell & Fesen 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Fesen & Milisavljevic 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A complementary high velocity out- flow also exists in the southwest (Fesen 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Knots of other chemical abundances of Cas A are more sym- metrical (see Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Hammell & Fesen 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Milisavljevic & Fesen 2013), and although there is a gap of O-rich material in the south of Cas A as seen in Fig- ure 9, sulfur-rich main shell ejecta at slower velocities is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Simulations and observations of clumpy, fil- amentary nebulosities have shown that Cas A likely in- teracted with an inhomogenous CSM environment (Weil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Overall, Cas A is a mixture of explosion asymmetry and an inhomogenous surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' CONCLUSION We present the first proper motion measurements of optical emitting ejecta of SNR N132D in the LMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The proper motions were measured using manual and automated procedures applied to two epochs of high resolution HST data taken 16 yr apart with the same ACS/F475W instrument+filter combination sensitive to [O III] λλ4959, 5007 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' With these proper mo- tion measurements, we have increased the number of young, O-rich SNRs with proper motion derived CoEs from three to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 12 Our measurement of the CoE made via visual in- spection converged on coordinates α=5h25m01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='71s δ=- 69◦38′41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′64 (J2000) with 1-σ uncertainty of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Our new CoE estimate is approximately 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′8 from previous estimates using geometric centers of emission (Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Combining this CoE estimate with the proper motion measurements leads to an age of 2770 ± 500 yr, consistent with recent age estimates of ≈ 2500 yr by 3D reconstructions of N132D (Vogt & Dopita 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Our new CoE and explosion age serves as a useful guide for searches to possibly locate the associated neu- tron star of the original core collapse explosion of N132D (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Holland-Ashford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Katsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' To date, no neutron star has been identified in N132D, and our CoE identifies a region where a targeted search can be performed with new 1 Msec Chandra observa- tions (PI: Plucinsky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Our CoE and age estimates of N132D can also effectively guide searches for a surviv- ing binary companion to the progenitor system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' To date, there have been no surviving stellar companions found for the population of nearby stripped-envelope SN rem- nants (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Kerzendorf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The nearby distance of N132D makes it possible to probe individual stars in the remnant’s stellar neighborhood and avoid distance uncertainties and source confusion encountered in studies at extragalactic distances (Fox et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' An attempt was recently made to identify the surviving companion of E0102 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021) using an updated CoE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' thus our new CoE for N132D makes the remnant an excellent opportunity for a similar analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank the anonymous referees and data editor for helping improving this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' acknowledges NSF support from grants PHY-1914448, PHY- 2209451, AST-2037297, and AST-2206532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' acknowledges funding from the National Science Foundation Grad- uate Research Fellowship under Grant DGE1745303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astron- omy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These observations are associated with HST programs 6052, 12001, 12858, and 13378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Support for program #13378 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', under NASA contract NAS 5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Plucinsky, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Hughes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', & Patnaude, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2019, ApJ, 874, 14, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3847/1538-4357/ab09ea 15 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' MORE DETAILED LOOK INTO THE COMPUTER VISION APPROACH A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Previous Automation Techniques In this paper, we used a new automated procedure for measuring the proper motions of supernova remnant ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Although manual inspection is a reliable method, using computer vision measurement techniques can allow for rapidly reproducible results, testing various quantitative thresholds, and scale to a large number of proper motions more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' One common technique is using a cross-correlation method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1996), that was been used for the proper motion measurements of SNRs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Winkler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2009), stellar ejecta (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Kiminki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2016), and protostellar jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Hartigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Bally et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This uses predefined box regions to outline a ‘clump’ in one epoch that is then translated to a new position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The translated image is then subtracted by the second epoch and the translation that produces the minimum sum of the differences is the translation that is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Other computer vision techniques for measuring proper motions in SNRs include measuring the optical flow, projection methods, and maximum likelihood functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=', Borkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The success of these procedures depends on the knots being bright and are best suited for large regions of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' However, with the high spatial resolution of HST, we are able to resolve individual knots and cover more of the periphery of the SNR where faint, high proper motion ejecta knots are expected to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Still, these knots can change in illumination and shape between epochs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Fesen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Patnaude & Fesen 2014), which can lead to errors in tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' One of the goals of our new procedure is to be able to track fainter, individual knots for an older O-rich remnant such as N132D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Automated Proper Motion Measurement Procedure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Image Preparation The first step in image preparation was to ensure that emission unrelated to N132D’s oxygen-rich material was corrected for and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Doing so ensured that our computer vision procedure only tracked [O III] λλ4959, 5007 emission and was not confused by CSM/ISM or stellar emission that was easily recognized in and avoided by the manually inspected proper motion measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We first scaled the F550M (continuum) and F658N (Hα) images taken in 2004 to the ACS/F475W images by matching the flux of common stars and then subtracting the F550M and F658N emission from the ACS/F475W image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This is to remove the stellar continuum and Hβ emission (using Hα as a tracer) from our images in order to only track the oxygen-rich ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We then manually removed any subtraction residuals outside of emission regions, and performed additional cleaning of the images using cosmicray lacosmic task from the astropy package ccdproc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Identifying Knots Next, small regions were identified for further individual processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We chose regions that had strong oxygen emission or showed evidence of proper motion measurements when blinking between epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We divide these regions into 2′′ by 2′′ boxes, or stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A1 shows all of the stamps while the first row of Figure A2 shows an enlarged example of a stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The stamp’s size was chosen for its ability to encapsulate knot motion between epochs and to sample the local background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This is important since N132D’s non-uniform emission properties makes using a global background value impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We allowed large overlap between the stamps to ensure that no knots were missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We then used the detect sources and deblend sources tasks from the photutils package in astropy to identify knots, shown in the second and third row of Figure A2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Detect sources creates a segmented image which identifies sources of emission above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' In our case, this was 2σ above the median background of the 2′′ by 2′′ stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We further restricted our analysis to those knots containing emission that spanned at least four adjacent pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This pixellimit was selected to account for smaller, fainter knots than those found through visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Deblend sources takes the segmented regions and finds the local maxima to separate potential overlapping knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' For this process, we assumed a Gaussian kernel and a low contrast between the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We then apply three initial filters to remove any embedded residuals or hot pixels that were missed with cosmicray lacosmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The first filter removes segments too close to the edges of the region, effectively reducing the region from 2′′ by 2′′ to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′9 by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The 16 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2004 ACS/F475W continuum- and hydrogen-subtracted image with vectors representing the automated procedure’s measured shifts (multiplied by a factor of 20 for visual clarity) shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The stamps used in the automated procedure are shown as blue boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Empty boxes represent areas where the automated procedure could not detect any motion and/or identify knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' next filter removes segments that do not move more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 mas yr−1 , to remove any remaining residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This lower limit was chosen based on the manually inspected lowest proper motions around 3 mas yr−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The last filter removes any segment that does not have a corresponding segment within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′3 around it in the other epoch, to remove residuals or hot pixels that appear in one epoch but not the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Matching Knots and Calculating Shifts The changing brightness and shape of individual knots over time, in addition to any residual hot pixels, can affect the local peak position and the geometric center determined for each knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' To mitigate these effects, the local peaks in the identified knots are calculated using a KDE fit of the stamp (fourth row of Figure A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' To achieve sub-pixel accuracy, we found the best results by resampling the stamp from 40x40 pixels to 100x100 before applying the KDE, resulting in an error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 pixels (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2) for the calculated centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We also found the best bandwidth for the KDE was the the Silverman kernel (Silverman 1982) through manual inspection of the resulting KDE centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The next step is to match the knots between the epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Each pair of centroids is found using an initial guess of the center of expansion to find the position angle between the knot and the guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We use the CoE calculated by the visually tracked knots for the initial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Each individual pair of knots between epochs is given a score, which is the shift between epochs multiplied by the difference between their trajectory and the position angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The pair with the lowest score is selected as the correct combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The combinations are then passed through two filters to 8422 中 V 5 1 pc17 Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Steps needed for the computer vision technique to identify and measure the proper motion of ejecta knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The 2004 and 2020 images of the same region as Figure 2 are shown on the left and right panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The top row shows the original cutout of the continuum and hydrogen subtracted image in gray scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The second and third rows show the results of the detect sources and deblend sources tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The fourth row shows the remaining segments after applying the KDE to the total region, identifying the localized peaks as shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The final row shows the matched centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The centers are shown in red (2004) and blue (2020), with a white vector showing the motion between the two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' mitigate outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Knot combinations are discarded if the difference between the trajectory and the position angle is greater than 90 degrees and if the shift is larger than 30 mas yr−1 , double the largest proper motion found using visual measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The effect of changing these parameters are explored in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' If multiple combinations use the same endpoint (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' the deblending process splits a knot into two between epochs), the combination with the lowest score is chosen and the other is discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The final row of Figure A2 shows an example final combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Finally, we clip the top and bottom 5% of the proper motion vectors to remove outliers, average duplicate mea- surements due to overlapping stamps, and remove measurements that are within 3 pixels of the mask for subtraction residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The remaining measurements are used for the proper motion analysis as was done for the manual inspection in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We ensured that the procedure is robust using a toy model simulation, detailed in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Cleaned Stamp Detect Sources Deblend Sources Filtered KDE Mask 2004 202018 Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Similar to Figure 8 with the addition of the automated procedure’s result of α=5h25m02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='771s and δ=-69◦38′38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′985 (J2000) with 1-σ uncertainty of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′47 in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Simulated Proper Motion We tested our procedure using a simulated, idealized SNR with ejecta moving ballisitically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We generated two 300 by 300 pixel images at a scale similar to HST with randomized background emission using the make noise image task in astropy’s photutils package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The first was an image of 45 knots randomly placed to concentrically surround a test CoE located at the center of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The knots were drawn from a sample of the visually inspected HST knots, the majority of which were in the top third of knot brightness, and were placed using the geometric center of each knot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The second image was then created using a different randomized background and placing these knots a certain distance, radially away from the simulated CoE with a proper motion with v ∝ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We implemented a scanning procedure to find regions containing the knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This procedure scanned the image using subdivided regions to 40x40 pixel boxes with overlap between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='7 Our automated procedure recovers all 45 knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' There was an average positional difference of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='12 pixels (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′1 with HST resolution), an average shift difference of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='53 pixels (≈3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 mas yr−1 ), and an average angle difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='12 radians between the inferred and true (simulated) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We calculated a CoE of (154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='77,155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='97) with a 1σ error radius of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='88 pixels (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′58) as compared to the simulated CoE of (150,150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Overall, we were able to find all of the 7 We did not pass regions through the filters for hot pixels and star removal, as neither of these features were present in the simulated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' WFC3/F502N 2020 Diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Rim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Center Computer Vision Visual Inspection TpC E19 knots, match them correctly, and calculate their speed and trajectory to return a CoE that is within 1-σ of the true value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Automated Procedure Results This procedure was able to identify and track 137 knots of ejecta with the error of the proper motions was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4 pixels (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′2), from the sub-pixel ratio of the KDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figures A4–A7 show the knot locations, trajectory, and proper motion trends using this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The procedure measured proper motions ranging from 2 to 17 mas yr−1 and an S of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′015 per km s−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Using the same CoE method as outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1, the 137 proper motions yields a CoE of α=5h25m02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='771s and δ=-69◦38′38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′985 (J2000) with 1-σ uncertainty of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='′′47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A3 in the Appendix shows this result as compared to the other center of expansion estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Utilizing this CoE and proper motions, we calculate an age of 3377 ± 2241 yr using all 137 knots, as shown in the left panel of Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This large discrepancy is most likely due to the procedure measuring the proper motions of artifacts or heavily decelerated knots, skewing the age estimates to higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' To account for the decelerated knots, we also calculated the age using knots above the median proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These 70 knots yield an age of 2497 ± 638 yr (right panel of Figure A5), much closer to that derived from visually measured proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Effect of Tuning Parameters Figures A4, A3, and A5 show the results of using the conservative metrics to measure the proper motions, their trajectories, and subsequent CoE and explosion ages, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' These results are highlighted in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The conservative parameter constraints used in the selection of knots adopted in our automated procedure were used to incorporate many degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' However, the associated proper motion measurement uncertainties were much larger than those associated with our manual procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' There are many ways that the parameters of knot selection can be further constrained to reduce the uncertainties and better match the manually measured CoE and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We explored limiting the difference angle between the trajectory and position angle of the input CoE to 45 degrees (from 90), increasing the minimum proper motion to 3 mas yr−1 (from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 mas yr−1 ), and using brighter knots by increasing the signal to noise of selected knots from 2σ to 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We found that by tightening the boundaries of these parameters, the CoE of the automated procedure and the age calculation were both within 1-σ of the visual inspection results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The following section contains a detailed discussion about the effect of each of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This procedure is ideal for ejecta proper motion analysis of other SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Two parameters that must be changed between SNRs are the KDE bandwidth and the arbitrary CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The KDE bandwidth is very sensitive and needs to be fine-tuned depending on the SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A bandwidth that is too small will identify many flux peaks within a knot while a bandwidth that is too large can miss fainter knots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Another parameter that we explored was the influence of the choice of the arbitrary CoE for the trajectory versus position angle cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' For our procedure outlined in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3, we chose the CoE calculated using the visual inspection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Assuming this CoE was not available, we could have chosen the [O III] geometric center for our arbitrary CoE (Morse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' We experimented with the 45 and 90 degree cutoff with this arbitrary CoE and found that the resulting CoE did not match that found with the visually inspected CoE using an arbitrary initial CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' However, we found that by iterating the arbitrary initial CoE calculation procedure, the CoE calculation converges to the same CoE as if using the visually inspected CoE for the arbitrary CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Running through 10 initial guess estimates of the CoE, we found that using a 45 degree cutoff would take 3–4 iterations, whereas the 90 degree cutoff would take 2 iterations before converging on the the same CoE as found by using the visual inspection CoE as the arbitrary CoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' As such, we would recommend anyone using our procedure to use the iteration method to verify results, especially if using a geometric center for the arbitrary CoE, as they are often offset from proper motion derived centers (see Thorstensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Katsuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Banovetz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Further Fine-tuning of Parameters Here we discuss how measurement and calculation of the CoE and age are affected by knot selection parameters in our automated procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A4 shows the results of using the conservative parameters for the automated procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A5 show the results of the age calculation when using all 137 knots, or only the fastest (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figures A6 and A7 show the results of changing the parameters of the automated procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The left, middle left, middle right, and right panels show the effect of changing the parameters to incorporate a higher minimum speed 20 (3 mas yr−1 ), only bright knots (3σ), difference of trajectory and position angle of 45 degrees, and all three of the changes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Similar to Figure 3 (left) and 5 (middle and right) but using the proper motions from the conservative parameters of the automated procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Similar to Figure 6 but using the proper motions from the automated procedure and including the results of only using the fastest ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' This results in an age of 3377 ± 2241 yr using all the knots and 2497 ± 638 yr using the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' The red shaded points correspond to the matching visually measured knots when applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' 30 30 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5 (mas/yr) 20 20 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 12 Knot ( Location (") 10 10 Location 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 Proper Motion of i 0 10 10 2 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0E 30 20 30 40 10 10 50 20 0 X location (") X location (") Distance away from Center of Expansion (")140 120 100 80 D Knot I 60 40 20 0 1000 2000 3000 4000 5000 6000 Explosion Age (years)140 120 100 80 D Knot I 60 40 20 Fast Eiecta 0 0 1000 2000 3000 4000 5000 6000 Explosion Age (years)21 Figure A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Similar to Figure A4 for the 3 mas yr−1 lower limit (left), increase to 3-σ of brightness (middle left), and reducing the trajectory and position angle difference of the knots to less than 45 degrees (middle right), and applying all three variations (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Figure A7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Similar to Figure 6 for the 3 mas yr−1 lower limit (top left), increase to 3-σ of brightness (top right),the trajectory and position angle difference of the knots to less than 45 degrees (bottom left), and using all the parameters (bottom right) that match closest to the visual inspection result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Proper Motion of Knot (mas/yr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Distance away from Center of Expansion (")18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Proper Motion of Knot (mas/yr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Distance away from Center of Expansion (")18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Proper Motion of Knot (mas/yr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Distance away from Center of Expansion (")18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Proper Motion of Knot (mas/yr) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Distance away from Center of Expansion (")20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Location (") ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='X location ("30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Y Location ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='OT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='X location ()30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Location (") ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='X location ()30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Y Location (") ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='OT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='X location ()140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Knot I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Explosion Age (years)140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Knot ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Explosion Age (years)140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Knot ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Fast Eiecta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Explosion Age (years)140 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Knot ID ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Fast Ejecta ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='Explosion Age (years)22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' ADDITIONAL TABLES Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Anchor Star Coordinates Star RA (J2000) Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' (J2000) 1 5h24m53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='7308s 69d38m15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='310s 2 5h24m58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='3686s 69d38m35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='298s 3 5h24m57.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='7550s 69d38m06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='447s 30 5h24m52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='6476s 69d38m01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='440s Table A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Manually Inspected Knot Measurements and Corresponding Automated Measurements Knot RA Dec Visual Procedure Automated Procedure µα σµα µδ σµδ µα µδ (J2000) (J2000) (mas yr−1) (mas yr−1) (mas yr−1) (mas yr−1) (mas yr−1) (mas yr−1) 1 5h25m06.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='293s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='301 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='693 Table A2 continued 26 Table A2 (continued) Knot RA Dec Visual Procedure Automated Procedure µα σµα µδ σµδ µα µδ (J2000) (J2000) (mas yr−1) (mas yr−1) (mas yr−1) (mas yr−1) (mas yr−1) (mas yr−1) 120 5h25m05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='432s 69d38m08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='588s 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='583 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='334 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content='528 NA NA Note—Positive values indicate direction to the north and east for RA and Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} +page_content=' Results of the automated procedure are matched when applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtA0T4oBgHgl3EQfMP8I/content/2301.02128v1.pdf'} diff --git a/yNE2T4oBgHgl3EQf3wh7/content/tmp_files/2301.04174v1.pdf.txt b/yNE2T4oBgHgl3EQf3wh7/content/tmp_files/2301.04174v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0312693973f777a7dbaec12ae15b45296395cdcc --- /dev/null +++ b/yNE2T4oBgHgl3EQf3wh7/content/tmp_files/2301.04174v1.pdf.txt @@ -0,0 +1,2115 @@ +To turn off the edits, from https://journals.aas.org/revision-history/ +Draft version January 12, 2023 +Typeset using LATEX twocolumn style in AASTeX62 +The Similar Seven: A set of very-alike exoplanets to test correlations between system parameters and atmospheric +properties +Chima D. McGruder,1, ∗ Mercedes López-Morales,1 Rafael Brahm,2, 3, 4 and Andrés Jordán2, 3, 4 +1Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA +2Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal las Torres 2640, Peñalolén, Santiago, Chile +3Millennium Institute for Astrophysics, Santiago, Chile +4Data Observatory Foundation +ABSTRACT +Studies of exoplanetary atmospheres have found no definite correlations between observed high alti- +tude aerosols and other system parameters. This could be, in part, because of the lack of homogeneous +exoplanet samples for which specific parameters can be isolated and inspected. Here we present a set +of seven exoplanets with very similar system parameters. We analyze existing photometric timeseries, +Gaia parallax, and high-resolution spectroscopic data to produce a new set of homogeneous stellar, +planetary, and orbital parameters for these systems. With this we confirm that most measured param- +eters for all systems are very similar, except for the host stars’ metallicities and possibly high energy +irradiation levels, which require UV and X-ray observations to constrain. From the sample, WASP-6b, +WASP-96b and WASP-110b, have observed transmission spectra that we use to estimate their aerosol +coverage levels using the Na I doublet 5892.9Å. We find a tentative correlation between the metallicity +of the host stars and the planetary aerosol levels. The trend we find with stellar metallicity can be +tested by observing transmission spectra of the remaining planets in the sample. Based on our predic- +tion, WASP-25b and WASP-55b should have higher levels of aerosols than WASP-124b and HATS-29b. +Finally, we highlight how targeted surveys of alike planets like the ones presented here might prove +key for identifying driving factors for atmospheric properties of exoplanets in the future and could be +used as a sample selection criterium for future observations with e.g. JWST, ARIEL, and the next +generation of ground-based telescopes. +Keywords: planets and satellites: atmospheres — stars: activity; starspots — techniques: spectro- +scopic; WASP-6b; WASP-25b; WASP-55b; WASP-95b; WASP-110b; WASP-124b, HATS- +29b +1. INTRODUCTION +The upper atmospheres of about 100 exoplanets have +been probed via transmission spectroscopy to date with +HST, Spitzer, ground-based telescopes (NASA Exo- +planet Archive 2019) 1, and now JWST (JWST Transit- +ing Exoplanet Community ERS team et al. 2022). Most +observations suggest that atmospheric features are heav- +ily muted by aerosols (e.g. Wakeford et al. 2019), with +only a few planets showing little to no aerosol coverage +Corresponding author: Chima D. McGruder +chima.mcgruder@cfa.harvard.edu +∗ NSF Graduate Research Fellow +1 Accessed on 2022-05-29 +(e.g. Kirk et al. 2019; Alam et al. 2021; Ahrer et al. 2022; +McGruder et al. 2022). +The formation of aerosols in planetary atmospheres +occurs via complex chemical and physical processes, +which are not yet fully understood, but work is in +progress (see e.g. Helling 2008; Marley et al. 2013; +Helling 2019; Gao et al. 2021). +For example, predic- +tions link cloud formation in exoplanets to the chemi- +cal composition of their atmospheres, where seed par- +ticles need to be lofted to high altitudes so gases can +condense on them and form clouds (e.g. Helling 2008, +and references therein). The composition, availability, +and altitude of potential seed particles has many de- +pendencies, including the composition of the protoplan- +etary disk (e.g. Mordasini et al. 2010), and atmospheric +arXiv:2301.04174v1 [astro-ph.EP] 10 Jan 2023 + +2 +McGruder et al. +differentiation, where heavier elements are expected to +sink into the lower layers of the atmosphere (e.g. Helling +2019). In the case of hazes, model predictions show that +high altitude hazes can form in exoplanets via UV-driven +photolysis (e.g. Moses et al. 2011, 2013), and laboratory +experiments have shown that UV radiation can form +photochemical hazes in H2 dominated atmospheres at +temperatures between 600K–1500K (Fleury et al. 2019). +Understanding what system parameters drive the +presence or absence of aerosols in exoplanetary atmo- +spheres will be key to understanding their formation +and evolution mechanisms. However, comparative stud- +ies of exoplanet atmospheres so far have yielded either +no correlations between the atmospheric properties of +the planets studied and other system parameters (see +e.g. Sing et al. 2016), or only tentative correlations +between planetary equilibrium temperature and atmo- +spheric aerosol levels (Heng 2016; Stevenson 2016; Fu +et al. 2017; Tsiaras et al. 2018; Dymont et al. 2021). +Yet, degeneracies between system parameters and dis- +agreeing observations leave any correlation uncertain +(Alam et al. 2020). +Possible reasons for why correlations between ob- +served exoplanet atmospheric properties and other sys- +tem parameters have not yet been found are the fact that +the parameter space being considered is too broad, with +multiple independent parameters typically being fitted +simultaneously (e.g. planetary equilibrium temperature, +density, log surface gravity, planetary/host star metal- +licity, etc), and over wide range of values; as well as the +parameter space being scarcely sampled, with typically +one-to-no pairs of similar planets being examined. +We present a way to potentially alleviate this problem +by identifying groups of planets with similar properties, +so that a reduced number of parameters can be isolated +and compared against observed atmospheric features of +the planets in detail. In particular, we have identified a +group of seven gas giants with very similar parameters, +except for the metallicity of their host stars and possibly +their high energy irradiation levels. We use this group +of planets to test whether aerosol properties are related +to those parameters. +Section 2 describes the identification of the sample. +Sections 3 and 4 present our reanalysis of system param- +eters and observed transmission spectra for comparative +purposes. Section 5 presents the analysis of correlations +between exoplanet aerosol level proxies and the host star +metallicity and high energy irradiation levels. Finally, +Section 6 summarizes our results. +2. THE SAMPLE +Using the NASA Exoplanet Archive(NASA Exoplanet +Archive 2021)2, we compiled system parameters for all +known exoplanets with an observed optical or near- +infrared transmission spectrum. We compiled the plan- +ets mass, radius, orbital period and semi-major axis sep- +aration, as well as the mass, radius, effective tempera- +ture, and metallicity of the host stars. With those values +in hand, we computed the planets gravity, density, equi- +librium temperature, and their stellar insolation levels +in earth units. We then searched in the compiled list for +planets with very similar system parameters and com- +pared their observed transmission spectra. +This is how we identified WASP-6b and WASP-96b, +two hot jupiters with very similar masses, radii, Teq, stel- +lar parameters, and insolation levels (see top six panels +of Figure 1 and Table 1), but strikingly different trans- +mission spectra (See bottom three panels of Figure 1 +and Section 4). From the parameters available for each +system, they only appear to differ significantly in the +host star metallicity, reported in the NExSci database as +[Fe/H] = -0.20±0.09 for WASP-6b (Gillon et al. 2009), +and [Fe/H] = 0.14±0.19 for WASP-96b (Hellier et al. +2014). +Next we compared the parameters of all known ex- +oplanets, including those without atmospheric observa- +tions, to those of WASP-6b and WASP-96b, assuming +the planets to be similar if all their planetary parameters +listed above agreed within about 1σ. Using this strategy +we found another five planets similar to WASP-6b and +WASP-96b: WASP-25b (Enoch et al. 2011), WASP-55b +(Hellier et al. 2012), WASP-110b (Anderson et al. 2014), +WASP-124b (Maxted et al. 2016), and HATS-29b (Es- +pinoza et al. 2016). This is how we arrived to the similar +seven planets sample described in the remaining of the +paper. The parameters of each system, re-derived ho- +mogeneously as described in the following section, are +summarized in Table 1. +3. DERIVATION OF HOMOGENEOUS STAR AND +PLANET PARAMETERS +The parameters used in Section 2 to identify the +sample were obtained from various literature sources. +Therefore, to minimize potential biases and systematics +between separate analyses, we re-derived the parameters +of each system homogeneously. Table 1 provides all the +newly derived parameters, with the derivation processes +of each set of parameters described below. +3.1. Stellar Parameters +2 Accessed on 2021-11-17 + +Twin Planets +3 +Table 1. Stellar and planetary parameters of all Similar Seven Systems +Param. +WASP-6 +WASP-25 +WASP-55 +WASP-96 +WASP-110 +WASP-124 +HATS-29 +M∗ +0.854+0.027 +−0.023 +0.962±0.027 +±0.021 +1.071±0.025 +1.03+0.031 +−0.036 +0.814+0.014 +−0.022 +1.157+0.016 +−0.015 +1.055+0.036 +−0.038 +R∗ +0.79+0.008 +−0.009 +0.884+0.008 +−0.009 +1.09+0.012 +−0.013 +1.055+0.018 +−0.017 +0.853±0.011 +1.074+0.013 +−0.015 +1.066+0.018 +−0.019 +Teff +5438±50 +5697±80 +6096±71 +5678±80 +5392±50 +6258±100 +5769±80 +log10(G∗) +4.565+0.022 +−0.016 +4.529±0.018 +±0.014 +4.393±0.015 +4.404+0.023 +−0.026 +4.487+0.013 +−0.018 +4.439+0.01 +−0.009 +4.406±0.024 +ρ∗ +1.73+0.08 +−0.07 +1.391+0.058 +−0.048 +0.822+0.036 +−0.035 +0.876+0.05 +−0.054 +1.31+0.055 +−0.062 +0.933+0.041 +−0.036 +0.869+0.055 +−0.054 +[Fe/H] +-0.15±0.05 +-0.2±0.05 +-0.03±0.04 +0.24±0.05 +-0.15±0.05 +0.16±0.05 +0.22±0.05 +v sin I +1.5±0.3 +2.5±0.3 +3.28±0.21 +3.2±0.3 +0.5±0.3 +5.9±0.3 +2.01±0.3 +Age +3.2+2.1 +−3.1 +1.9+1.3 +−1.8 +3.05+0.92 +−0.99 +5.2±1.9 +11.0+2.4 +−1.6 +0.4+0.2 +−0.4 +4.2+1.8 +−1.9 +MagNUV +18.21±0.04 +17.63±0.03 +17.02±0.03 +18.69±0.05 +17.03±0.02 +17.53±0.02 +18.56±0.05 +FNUV +844+45 +−42 +1306+55 +−52 +3142+148 +−141 +1400+95 +−94 +4147+195 +−190 +7165+339 +−316 +1016+76 +−67 +log10(R’hk) +-4.476±0.091 +-4.507±0.119 +-4.844±0.146 +-4.781±0.028 +-4.674±0.089 +-4.765±0.056 +-4.455±0.154 +Prot1 +26.37+7.17 +−4.89 +17.16+3.06 +−2.62 +16.75+1.41 +−4.79 +16.62+2.1 +−5.12 +85.98+132.58 +−45.19 +9.17+0.65 +−2.55 +26.73+5.34 +−9.16 +Prot2 +29.69+5.00 +−5.78 +16.20+3.06 +−0.49 +19.63+3.51 +−5.39 +25.69+3.15 +−8.56 +134.42+46.32 +−58.26 +13.51+5.77 +−7.40 +27.07+3.96 +−6.85 +Prot3 +9.51±2.74 +11.65±4.17 +25.25±11.28 +24.3±5.12 +22.8±6.91 +24.08±5.85 +8.29±3.8 +Prot +28.28+3.99 +−3.97 +16.93+2.02 +−1.55 +17.45+2.4 +−2.85 +20.16+2.88 +−4.22 +120.1+60.46 +−37.54 +10.65+3.27 +−3.01 +25.7+4.16 +−4.71 +Mp +0.467+0.024 +−0.023 +0.564+0.026 +−0.025 +0.586+0.033 +−0.032 +0.47+0.034 +−0.036 +0.487+0.052 +−0.054 +0.577+0.057 +−0.056 +0.65+0.06 +−0.062 +Rp +1.119+0.017 +−0.018 +1.232+0.014 +−0.017 +1.314+0.021 +−0.025 +1.23+0.027 +−0.025 +1.177+0.028 +−0.024 +1.337+0.028 +−0.033 +1.201+0.033 +−0.028 +Teq +1167±96 +1217±101 +1342+111 +−110 +1350±112 +1158±95 +1481±123 +1230±102 +Gp +9.65+0.59 +−0.56 +9.62+0.52 +−0.48 +8.8+0.59 +−0.56 +8.04+0.67 +−0.71 +9.1+1.0 +−1.1 +8.36+0.92 +−0.88 +11.7+1.2 +−1.3 +ρp +0.332+0.023 +−0.022 +0.301+0.019 +−0.017 +0.258+0.02 +−0.019 +0.252+0.024 +−0.025 +0.299+0.037 +−0.039 +0.241+0.03 +−0.028 +0.375+0.044 +−0.047 +Ip +288+47 +−45 +341±68 +504+96 +−93 +517+127 +−128 +279±51 +748+178 +−174 +356+93 +−87 +Porb +3.3610026 +3.7648337 +4.465631 +3.4252567 +3.7784022 +3.3726511 +4.6058827 ++6.1e−7−6.3e−7 +±1.2e−6 +±1.3e−6 +±1.2e−6 +±1.6e−6 ++2.7e−6−2.9e−6 +±1.1e−6 +a/R∗ +11.21+0.13 +−0.14 +11.33±0.14 +10.66+0.14 +−0.15 +9.13±0.17 +11.2+0.17 +−0.18 +9.22±0.13 +11.36+0.2 +−0.26 +t0 +2454596.43260 +2455274.99649 +2455737.93919 +2456258.06272 +2456502.72415 +2457028.58329 +2457031.95666 ++0.000762−0.00075 ++0.00100−0.00103 ++0.00097−0.00095 ++0.00084−0.00088 +±0.00102 ++0.00103−0.00101 ++0.00025−0.00026 +b +0.195+0.077 +−0.114 +0.357+0.035 +−0.042 +0.235+0.067 +−0.105 +0.724+0.019 +−0.02 +0.319+0.059 +−0.072 +0.619+0.027 +−0.033 +0.395+0.065 +−0.055 +i +89.0+0.59 +−0.41 +88.19+0.23 +−0.2 +88.74+0.57 +−0.38 +85.45±0.2 +88.37+0.38 +−0.33 +86.15+0.24 +−0.21 +88.01+0.3 +−0.38 +K +69.6+2.6 +−2.7 +75.3+2.5 +−2.6 +68.7±3.2 +62.1+3.8 +−3.9 +73.1+7.6 +−7.7 +70.8±6.7 +78.3+6.3 +−6.6 +Note—The top 15 parameters, delineated by a thicker line, are for the host stars. In order, they are mass [M⊙], radius [R⊙], +effective temperature [K], log10 of surface gravity [cgs], density [ρ⊙], log10 of iron to hydrogen abundance relative to the +sun [dex], radial velocity [km/s], age [Gy], NUV magnitude, NUV flux from the star at the semi-major axis of the planet +[Watts/m2], log10 of the Calcium H and K indices [dex], rotational period derive from v sin I (Prot1), photometry (Prot2), R’hk +(Prot3), and the adopted rotational period (a weighted mean of the v sin I and photometry periods, Prot) [days]. The following +12 parameters are for the corresponding planet and are mass [Mj], radius [Rj], equilibrium temperature [K], surface gravity +[m/s2], density [ρj], insolation [I⊕], and orbital period [days], semi-major axes relative to the host stars’ radii, reference mid +transit time transit [BJD], impact parameters, orbital inclinations [degrees], and RV semi-amplitude [m s−1], respectively. +We re-derived the stellar parameters of each host star +using archival HARPS (Mayor et al. 2003) and FEROS +(Stahl et al. 1999) spectra3. Each spectra was homoge- +neously reduced using CERES (Brahm et al. 2017a), and +the stellar parameters (Teff, log g, [Fe/H], v sin I) were +3 archive.eso.org +derived using ZASPE (Brahm et al. 2017b), as described +in detail in Brahm et al. (2019, 2020). +Once ZASPE +had obtained the stellar atmospheric parameters, the re- +maining physical parameters were computed comparing +synthetic values generated using PARSEC isochrones +(Bressan et al. 2012), and Gaia DR2 (Gaia Collabo- +ration et al. 2018) parallaxes. +For this step we fixed +the stellar metallicity to the values obtained by ZASPE, + +4 +McGruder et al. +Figure 1. Top: Mass versus equilibrium temperature, gravity, radius, density, stellar irradiation, and star’s metallicity for +WASP-6b (blue diamonds), WASP-96b (blue stars), WASP-110b (cyan circles), WASP-124b (red ’X’), WASP-55b (red squares), +WASP-25b (red triangle), HATS-29b (red pentagon), the PanCET planets (green crosses), and all exoplanets with measured +parameters (grey circles). +The Equilibrium Temperature, Insolation, and metallicity of WASP-110b and WASP-6b overlap +(see Table 1), making it hard to see the cyan circles of WASP-110b underneath the blue diamonds of WASP-6b. Bottom: +Transmission spectrum of WASP-6b (red/magenta, Nikolov et al. 2015; Carter et al. 2020), WASP-96b (blue, Nikolov et al. +2018; Yip et al. 2021; Nikolov et al. 2022; McGruder et al. 2022), and WASP-110b (green circles, Nikolov et al. 2021), with the +Platon best-fit retrieval model (black line) and 1-σ confidence interval (cyan shaded regions) overplotted. In this figure, the +parameters for the similar seven planets are the re-derived parameters discussed in Section 3. + +('w) +1.5 +Mass +1.0 +0.5 +500 +1000 +1500 +2000 +2500 +Equilibrium Temp (K) +(M) +1.5 +Mass +1.0 +0.5 +5 +10 +15 +20 +25 +30 +Planet Gravity (m/s2) +(M) +1.5 +Mass +1.0 +0.5 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Radius (R;) +(w) +1.5 +Mass +1.0 +0.5 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +Planet Density (pi) +(w) +1.5 +Mass +1.0 +0.5 +1000 +2000 +3000 +4000 +5000 +Insolation (Earth Units) +(M) +1.5 +Mass +1.0 +0.5 +-0.4 +-0.2 +0.0 +0.2 +0.4 +stellar metallicity ([Fe/H])0.150 +WASP-6: 「Fe/H|=-0.15±0.05 +FORS2 +STIS/430 +0.148 +STIS/750 +WFC3/G141 +R +0.146 +R +0.144 +0.142 +0.122 +FORS2+IMACS +WASP-96:[Fe/H|=+0.24±0.05 +HST/G102 +0.120 +HST/G141 +0.118 +R +0.116 +0.114 +WASP-110:[Fe/H|=-0.15±0.05 +FORS2 +0.144 +0.142 +R +0.140 +R +0.138 +0.136 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +Wavelength (μm)Twin Planets +5 +and used Teff from ZASPE as a prior to obtain posterior +distributions for the stellar age, mass, and interstellar +extinction using emcee (Foreman-Mackey et al. 2013). +From these values we also derived stellar radii and log g, +which was then fed back into ZASPE as a fixed param- +eter. +The process described above was iterated until +reaching convergence in log g. For WASP-55, which has +both HARPS and FEROS observations, we weighted av- +eraged the values from each instrument. +3.1.1. R′ +HK Activity Index +We derived R′ +HK values for each host star from the +Ca II H&K lines in their HARPS and FEROS spectra, +following the methods in Noyes et al. (1984), and cali- +brated them to the standard Mount Wilson scale follow- +ing Lovis et al. (2011). To calibrate the R′ +HK indexes +from HARPS and FEROS to the Mount Wilson scale +we used six of the seven (all but HD 219834) reference +stars used by Lovis et al. (2011) that have both HARPS +and FEROS spectra. For HARPS we found a conversion +of the form SMW = 1.118 · SHARP S + 0.0135, with a +0.0035 fit dispersion. For FEROS we found the conver- +sion SMW = 1.2121 · SF EROS + 0.0072, with a 0.0275 +fit dispersion. +3.1.2. Rotational Period +We estimated the rotation period, Prot, of each star +using three proxies: 1) v sin I from section 3.1 using eq. +Prot = 2πR∗/v sin (i − λ), +(1) +where R∗ is the stellar radius, i is the inclination of the +planet’s orbital axis, and λ is the rotation axis align- +ments – λ has been measured for WASP-6 and WASP- +25 (Gillon et al. 2009; Tregloan-Reed et al. 2015; Brown +et al. 2012), and we estimated it for the remaining +stars using probability distributions (see Appendix A), +2) TESS (Ricker et al. 2014) and ASAS-SN (Shappee +et al. 2014; Kochanek et al. 2017) light curves (see Ap- +pendix B), and 3) the R′ +HK indexes derived above, com- +bined with Table 3 and eq. 6 of Suárez Mascareño et al. +(2016). +The values of Prot obtained via each method are sum- +marized in Table 1. The values obtained with the first +two methods are consistent with each other, while the +values obtained using R′ +HK are not fully consistent. Be- +cause the periods obtained from the first two methods +are more direct measurements, we used the weighted +mean from these two methods as our adopted values, +reported in Table 1. +3.1.3. Near-UV Flux +We derived Near-Ultraviolet (NUV) fluxes for each +host +star +using +their +Galaxy +Evolution +Explorer +(GALEX; Bianchi & GALEX Team 1999) observations. +We obtained GALEX NUV magnitudes from VizieR, +which can also be accessed via the Mikulski Archive +for Space Telescopes (MAST) 10.17909/T9H59D. The +magnitude was converted to total NUV luminosity using +eq. 6 of Schneider & Shkolnik (2018), and Gaia D3 par- +allaxes, assuming a central mean wavelength of 2267Å. +We then converted those magnitudes to flux density as +FNUV [ +erg +s cm2 Å] = 2.06 ∗ 10−16 ∗ 10 +20.08−mNUV +2.5 +, +(2) +where mNUV is the observed GALEX AB NUV magni- +tude. This equation was derived from gsfc.nasa.gov. +3.2. Planetary parameters +We re-derived the parameters of each planet using +the SPOC4 TESS light curves downloaded from MAST +10.17909/fwdt-2x66 and described in Appendix B, and +radial velocity observations from CORALIE (Queloz +et al. 2000), HARPS (Mayor et al. 2003), and CY- +CLOPS (Horton et al. 2012). We also used HAT-South +(Bakos et al. 2013) data with TESS to fit the transit of +HATS-29b because of contamination from a background +RR Lyrae star in the TESS data. After removing that +contamination (see Appendix C), the TESS and HAT- +South transit light curves were modeled using the same +procedure as the other six targets. A table summarizing +the RV and photometric data is provided in Appendix +B. +We jointly fitted the light curves and RVs of each tar- +get with Juliet (Espinoza et al. 2018), initially assum- +ing circular orbits and quadratic limb darkening coeffi- +cients with uniform priors from 0 to 1. All other orbital +parameters (P, t0, a/R∗, Rp/R∗, b, i, K, and ρp) were fit- +ted with Gaussian priors set off of the discovery papers’ +mean and uncertainty values. Then we phase folded the +photometric light curves based on the period of that fit, +and removed points that were 3σ deviant from a moving +average of 20 points. This resulted in a few percent of +data-points being removed per sector. The clipped data +was then used for the final Juliet joint RV-transit fit. +The equilibrium temperature of each planet, Teq, was +computed using eq. +1 from López-Morales & Seager +(2007) and assuming each planet had the same atmo- +spheric albedo and energy redistribution factors of AB += 0.2±0.1 and f = 1/3±0.15. To approximate the inso- +4 Science Processing Operations Center (Jenkins et al. 2016) +5 AB is based on geometric albedos measured for other hot +jupiters (e.g. Mallonn et al. 2019; Adams et al. 2022; Blažek et al. +2022), and f is based on the expectation that gas giants with lower +incident flux tend to have more efficient heat redistribution (Perez- +Becker & Showman 2013; Komacek & Showman 2016; Komacek +et al. 2017) + +6 +McGruder et al. +lation levels, Ip, reaching each planet we calculated the +bolometric luminosity assuming the stars emit as black- +bodies. To compensate for this oversimplification, we +increased the obtained uncertainties in each Ip by a fac- +tor of three. Given that we can also obtain information +about stellar densities from the transits, we weighted av- +eraged the density obtained from the spectral analysis +discussed in Section 3.1 and the density obtained from +Juliet to produce the values in Table 1. Our updated +parameters are on average 2.7 times more precise than +previous literature discovery parameters. Figures 2 and +3 show the resulting light curve and radial velocity fits +for each planet. +4. HOMOGENEOUS ANALYSIS OF +TRANSMISSION SPECTRA +Our last step in the process of obtaining as homo- +geneous as possible parameters for all seven systems +was to re-derive atmospheric parameters for the three +planets in the sample with observed transmission spec- +tra. Those are WASP-6b, observed with VLT/FORS2, +HST/STIS G430 and G750, and HST/WFC3 G141 be- +tween 0.32 and 1.65 µm (Nikolov et al. 2015; Carter et al. +2020), WASP-96b observed with VLT/FORS2, Magel- +lan/IMACS, and HST/WFC3 G102 and G141 between +0.4 and 1.64 µm (Nikolov et al. 2018; Yip et al. 2021; +Nikolov et al. 2022; McGruder et al. 2022), and WASP- +110b observed with VLT/FORS2 between 0.4 and 0.83 +µm (Nikolov et al. 2021). +A retrieval analysis of the WASP-96b data was re- +cently done by McGruder et al. (2022), using Platon +(Zhang et al. 2019) and Exoretrievals (Espinoza et al. +2019). +For consistency, we did a similar analysis for +the transmission spectra of WASP-6b and WASP-110b. +That is, with Exoretrievals we tested models includ- +ing water, potassium, sodium, stellar activity, or scat- +tering features and the different combinations of each. +With Platon we tested models with scatters or stel- +lar activity, where Platon assumes equilibrium chem- +istry and fits for the C/O ratio and planetary metallic- +ity to extrapolate the abundances of atomic/molecular +species. lnZ Bayesian evidences were used to favor one +model over another. We considered a difference in lnZ +greater than 2.5 between two models to be moderately +significant, and greater than 5 to strongly support the +higher lnZ model (Trotta 2008; Benneke & Seager 2013). +Additionally, we used Table 2 and eq. 2 from Rackham +et al. (2019), to limit the contribution of stellar activity +to inhomogeneity covering fractions of 4.1 ± 4.1 % when +both spots and faculae are present. +The priors of each retrieval for both WASP-6b and +WASP110b and the Bayesian evidences relative to a +flat (for Exoretrievals) or clear (for Platon) spec- +trum are shown in Appendix D. The highest evidence +models for both targets with both retrievals were ones +that included activity, which is consistent with what was +found in previous analyses (Nikolov et al. 2015; Carter +et al. 2020; Nikolov et al. 2021). Exoretrievals also +found significant evidence for water and a sodium fea- +ture in the WASP-6b spectrum, which was not found for +WASP-110b. The features found in the WASP-6 data +are muted, indicative of high altitude aerosols. Activ- +ity could not explain the muted features, in fact, with +the unocculted cooler spots that the retrievals find, the +sodium signal would be enhanced. +This can be seen +in Figure 9 of Carter et al. (2020). Furthermore, the +cloud deck pressures of ∼ 0.1 bars suggest WASP-6b +has substantially more aerosols than WASP-96b, where +its models favor a cloud deck pressure of ∼ 20 bars. +WASP-110b’s spectrum is more extreme than WASP- +6b’s, where all atomic features are missing and a cloud +deck pressure of ∼ 0.03 mbars is suggested, albeit not +well constrained due to the lack of features. +5. SEARCH FOR TRENDS +Using the new set of homogeneously derived parame- +ters described in sections 3 and 4, we searched for corre- +lations between system parameters and what we define +as aerosol levels in the transmission spectrum of the +planets. +We quantify aerosol levels using as proxy the ampli- +tude of the Na I doublet at 5892.9Å in the transmis- +sion spectra of WASP-6b, WASP-96b, and WASP-110b. +This was calculated as the sum of transit bins within +a narrow range centered on the Na I doublet (5862.9 +to 5892.9 Å) minus the sum of bins blue-ward and red- +ward of this region, while also being outside of the wings +of the sodium feature. For WASP-96b this was 4880 to +5380Å and 6200 to 6700Å, but 5340 to 5820Å and 5960 +to 6440Å for WASP-6b and WASP-110b which did not +have notable absorption wings. +The result of our search for correlations between +aerosol levels and system parameters is summarized in +Figure 4. We find a significant correlation between the +amplitude of the Na I feature in the transmission spec- +trum of the planet and the overall [Fe/H] of the host star. +The linear fit to this trend has a Pearson correlation co- +efficient of r=0.83, corresponding to a 99% confidence of +a found correlation. To examine if there could be a cor- +relation with stellar activity, we used log10(R’HK) (see +Section 3.1.1). The best linear fit to log10(R’HK) and +the Na I signal has r = -0.28, corresponding to about +24% confidence that there is such a correlation. There- +fore, we find no correlation with log10(R’HK). However, + +Twin Planets +7 +Figure 2. The phase folded transit data for each of the similar seven systems and their Juliet best fit transit models (dotted +lines). 1-σ uncertainties of the fits are shaded in the same color as the transit models. The plotted ’bin’ data is with 30 points +binning, aside for WASP-110, which had 8 point binning. + +WASP-6 +WASP-25 +1.000 +0.995 +0.990 +0.985 +0.980 +TESS +OTESs = 392 [ppm] +0.980 +OTEss = 433 [ppm] +TESS bin +0.975 +0.001 +0.001- +0.000 +0.000- +-0.001 +-0.001 +-0.04 +-0.02 +0.00 +0.02 +0.04 +-0.04 +-0.02 +0.00 +0.02 +0.04 +1.0025 +WASP-55 +1.000 + ++++++++! +WASP-96 +1.0000 +0.9975 +0.9950 +0.9925 +Reli +0.9900 +0.985 +0.9875 +OTEss = 495 [ppm] +0.9850 +OTESs = 547 [ppm] +0.001 +0.000 +0.000 +-0.001 +-0.04 +-0.02 +0.00 +0.02 +0.04 +-0.04 +-0.02 +0.00 +0.02 +0.04 +WASP-110 +1.000 +WASP-124 +111 +1.000 +1 +0.995 +0.990 +Rel +0.980 +0.985 +OTESs = 634 [ppm] +OTESs = 723 [ppm] +0.975 +0.980 +0.002 +0.002 +0.001 +0.000 +0.000 +-0.001 +-0.002 +-0.002 +-0.04 +-0.02 +0.00 +0.02 +0.04 +-0.04 +-0.02 +0.00 +0.02 +0.04 +1.005 +Phase +.. +1.000 +HATS-29 +xni +0.995 +HATS +R 0.985 +OTEss = 762 [ppm] + OHATS = 1954 [ppm] +0.980 +HATSbin +0.005 +0.000 +-0.005 +0.04 +-0.02 +0.00 +0.02 +0.04 +Phase8 +McGruder et al. +the chromospheric activity measured from log10(R’HK) +is not a direct measurement of total high energy flux +(e.g. see Zhang et al. 2020; Johnstone et al. 2021), where +total high energy emission is likely the more important +parameter affecting aerosol formation rates (Moses et al. +2011, 2013; Fleury et al. 2019). Thus, we need more di- +rect measurements of the host stars high energy levels +(e.g. HST/UVIS or XMM-Newton observations) to con- +fidently rule out such a correlation. +6. SUMMARY AND CONCLUSIONS +We have identified seven systems that have very sim- +ilar characteristics to one another, the ’Similar Seven’, +where the host star metallicity is the only stark differ- +ence between the parameters measured for these sys- +tems. +Three of the planets in this sample already +have transmission spectra observed, and though they +have similar parameters, their transmission spectra have +widely varying amounts of high altitude aerosols obscur- +ing features. To thoroughly search for correlations be- +tween the observed spectra aerosol levels and system +parameters, we homogenously reanalyze HARPS and +FEROS stellar spectra, and HARPS, CORALIE, and +CYCLOPS RV data with TESS and HAT-South transit +data to refined the stellar and transit parameters. We +found that host star metallicity seems to correlate with +the observed aerosol levels, with a 99% confidence that +a linear correlation exists, implying that planets around +higher metallicity stars would have lower high altitude +aerosol levels. If this holds, it could be explained by the +requirement of viable seed particles needing to be lofted +to high enough altitudes for cloud forming gases to con- +dense on (e.g. Helling 2008, and references therein). The +higher metallicity might cause the formed seed particles +to be more dense and subsequently differentiate lower +in the atmosphere. However, given that the potential +trend was found with only the three observed transmis- +sion spectra, the correlation is tentative, pending on fur- +ther observations of the other sample planets. +We also use log10(R’HK) as a proxy to explore if high +energy irradiation could be correlated to the differences +in the transmission spectra, given that there are no di- +rect measurements of the stars’ high energy emissions. +We found no clear signs of a correlation with this param- +eter. However, correlation to the host star’s high energy +levels may still be present and require more direct mea- +surements, i.e. with XMM-Newton and/or HST/UVIS. +Regardless of if metallicity or high energy irradiation +is found to be a contributing factor to high altitude +aerosols in these system, the similar seven planets are +ideal targets for understanding the unique physical and +chemical processes undergoing in these class of planets. +This is because the similarity of most parameters act as +a controlled sample. This approach of specifically select- +ing very similar targets should be a common practice in +exoplanet atmosphere studies, and has the potential to +isolate key physical or chemical phenomena. +APPENDIX +A. STELLAR ROTATION AXIS DISTRIBUTION +For WASP-6 there are two measurements of the rotation axis alignment, λ, i.e. the angle between the planet’s orbital +axis and the rotation axis of the star: λ = 11+14 +−18 +◦, obtained using the Rossiter-McLaughlin effect (Gillon et al. 2009) +and λ =7.2±3.7◦ (which we adopt), obtained using occulted star spots (Tregloan-Reed et al. 2015). For WASP-25 +Brown et al. (2012) measured a λ of 14.6±6.7◦ via Rossiter-McLaughlin. The other five stars in our sample do not +have direct λ measurements, so instead we calculated their most likely λ values using the distribution of λ values +measured for G-type stars, i.e. with Teff ∈ [5300,6300] K, as shown in Figure 5. 75% of the λ values are less that +±20◦, suggesting that the bulk of exoplanet systems with G-type host stars are aligned. This is in agreement with the +findings of Triaud (2018) (see their Fig. 6). Based on the distribution of values in Fig. 5, we adopt a λ = 0 ± 30◦ for +the remaning five systems in our sample. The estimated rotation periods for all our targets, computed using eq. 1, +are listed in Table 1. +B. STELLAR ROTATION PERIODS FROM TESS AND ASAS-SN LIGHT CURVES +We downloaded the TESS SPOC 6 light curves for each target from MAST 10.17909/fwdt-2x66. The TESS sectors +and observed number of transits for each target are summarized in Table 2. We also downloaded ASAS-SN (Shappee +et al. 2014; Kochanek et al. 2017) time series observations for each target in V- and g-bands, which are treated as +separate photometric monitoring campaigns. The number of ASAS-SN observations per photometric band, for each +target are also summarized in Table 2. +6 Science Processing Operations Center (Jenkins et al. 2016) + +Twin Planets +9 +Figure 3. The phase folded RV data of each of the similar seven systems and their Juliet best fit RV models (solid black +lines). The HARPS RV measurements that were during transit were omitted. For WASP-124b, ’CORALIE07’ and ’CORALIE14’ +represents observations taken before/after the CORALIE upgrade (see Maxted et al. 2016). + +100 +WASP-6 +WASP-25 +75 +K=69.55±2.66[m/s] +100 +K=75.28±2.61[m/s] +P=3.3610026±6e-07[days] + 50 +P=3.7648337±1.2e-06[days] +25 +50 +Relative +-25 +-50 +-50 +-75 +CORALIE, 0 = 13 [m/s] +CORALIE, g = 20 [m/sl +-100 +HARPS, 0 = 11 [m/s] +HARPS, = 9[m/s] +198 +50 +25 +25 +11.1 +0 +0 +。。1 +-25 +-25 +-50 +-50 +-0.4 +-0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0'0 +0.2 +0.4 +100 +WASP-55 +100 +WASP-96 +K=68.73±3.22[m/s] +75 +P=4.465631±1.3e-06[days] +K=62.1±3.93[m/s] +50 +P=3.4252567±1.2e-06[days] +Flux +50 +Relative +25 +-50 +-25 +↑ CORALIE, = 14 [m/s] +-50 +-100 +HARPS, = 10 [m/s] +CORALIE, 0 = 12 [m/s] +-75 +40 +42 +20 +0 +0 +-20 +-20 +-40 +-40 - +0.0 +0.2 +0.4 +-0.4 +-0.2 +-0.4 +-0.2 +0.0 +0.2 +0.4 +WASP-110 +WASP-124 +100 +150 +K=73.1±7.7[m/s] +100 +K=70.85±6.73[m/s] +Flux +P=3.7784022±1.6e-06[days] + 50 +P=3.3726511±2.9e-06[days] +50 +Relative +-50 +-50 +100 +↑ CORALIE07, 0 = 31 [m/s] +-150 +-100 +CORALIE, = 25 [m/s] +CORALIE14, 0 = 39 [m/s] +50 +100 +0. +-50 +100 +-0.4 +0.2 +0.0 +0.2 +0.4 +0.4 +-0.2 +0'0 +0.2 +0.4 +Phase +HATS-29 +100 +K=78.27±6.64[m/s] +Flux +P=4.6058827±1.1e-06[days] +50 +Relative [ +-50 +↑ HARPS, = 10 [m/s] +↑ CYCLOPS, = 37 [m/s] +100 +CORALIE, = 10 [m/s] +50 +1 +0 +d +50 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Phase10 +McGruder et al. +Figure 4. +Left: Sodium amplitude versus host star metallicity from the observed WASP-6b (red diamond), WASP-96b +(blue star), and WASP-110b (green circle) transmission spectra. The best fit linear regression (black dashes) has a Pearson’s +correlation coefficient, r-value, of 0.831. The metallicty range WASP-25b, WASP-55b, WASP-124b, and HATS-29b cover are +plotted as yellow shaded regions. Right: Same as left, but with Na amplitude versus log10(R’HK). Here the r-value to the best +fit linear regression does not support a correlation of the Na amplitude to log10(R’HK). +Figure 5. The distribution of observed λ values of 80 exoplanet systems with the host star between 5300 and 6300K, obtained +through TEPCatb. Here λ = 0◦ means the axes are fully aligned. The bulk of the λ values are smaller than ±20◦ suggesting +alignment of the orbital and spin axes. Two gaussians are overplotted: one with the wide distribution (σ = 41.3, black) and one +with a narrow distribution (σ = 20, red). We adopt a mean distribution of λ = 0 ± 30◦. +ahttps://www.astro.keele.ac.uk/jkt/tepcat/obliquity.html +bhttps://www.astro.keele.ac.uk/jkt/tepcat/obliquity.html + +0.004 +fit:p-value=0.375, +fit: p-value=0.817, +r-value=0.831 +WASP96b +r-value=-0.283 +WASP96b +0.003 +0.002 +WASP6b +WASP6b +0.001 +Na +0.000 +WASP110b +WASP110b +WASP124b +WASP25b +-0.001 +WASP55b +HATS29b +0.2 +0.1 +0.0 +0.1 +0.2 +0.3 +-4.8 +4.7 +4.6 +-4.5 +-4.4 +Host star [Fe/H] +log1oR'hkμ=-1.09,=41.27 +μ=0g=20 +0.020 +I counts +0.015 +Normalized +0.010 +0.005 +0.000 +-150 +-100 +-50 +0 +50 +100 +150 +^[degrees]Twin Planets +11 +We used the time series observations above to estimate the rotation period of each star following a similar analysis +to the one described in McGruder et al. (2020). For the TESS data, we masked all the in-transit points using the +known ephemerides of each planet, to search for photometric modulations of the stars themselves. Before searching +for photometric modulations, we binned the data for each target in 3.33-hour bins (16.66-hour bins for WASP-110). +Given that the expected rotation periods for all the stars are longer than 10 days, those binning levels should not +affect results. For the ASAS-SN data, we sigma-clipped observations that deviated by more than 3σ from the overall +mean of each light curve, weighted averaged all observations for a given night (typically 3 observations per night), and +removed observations with uncertainties 3 times larger than the mean uncertainty. +To search for the Prot of each star, we jointly modelled all the photometric datasets for each target using Juliet’s +(Espinoza et al. 2018) gaussian processes (GP) semi-periodic kernel, setting the GP characteristic time-scale and period +as common terms between all datasets. The other parameters (jitter term, GP amplitude, and GP constant scale term) +were specific to the individual datasets. Also, when data from more than one TESS sector were available, we combined +them and modeled them together. We modeled activity as a semi-periodic GP instead of using periodograms because +it has been found that peaks non-consistent with the rotational period of stars can appear in the latter (Haywood +et al. 2014; Nava et al. 2020). We assume the main driving factor for the GP period is stellar inhomogeneities coming +in and out of view as the star rotates, as such we call this the rotation periods, Prot. The priors for Prot were set using +the v sin I of each star from section 3.1. For all the targets but WASP-124 and HATS-29 we used normal priors with +mean and standard deviations near the values derived from v sin I. For WASP-124 we used a normal prior truncated +at 5 days to prevent sampling of unrealistic low periods driven by the TESS data. For HATS-29 we used a uniform +distribution between 17 and 33 days for the same reason. +The advantage of the TESS data is the continuous monitoring with high photometric precision. However, given +that it only monitors a sector for about 24 days, the long baseline of the ASAS-SN data complements TESS well. +This emphasizes the advantage of a joint fit with all photometric data, but we also run the TESS and ASAS-SN data +separately (with the same priors) to outline the contribution from each monitoring source. Table 3 shows the period, +amplitude, and median absolute deviations (MAD) of both photometric monitoring sources. +Finally, to ensure that the found rotation periods were not driven by the sampling of the data we tested their +window functions, where we used all timestamps of monitoring data but set the flux and uncertainties to 0. Doing +this suggested no periodic signal due to the observing cadence. The Prot values for each system are listed in Table 1. +C. HATS-29 TESS LIGHT CURVE +DECONTAMINATION +Because of the low image resolution of TESS, the +light curve of HATS-29 is contaminated by a back- +ground RR Lyrae star (see top panel of Figure 6). We +isolated the RR-Lyrae signal by first excluding the in +transit data (using transit parameters from (Espinoza +et al. 2016)), then applying a Lomb-Scargle periodigram +(Lomb 1976; Scargle 1982) analysis to this out-of-transit +data to find a period of 0.631 days. +Next, we phase +folded the data to the 0.631 d period, binned the data by +100 points, and finally smoothed the binned data with +scipy.signal.savgol_filter (Virtanen et al. 2020) 7 +that had a window set to 51 and order of 10. Our phase +folded data and corresponding RR-Lyrae model can be +seen in the second panel of Figure 6. +We then sub- +tracted (in magnitude space) the best RR-Lyrae light +curve model from the TESS data, and reduced the cor- +rected TESS data with the same procedures of the other +TESS observations (see Appendix B). +7 We used an older version, ’savitzky_golay,’ which is the same +algorithms before it was included in scipy. +HAT-South (Bakos et al. 2013) has public light curves +for HATS-29, which we downloaded from the survey’s +website 8, and is not contaminated by the background +star. +We compared the best fit Juliet transit with +TESS against a fit with the HAT-South data. +Upon +confirming that the transit parameters - aside from tran- +sit depth, which one would expect to differ due to the +different photometric bands - were consistent with each +other, we ran a joint Juliet fit with all the transit and +RV data to obtain our final planetary parameters of this +system. See the bottom panel of Figure 6 for an overlay +of the TESS and HAT-South data. +D. ATMOSPHERIC RETRIEVAL RESULTS +Table 4 has the priors used for each model and the +∆ ln Z of each retrieval run are in Table 5 +We thank the anonymous referee for helpful com- +ments to the manuscript. +This work has been sup- +ported by the National Aeronautics and Space Admin- +istration’s Exoplanet Research Program via grant No. +8 hatsouth.org + +12 +McGruder et al. +Table 2. Summary of the photometric and RV data +Transit data +ASAS-SN +RV data +Star +Sectors +Transits +Filter +date range +Obs. +Spectrograph +Obs. +Source +WASP-6 +2, 29 +13 +V +2013-11-25 to 2018-11-26 +883 +CORALIE +35 +Gillon et al. (2009) +g +2017-09-16 to 2022-04-18 +2073 +HARPS +55 +Trifonov et al. (2020) +WASP-25 +10 +6 +V +2012-01-24 to 2018-08-20 +820 +CORALIE +28 +Brown et al. (2012) +g +2017-12-21 to 2022-04-18 +2401 +HARPS +31 +Trifonov et al. (2020) +WASP-55 +10, 37 +8 +V +2012-02-17 to 2018-08-19 +961 +CORALIE +20 +Hellier et al. (2012) +g +2017-12-16 to 2022-04-19 +2156 +HARPS +19 +Trifonov et al. (2020) +WASP-96 +2, 29 +13 +V +2014-04-30 to 2018-09-24 +921 +CORALIE +21 +Hellier et al. (2014)∗ +g +2017-09-05 to 2022-02-10 +2592 +WASP-110 +27 +6 +V +2014-04-29 to 2018-09-24 +916 +CORALIE +15 +Anderson et al. (2014)∗ +g +2017-09-05 to 2022-04-18 +2286 +WASP-124 +1 +8 +V +2014-04-30 to 2018-09-19 +1577 +CORALIE +39 +Maxted et al. (2016)∗ +g +2017-09-07 to 2022-04-18 +4217 +HATS-29 +13 +6 +V +2014-05-17 to 2018-09-24 +989 +HARPS +3 +Espinoza et al. (2016) +(HAT-South +g +2017-10-03 to 2022-04-18 +2165 +CYCLOPS +9 +Espinoza et al. (2016) +data:) +– +23 +CORALIE +4 +Espinoza et al. (2016) +Note—The HAT-South photometric data for HATS-29 (first columns, last row) was acquired from Bakos et al. (2013). Trifonov +et al. (2020) reanalyzed archival HARPS data using SERVAL (Zechmeister et al. 2018). The number of observations is denoted +’Obs.’, and is the unbinned/unclipped observations for the ASAS-SN data. +∗data obtained through DACE +Table 3. Summary of TESS and ASAS-SN photometric monitoring Juliet fits +Star +TESS +ASAS-SN +Joint +MAD +Period +Amplitude +MADg +MADv +Period +Amplitude +Period +Amplitude +[ppm] +[days] +[ppm] +[ppm] +[ppm] +[days] +[ppm] +[days] +[ppm] +WASP-6 +348.5 +28.93+6.63 +−6.9 +0.23 +2407.3 +2266.2 +28.37+5.58 +−7.19 +1383.1 +29.69+5.00 +−5.78 +57.3 +WASP-25 +376.8 +18.11+2.82 +−2.99 +187.9 +1847.6 +1459.3 +16.11+1.73 +−0.63 +947.2 +16.20+3.06 +−0.49 +199.5 +WASP-55 +241.2 +19.46+3.77 +−3.87 +132.4 +1516.3 +2494.4 +15.54+6.36 +−3.21 +1181.2 +19.63+3.51 +−5.39 +49.1 +WASP-96 +230.0 +19.85+5.05 +−5.84 +58.7 +1955.89 +1761.3 +15.38+10.78 +−2.78 +703.8 +25.69+3.15 +−8.56 +98.1 +WASP-110 +330.6 +112.2+52.4 +−47.6 +48.0 +2063.9 +2220.8 +132.9+43.6 +−71.1 +488.1 +134.42+46.32 +−58.26 +58.8 +WASP-124 +447.3 +12.04+4.84 +−6.0 +256.4 +1427.5 +1441.3 +10.98+4.44 +−3.21 +714.6 +13.51+5.77 +−7.40 +340.1 +HATS-29 +737.5 +26.06+4.79 +−5.78 +78.3 +1775.7 +2340.9 +20.78+9.063 +−0.585 +1654.5 +27.07+3.96 +−6.85 +90.1 +Note—The MAD is for the binned data. The amplitdues are obtained by using scipy.optimize.minimize (Virtanen et al. +2020) to fit a sine curve to the data phase folded on the Juliet best fit period. The subscripts g and v on the ASAS-SN MAD +correspond to the g and V band filters. +20-XRP20_2.0091. We thank N. Espinoza for provid- +ing access to Exoretrievals, S. Blanco-Cuaresma for +continuous support using iSpec, E. Shkolnik for help- +ful discussion regarding GALEX data, and V. DiTo- +masso for discussion regarding analysis of RV data. We +also appreciate the support from the NSF Graduate +Research Fellowship (GRFP), grant No. DGE1745303. +RB and AJ acknowledge support from ANID – Millen- +nium Science Initiative – ICN12_009. AJ acknowledges +additional support from FONDECYT project 1210718. +RB acknowledges support from FONDECYT project +11200751. + +Twin Planets +13 +Table 4. The priors for Exoretrievals and PLATON +Exoretrievals +PLATON +parameter +function +bounds +parameter +function +bounds +reference pressure (P0, bars) +log-uniform +-8 to 3 +reference pressure (Pclouds, Pa) +log-uniform +-3.99 to 7.99 +planetary atmospheric +uniform +600 to 1800K +planetary atmospheric +uniform +600 to 1800K +temperature (Tp) +temperature (Tp) +stellar temperature +uniform +Teff-240 to Teff+240K +stellar temperature +gaussian +µ=Teff, σ=150K +(Tocc) +(Tstar) +stellar heterogeneities +uniform +Teff-3000 to Teff+3000K +stellar heterogeneities +uniform +Teff-3000 to Teff+3000K +temperature (Thet) +temperature (Tspot) +heterogeneity covering +gaussian +µ=0.041, σ=0.041 +heterogeneity covering +gaussian +µ=0.041, σ=0.041 +fraction (fhet) +fraction (fspot) +haze amplitude (a) +log-uniform +-30 to 30 +scattering factor +log-uniform +-10 to 10 +haze power law (γ) +uniform +-14 to 4 +scattering slope (α) +uniform +-4 to 14 +log cloud absorbing +uniform +-80 to 80 +metallicity (Z/Z⊙) +log-uniform +-1 to 3 +cross-section (σcloud) +trace molecules’ +log-uniform +-30 to 0 +C/O +uniform +0.05 to 2 +mixing ratios +reference radius factor(f) +uniform +0.8 to 1.2 +1 bar, reference radius (R0) +uniform +Rp-.2Rp to Rp+.2Rp +Note. These priors were set to allow for a wide parameter space to be surveyed, but contained within physical regimes. Not +all parameters were included in each model fit (see Tab. 5). We used 5000 live points for all runs. For further description of +the parameters of Exoretrievals, please refer to the Appendix D of Espinoza et al. (2019). Teff is the effective temperature +of the host star, which is 5438K and 5392K for WASP-6 and WASP-110, respectively. γ is the exponent of the scattering slope +power law, where −4 is a Rayleigh scattering slope. α is the wavelength dependence of scattering, with 4 being Rayleigh. f +is a factor multiplied by the inputted planetary radius to produce the reference radius, i.e. R0 = fRp, Rp is the radius of the +planet, corresponding to 1.119Rj and 1.177Rj for WASP-6b and WASP-110b, respectively. +Facilities: +Magellan:Baade +(IMACS), +Smithso- +nian Institution High Performance Cluster (SI/HPC), +HST(STIS/WFC3), All-Sky Automated Survey for Su- +pernovae (ASAS-SN), VLT(FORS2),TESS, ESO La +Silla 3.6m (HARPS), Swiss 1.2-metre Leonhard Eu- +ler Telescope (CORALIE), Anglo-Australian Telescope +(CYCLOPS), MPG/ESO2.2(FEROS), and Gaia space- +craft +Software: +Astropy (Astropy Collaboration et al. +2013), +corner +(Foreman-Mackey +2016), +Matplotlib +(Hunter 2007), NumPy (Harris et al. 2020), Multinest +(Feroz et al. 2009), PyMultiNest (Buchner et al. 2014), +SciPy (Virtanen et al. 2020), batman (Kreidberg 2015), +george (Ambikasaran et al. 2015) +dynesty (Speagle +2020), Platon (Zhang et al. 2019), Juliet (Espinoza +et al. 2018), CERES (Brahm et al. 2017a), ZASPE +(Brahm et al. 2017b), iSpec (Blanco-Cuaresma et al. +2014; Blanco-Cuaresma 2019) +REFERENCES +Adams, D. 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A. 2019, PASP, 131, 034501 + diff --git a/yNE2T4oBgHgl3EQf3wh7/content/tmp_files/load_file.txt b/yNE2T4oBgHgl3EQf3wh7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f2d79787c68a8f0f072763184dd098947ee62906 --- /dev/null +++ b/yNE2T4oBgHgl3EQf3wh7/content/tmp_files/load_file.txt @@ -0,0 +1,1961 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf,len=1960 +page_content='To turn off the edits, from https://journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='aas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='org/revision-history/ Draft version January 12, 2023 Typeset using LATEX twocolumn style in AASTeX62 The Similar Seven: A set of very-alike exoplanets to test correlations between system parameters and atmospheric properties Chima D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' McGruder,1, ∗ Mercedes López-Morales,1 Rafael Brahm,2, 3, 4 and Andrés Jordán2, 3, 4 1Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA 2Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Diagonal las Torres 2640, Peñalolén, Santiago, Chile 3Millennium Institute for Astrophysics, Santiago, Chile 4Data Observatory Foundation ABSTRACT Studies of exoplanetary atmospheres have found no definite correlations between observed high alti- tude aerosols and other system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This could be, in part, because of the lack of homogeneous exoplanet samples for which specific parameters can be isolated and inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Here we present a set of seven exoplanets with very similar system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We analyze existing photometric timeseries, Gaia parallax, and high-resolution spectroscopic data to produce a new set of homogeneous stellar, planetary, and orbital parameters for these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' With this we confirm that most measured param- eters for all systems are very similar, except for the host stars’ metallicities and possibly high energy irradiation levels, which require UV and X-ray observations to constrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' From the sample, WASP-6b, WASP-96b and WASP-110b, have observed transmission spectra that we use to estimate their aerosol coverage levels using the Na I doublet 5892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We find a tentative correlation between the metallicity of the host stars and the planetary aerosol levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The trend we find with stellar metallicity can be tested by observing transmission spectra of the remaining planets in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Based on our predic- tion, WASP-25b and WASP-55b should have higher levels of aerosols than WASP-124b and HATS-29b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Finally, we highlight how targeted surveys of alike planets like the ones presented here might prove key for identifying driving factors for atmospheric properties of exoplanets in the future and could be used as a sample selection criterium for future observations with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' JWST, ARIEL, and the next generation of ground-based telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Keywords: planets and satellites: atmospheres — stars: activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' starspots — techniques: spectro- scopic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-6b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-25b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-55b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-95b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-110b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-124b, HATS- 29b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' INTRODUCTION The upper atmospheres of about 100 exoplanets have been probed via transmission spectroscopy to date with HST, Spitzer, ground-based telescopes (NASA Exo- planet Archive 2019) 1, and now JWST (JWST Transit- ing Exoplanet Community ERS team et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Most observations suggest that atmospheric features are heav- ily muted by aerosols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019), with only a few planets showing little to no aerosol coverage Corresponding author: Chima D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' McGruder chima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='mcgruder@cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='edu ∗ NSF Graduate Research Fellow 1 Accessed on 2022-05-29 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Ahrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The formation of aerosols in planetary atmospheres occurs via complex chemical and physical processes, which are not yet fully understood, but work is in progress (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Helling 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Marley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Helling 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For example, predic- tions link cloud formation in exoplanets to the chemi- cal composition of their atmospheres, where seed par- ticles need to be lofted to high altitudes so gases can condense on them and form clouds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Helling 2008, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The composition, availability, and altitude of potential seed particles has many de- pendencies, including the composition of the protoplan- etary disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Mordasini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2010), and atmospheric arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04174v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='EP] 10 Jan 2023 2 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' differentiation, where heavier elements are expected to sink into the lower layers of the atmosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Helling 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' In the case of hazes, model predictions show that high altitude hazes can form in exoplanets via UV-driven photolysis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2011, 2013), and laboratory experiments have shown that UV radiation can form photochemical hazes in H2 dominated atmospheres at temperatures between 600K–1500K (Fleury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Understanding what system parameters drive the presence or absence of aerosols in exoplanetary atmo- spheres will be key to understanding their formation and evolution mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' However, comparative stud- ies of exoplanet atmospheres so far have yielded either no correlations between the atmospheric properties of the planets studied and other system parameters (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016), or only tentative correlations between planetary equilibrium temperature and atmo- spheric aerosol levels (Heng 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Stevenson 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Tsiaras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Dymont et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Yet, degeneracies between system parameters and dis- agreeing observations leave any correlation uncertain (Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Possible reasons for why correlations between ob- served exoplanet atmospheric properties and other sys- tem parameters have not yet been found are the fact that the parameter space being considered is too broad, with multiple independent parameters typically being fitted simultaneously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' planetary equilibrium temperature, density, log surface gravity, planetary/host star metal- licity, etc), and over wide range of values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' as well as the parameter space being scarcely sampled, with typically one-to-no pairs of similar planets being examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We present a way to potentially alleviate this problem by identifying groups of planets with similar properties, so that a reduced number of parameters can be isolated and compared against observed atmospheric features of the planets in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' In particular, we have identified a group of seven gas giants with very similar parameters, except for the metallicity of their host stars and possibly their high energy irradiation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We use this group of planets to test whether aerosol properties are related to those parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Section 2 describes the identification of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Sections 3 and 4 present our reanalysis of system param- eters and observed transmission spectra for comparative purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Section 5 presents the analysis of correlations between exoplanet aerosol level proxies and the host star metallicity and high energy irradiation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Finally, Section 6 summarizes our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' THE SAMPLE Using the NASA Exoplanet Archive(NASA Exoplanet Archive 2021)2, we compiled system parameters for all known exoplanets with an observed optical or near- infrared transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We compiled the plan- ets mass, radius, orbital period and semi-major axis sep- aration, as well as the mass, radius, effective tempera- ture, and metallicity of the host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' With those values in hand, we computed the planets gravity, density, equi- librium temperature, and their stellar insolation levels in earth units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We then searched in the compiled list for planets with very similar system parameters and com- pared their observed transmission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This is how we identified WASP-6b and WASP-96b, two hot jupiters with very similar masses, radii, Teq, stel- lar parameters, and insolation levels (see top six panels of Figure 1 and Table 1), but strikingly different trans- mission spectra (See bottom three panels of Figure 1 and Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' From the parameters available for each system, they only appear to differ significantly in the host star metallicity, reported in the NExSci database as [Fe/H] = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='20±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='09 for WASP-6b (Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2009), and [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='19 for WASP-96b (Hellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Next we compared the parameters of all known ex- oplanets, including those without atmospheric observa- tions, to those of WASP-6b and WASP-96b, assuming the planets to be similar if all their planetary parameters listed above agreed within about 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Using this strategy we found another five planets similar to WASP-6b and WASP-96b: WASP-25b (Enoch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2011), WASP-55b (Hellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2012), WASP-110b (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014), WASP-124b (Maxted et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016), and HATS-29b (Es- pinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This is how we arrived to the similar seven planets sample described in the remaining of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The parameters of each system, re-derived ho- mogeneously as described in the following section, are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' DERIVATION OF HOMOGENEOUS STAR AND PLANET PARAMETERS The parameters used in Section 2 to identify the sample were obtained from various literature sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Therefore, to minimize potential biases and systematics between separate analyses, we re-derived the parameters of each system homogeneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Table 1 provides all the newly derived parameters, with the derivation processes of each set of parameters described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Stellar Parameters 2 Accessed on 2021-11-17 Twin Planets 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Stellar and planetary parameters of all Similar Seven Systems Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-6 WASP-25 WASP-55 WASP-96 WASP-110 WASP-124 HATS-29 M∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='854+0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='8±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6 Note—The top 15 parameters, delineated by a thicker line, are for the host stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' In order,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' they are mass [M⊙],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' radius [R⊙],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' effective temperature [K],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' log10 of surface gravity [cgs],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' density [ρ⊙],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' log10 of iron to hydrogen abundance relative to the sun [dex],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' radial velocity [km/s],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' age [Gy],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' NUV magnitude,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' NUV flux from the star at the semi-major axis of the planet [Watts/m2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' log10 of the Calcium H and K indices [dex],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' rotational period derive from v sin I (Prot1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' photometry (Prot2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' R’hk (Prot3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' and the adopted rotational period (a weighted mean of the v sin I and photometry periods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Prot) [days].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The following 12 parameters are for the corresponding planet and are mass [Mj], radius [Rj], equilibrium temperature [K], surface gravity [m/s2], density [ρj], insolation [I⊕], and orbital period [days], semi-major axes relative to the host stars’ radii, reference mid transit time transit [BJD], impact parameters, orbital inclinations [degrees], and RV semi-amplitude [m s−1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We re-derived the stellar parameters of each host star using archival HARPS (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2003) and FEROS (Stahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 1999) spectra3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Each spectra was homoge- neously reduced using CERES (Brahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2017a), and the stellar parameters (Teff, log g, [Fe/H], v sin I) were 3 archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='org derived using ZASPE (Brahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2017b), as described in detail in Brahm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Once ZASPE had obtained the stellar atmospheric parameters, the re- maining physical parameters were computed comparing synthetic values generated using PARSEC isochrones (Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2012), and Gaia DR2 (Gaia Collabo- ration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018) parallaxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For this step we fixed the stellar metallicity to the values obtained by ZASPE, 4 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Top: Mass versus equilibrium temperature, gravity, radius, density, stellar irradiation, and star’s metallicity for WASP-6b (blue diamonds), WASP-96b (blue stars), WASP-110b (cyan circles), WASP-124b (red ’X’), WASP-55b (red squares), WASP-25b (red triangle), HATS-29b (red pentagon), the PanCET planets (green crosses), and all exoplanets with measured parameters (grey circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The Equilibrium Temperature, Insolation, and metallicity of WASP-110b and WASP-6b overlap (see Table 1), making it hard to see the cyan circles of WASP-110b underneath the blue diamonds of WASP-6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Bottom: Transmission spectrum of WASP-6b (red/magenta, Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020), WASP-96b (blue, Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Yip et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022), and WASP-110b (green circles, Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021), with the Platon best-fit retrieval model (black line) and 1-σ confidence interval (cyan shaded regions) overplotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' In this figure, the parameters for the similar seven planets are the re-derived parameters discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=" ('w) 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 Mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 500 1000 1500 2000 2500 Equilibrium Temp (K) (M) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 Mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 5 10 15 20 25 30 Planet Gravity (m/s2) (M) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 Mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 Radius (R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=') (w) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 Mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 Planet Density (pi) (w) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 Mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 1000 2000 3000 4000 5000 Insolation (Earth Units) (M) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 Mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 stellar metallicity ([Fe/H])0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='150 WASP-6: 「Fe/H|=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='05 FORS2 STIS/430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='148 STIS/750 WFC3/G141 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='146 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='122 FORS2+IMACS WASP-96:[Fe/H|=+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='05 HST/G102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='120 HST/G141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='118 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='114 WASP-110:[Fe/H|=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='05 FORS2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='142 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='140 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='8 Wavelength (μm)Twin Planets 5 and used Teff from ZASPE as a prior to obtain posterior distributions for the stellar age, mass, and interstellar extinction using emcee (Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' From these values we also derived stellar radii and log g, which was then fed back into ZASPE as a fixed param- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The process described above was iterated until reaching convergence in log g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-55, which has both HARPS and FEROS observations, we weighted av- eraged the values from each instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' R′ HK Activity Index We derived R′ HK values for each host star from the Ca II H&K lines in their HARPS and FEROS spectra, following the methods in Noyes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (1984), and cali- brated them to the standard Mount Wilson scale follow- ing Lovis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' To calibrate the R′ HK indexes from HARPS and FEROS to the Mount Wilson scale we used six of the seven (all but HD 219834) reference stars used by Lovis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2011) that have both HARPS and FEROS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For HARPS we found a conversion of the form SMW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='118 · SHARP S + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0135, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0035 fit dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For FEROS we found the conver- sion SMW = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2121 · SF EROS + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0072, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0275 fit dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Rotational Period We estimated the rotation period, Prot, of each star using three proxies: 1) v sin I from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Prot = 2πR∗/v sin (i − λ), (1) where R∗ is the stellar radius, i is the inclination of the planet’s orbital axis, and λ is the rotation axis align- ments – λ has been measured for WASP-6 and WASP- 25 (Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Tregloan-Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2012), and we estimated it for the remaining stars using probability distributions (see Appendix A), 2) TESS (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014) and ASAS-SN (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2017) light curves (see Ap- pendix B), and 3) the R′ HK indexes derived above, com- bined with Table 3 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 6 of Suárez Mascareño et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The values of Prot obtained via each method are sum- marized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The values obtained with the first two methods are consistent with each other, while the values obtained using R′ HK are not fully consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Be- cause the periods obtained from the first two methods are more direct measurements, we used the weighted mean from these two methods as our adopted values, reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Near-UV Flux We derived Near-Ultraviolet (NUV) fluxes for each host star using their Galaxy Evolution Explorer (GALEX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Bianchi & GALEX Team 1999) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We obtained GALEX NUV magnitudes from VizieR, which can also be accessed via the Mikulski Archive for Space Telescopes (MAST) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='17909/T9H59D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The magnitude was converted to total NUV luminosity using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 6 of Schneider & Shkolnik (2018), and Gaia D3 par- allaxes, assuming a central mean wavelength of 2267Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We then converted those magnitudes to flux density as FNUV [ erg s cm2 Å] = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='06 ∗ 10−16 ∗ 10 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='08−mNUV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 , (2) where mNUV is the observed GALEX AB NUV magni- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This equation was derived from gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Planetary parameters We re-derived the parameters of each planet using the SPOC4 TESS light curves downloaded from MAST 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='17909/fwdt-2x66 and described in Appendix B, and radial velocity observations from CORALIE (Queloz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2000), HARPS (Mayor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2003), and CY- CLOPS (Horton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We also used HAT-South (Bakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2013) data with TESS to fit the transit of HATS-29b because of contamination from a background RR Lyrae star in the TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' After removing that contamination (see Appendix C), the TESS and HAT- South transit light curves were modeled using the same procedure as the other six targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' A table summarizing the RV and photometric data is provided in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We jointly fitted the light curves and RVs of each tar- get with Juliet (Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018), initially assum- ing circular orbits and quadratic limb darkening coeffi- cients with uniform priors from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' All other orbital parameters (P, t0, a/R∗, Rp/R∗, b, i, K, and ρp) were fit- ted with Gaussian priors set off of the discovery papers’ mean and uncertainty values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Then we phase folded the photometric light curves based on the period of that fit, and removed points that were 3σ deviant from a moving average of 20 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This resulted in a few percent of data-points being removed per sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The clipped data was then used for the final Juliet joint RV-transit fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The equilibrium temperature of each planet, Teq, was computed using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 1 from López-Morales & Seager (2007) and assuming each planet had the same atmo- spheric albedo and energy redistribution factors of AB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 and f = 1/3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' To approximate the inso- 4 Science Processing Operations Center (Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016) 5 AB is based on geometric albedos measured for other hot jupiters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Mallonn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Blažek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022), and f is based on the expectation that gas giants with lower incident flux tend to have more efficient heat redistribution (Perez- Becker & Showman 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Komacek & Showman 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Komacek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2017) 6 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' lation levels, Ip, reaching each planet we calculated the bolometric luminosity assuming the stars emit as black- bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' To compensate for this oversimplification, we increased the obtained uncertainties in each Ip by a fac- tor of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Given that we can also obtain information about stellar densities from the transits, we weighted av- eraged the density obtained from the spectral analysis discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 and the density obtained from Juliet to produce the values in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Our updated parameters are on average 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7 times more precise than previous literature discovery parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Figures 2 and 3 show the resulting light curve and radial velocity fits for each planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' HOMOGENEOUS ANALYSIS OF TRANSMISSION SPECTRA Our last step in the process of obtaining as homo- geneous as possible parameters for all seven systems was to re-derive atmospheric parameters for the three planets in the sample with observed transmission spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Those are WASP-6b, observed with VLT/FORS2, HST/STIS G430 and G750, and HST/WFC3 G141 be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='32 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='65 µm (Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020), WASP-96b observed with VLT/FORS2, Magel- lan/IMACS, and HST/WFC3 G102 and G141 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='64 µm (Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Yip et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2022), and WASP- 110b observed with VLT/FORS2 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='83 µm (Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' A retrieval analysis of the WASP-96b data was re- cently done by McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2022), using Platon (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019) and Exoretrievals (Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For consistency, we did a similar analysis for the transmission spectra of WASP-6b and WASP-110b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' That is, with Exoretrievals we tested models includ- ing water, potassium, sodium, stellar activity, or scat- tering features and the different combinations of each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' With Platon we tested models with scatters or stel- lar activity, where Platon assumes equilibrium chem- istry and fits for the C/O ratio and planetary metallic- ity to extrapolate the abundances of atomic/molecular species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' lnZ Bayesian evidences were used to favor one model over another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We considered a difference in lnZ greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 between two models to be moderately significant, and greater than 5 to strongly support the higher lnZ model (Trotta 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Benneke & Seager 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Additionally, we used Table 2 and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2 from Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2019), to limit the contribution of stellar activity to inhomogeneity covering fractions of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 % when both spots and faculae are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The priors of each retrieval for both WASP-6b and WASP110b and the Bayesian evidences relative to a flat (for Exoretrievals) or clear (for Platon) spec- trum are shown in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The highest evidence models for both targets with both retrievals were ones that included activity, which is consistent with what was found in previous analyses (Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Exoretrievals also found significant evidence for water and a sodium fea- ture in the WASP-6b spectrum, which was not found for WASP-110b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The features found in the WASP-6 data are muted, indicative of high altitude aerosols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Activ- ity could not explain the muted features, in fact, with the unocculted cooler spots that the retrievals find, the sodium signal would be enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This can be seen in Figure 9 of Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Furthermore, the cloud deck pressures of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 bars suggest WASP-6b has substantially more aerosols than WASP-96b, where its models favor a cloud deck pressure of ∼ 20 bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-110b’s spectrum is more extreme than WASP- 6b’s, where all atomic features are missing and a cloud deck pressure of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='03 mbars is suggested, albeit not well constrained due to the lack of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' SEARCH FOR TRENDS Using the new set of homogeneously derived parame- ters described in sections 3 and 4, we searched for corre- lations between system parameters and what we define as aerosol levels in the transmission spectrum of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We quantify aerosol levels using as proxy the ampli- tude of the Na I doublet at 5892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9Å in the transmis- sion spectra of WASP-6b, WASP-96b, and WASP-110b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This was calculated as the sum of transit bins within a narrow range centered on the Na I doublet (5862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9 to 5892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9 Å) minus the sum of bins blue-ward and red- ward of this region, while also being outside of the wings of the sodium feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-96b this was 4880 to 5380Å and 6200 to 6700Å, but 5340 to 5820Å and 5960 to 6440Å for WASP-6b and WASP-110b which did not have notable absorption wings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The result of our search for correlations between aerosol levels and system parameters is summarized in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We find a significant correlation between the amplitude of the Na I feature in the transmission spec- trum of the planet and the overall [Fe/H] of the host star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The linear fit to this trend has a Pearson correlation co- efficient of r=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='83, corresponding to a 99% confidence of a found correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' To examine if there could be a cor- relation with stellar activity, we used log10(R’HK) (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The best linear fit to log10(R’HK) and the Na I signal has r = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='28, corresponding to about 24% confidence that there is such a correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' There- fore, we find no correlation with log10(R’HK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' However, Twin Planets 7 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The phase folded transit data for each of the similar seven systems and their Juliet best fit transit models (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 1-σ uncertainties of the fits are shaded in the same color as the transit models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The plotted ’bin’ data is with 30 points binning, aside for WASP-110, which had 8 point binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-6 WASP-25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='990 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='980 TESS OTESs = 392 [ppm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='980 OTEss = 433 [ppm] TESS bin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0025 WASP-55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 + +++++++!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' WASP-96 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9950 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9925 Reli 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9875 OTEss = 495 [ppm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9850 OTESs = 547 [ppm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 WASP-110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 WASP-124 111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 +1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='990 Rel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='985 OTESs = 634 [ppm] OTESs = 723 [ppm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='005 Phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='. 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 HATS-29 xni 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='995 HATS R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='985 OTEss = 762 [ppm] OHATS = 1954 [ppm] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='980 HATSbin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='04 Phase8 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' the chromospheric activity measured from log10(R’HK) is not a direct measurement of total high energy flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' see Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2021), where total high energy emission is likely the more important parameter affecting aerosol formation rates (Moses et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2011, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Fleury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Thus, we need more di- rect measurements of the host stars high energy levels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' HST/UVIS or XMM-Newton observations) to con- fidently rule out such a correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS We have identified seven systems that have very sim- ilar characteristics to one another, the ’Similar Seven’, where the host star metallicity is the only stark differ- ence between the parameters measured for these sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Three of the planets in this sample already have transmission spectra observed, and though they have similar parameters, their transmission spectra have widely varying amounts of high altitude aerosols obscur- ing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' To thoroughly search for correlations be- tween the observed spectra aerosol levels and system parameters, we homogenously reanalyze HARPS and FEROS stellar spectra, and HARPS, CORALIE, and CYCLOPS RV data with TESS and HAT-South transit data to refined the stellar and transit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We found that host star metallicity seems to correlate with the observed aerosol levels, with a 99% confidence that a linear correlation exists, implying that planets around higher metallicity stars would have lower high altitude aerosol levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' If this holds, it could be explained by the requirement of viable seed particles needing to be lofted to high enough altitudes for cloud forming gases to con- dense on (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Helling 2008, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The higher metallicity might cause the formed seed particles to be more dense and subsequently differentiate lower in the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' However, given that the potential trend was found with only the three observed transmis- sion spectra, the correlation is tentative, pending on fur- ther observations of the other sample planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We also use log10(R’HK) as a proxy to explore if high energy irradiation could be correlated to the differences in the transmission spectra, given that there are no di- rect measurements of the stars’ high energy emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We found no clear signs of a correlation with this param- eter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' However, correlation to the host star’s high energy levels may still be present and require more direct mea- surements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' with XMM-Newton and/or HST/UVIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Regardless of if metallicity or high energy irradiation is found to be a contributing factor to high altitude aerosols in these system, the similar seven planets are ideal targets for understanding the unique physical and chemical processes undergoing in these class of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This is because the similarity of most parameters act as a controlled sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This approach of specifically select- ing very similar targets should be a common practice in exoplanet atmosphere studies, and has the potential to isolate key physical or chemical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' STELLAR ROTATION AXIS DISTRIBUTION For WASP-6 there are two measurements of the rotation axis alignment, λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' the angle between the planet’s orbital axis and the rotation axis of the star: λ = 11+14 −18 , obtained using the Rossiter-McLaughlin effect (Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2009) and λ =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7◦ (which we adopt), obtained using occulted star spots (Tregloan-Reed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-25 Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2012) measured a λ of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7◦ via Rossiter-McLaughlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The other five stars in our sample do not have direct λ measurements, so instead we calculated their most likely λ values using the distribution of λ values measured for G-type stars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' with Teff ∈ [5300,6300] K, as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 75% of the λ values are less that ±20◦, suggesting that the bulk of exoplanet systems with G-type host stars are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This is in agreement with the findings of Triaud (2018) (see their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Based on the distribution of values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 5, we adopt a λ = 0 ± 30◦ for the remaning five systems in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The estimated rotation periods for all our targets, computed using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 1, are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' STELLAR ROTATION PERIODS FROM TESS AND ASAS-SN LIGHT CURVES We downloaded the TESS SPOC 6 light curves for each target from MAST 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='17909/fwdt-2x66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The TESS sectors and observed number of transits for each target are summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We also downloaded ASAS-SN (Shappee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2017) time series observations for each target in V- and g-bands, which are treated as separate photometric monitoring campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The number of ASAS-SN observations per photometric band, for each target are also summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 6 Science Processing Operations Center (Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016) Twin Planets 9 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The phase folded RV data of each of the similar seven systems and their Juliet best fit RV models (solid black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The HARPS RV measurements that were during transit were omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-124b, ’CORALIE07’ and ’CORALIE14’ represents observations taken before/after the CORALIE upgrade (see Maxted et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 100 WASP-6 WASP-25 75 K=69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='55±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='66[m/s] 100 K=75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='28±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='61[m/s] P=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3610026±6e-07[days] 50 P=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7648337±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2e-06[days] 25 50 Relative 25 50 50 75 CORALIE, 0 = 13 [m/s] CORALIE, g = 20 [m/sl 100 HARPS, 0 = 11 [m/s] HARPS, = 9[m/s] 198 50 25 25 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 0 0 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 25 25 50 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content="2 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 100 WASP-55 100 WASP-96 K=68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='73±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='22[m/s] 75 P=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='465631±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3e-06[days] K=62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='93[m/s] 50 P=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4252567±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2e-06[days] Flux 50 Relative 25 50 25 ↑ CORALIE, = 14 [m/s] 50 100 HARPS, = 10 [m/s] CORALIE, 0 = 12 [m/s] 75 40 42 20 0 0 20 20 40 40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 WASP-110 WASP-124 100 150 K=73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7[m/s] 100 K=70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='85±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='73[m/s] Flux P=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7784022±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6e-06[days] 50 P=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3726511±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9e-06[days] 50 Relative 50 50 100 ↑ CORALIE07, 0 = 31 [m/s] 150 100 CORALIE, = 25 [m/s] CORALIE14, 0 = 39 [m/s] 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 50 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content="2 0'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 Phase HATS-29 100 K=78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='27±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='64[m/s] Flux P=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6058827±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1e-06[days] 50 Relative [ 50 ↑ HARPS, = 10 [m/s] ↑ CYCLOPS, = 37 [m/s] 100 CORALIE, = 10 [m/s] 50 1 0 d 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4 Phase10 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Left: Sodium amplitude versus host star metallicity from the observed WASP-6b (red diamond), WASP-96b (blue star), and WASP-110b (green circle) transmission spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The best fit linear regression (black dashes) has a Pearson’s correlation coefficient, r-value, of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The metallicty range WASP-25b, WASP-55b, WASP-124b, and HATS-29b cover are plotted as yellow shaded regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Right: Same as left, but with Na amplitude versus log10(R’HK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Here the r-value to the best fit linear regression does not support a correlation of the Na amplitude to log10(R’HK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The distribution of observed λ values of 80 exoplanet systems with the host star between 5300 and 6300K, obtained through TEPCatb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Here λ = 0◦ means the axes are fully aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The bulk of the λ values are smaller than ±20◦ suggesting alignment of the orbital and spin axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Two gaussians are overplotted: one with the wide distribution (σ = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3, black) and one with a narrow distribution (σ = 20, red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We adopt a mean distribution of λ = 0 ± 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' ahttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='keele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='uk/jkt/tepcat/obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='html bhttps://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='keele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='uk/jkt/tepcat/obliquity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='html 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='004 fit:p-value=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='375, fit: p-value=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='817, r-value=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='831 WASP96b r-value=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='283 WASP96b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='002 WASP6b WASP6b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 Na 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 WASP110b WASP110b WASP124b WASP25b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='001 WASP55b HATS29b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content="4 Host star [Fe/H] log1oR'hkμ=-1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='09,=41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='27 μ=0g=20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='020 I counts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='015 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 150 100 50 0 50 100 150 ^[degrees]Twin Planets 11 We used the time series observations above to estimate the rotation period of each star following a similar analysis to the one described in McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For the TESS data, we masked all the in-transit points using the known ephemerides of each planet, to search for photometric modulations of the stars themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Before searching for photometric modulations, we binned the data for each target in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='33-hour bins (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='66-hour bins for WASP-110).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Given that the expected rotation periods for all the stars are longer than 10 days, those binning levels should not affect results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For the ASAS-SN data, we sigma-clipped observations that deviated by more than 3σ from the overall mean of each light curve, weighted averaged all observations for a given night (typically 3 observations per night), and removed observations with uncertainties 3 times larger than the mean uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' To search for the Prot of each star, we jointly modelled all the photometric datasets for each target using Juliet’s (Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018) gaussian processes (GP) semi-periodic kernel, setting the GP characteristic time-scale and period as common terms between all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The other parameters (jitter term, GP amplitude, and GP constant scale term) were specific to the individual datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Also, when data from more than one TESS sector were available, we combined them and modeled them together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We modeled activity as a semi-periodic GP instead of using periodograms because it has been found that peaks non-consistent with the rotational period of stars can appear in the latter (Haywood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Nava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We assume the main driving factor for the GP period is stellar inhomogeneities coming in and out of view as the star rotates, as such we call this the rotation periods, Prot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The priors for Prot were set using the v sin I of each star from section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For all the targets but WASP-124 and HATS-29 we used normal priors with mean and standard deviations near the values derived from v sin I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-124 we used a normal prior truncated at 5 days to prevent sampling of unrealistic low periods driven by the TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For HATS-29 we used a uniform distribution between 17 and 33 days for the same reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The advantage of the TESS data is the continuous monitoring with high photometric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' However, given that it only monitors a sector for about 24 days, the long baseline of the ASAS-SN data complements TESS well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This emphasizes the advantage of a joint fit with all photometric data, but we also run the TESS and ASAS-SN data separately (with the same priors) to outline the contribution from each monitoring source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Table 3 shows the period, amplitude, and median absolute deviations (MAD) of both photometric monitoring sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Finally, to ensure that the found rotation periods were not driven by the sampling of the data we tested their window functions, where we used all timestamps of monitoring data but set the flux and uncertainties to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Doing this suggested no periodic signal due to the observing cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The Prot values for each system are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' HATS-29 TESS LIGHT CURVE DECONTAMINATION Because of the low image resolution of TESS, the light curve of HATS-29 is contaminated by a back- ground RR Lyrae star (see top panel of Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We isolated the RR-Lyrae signal by first excluding the in transit data (using transit parameters from (Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2016)), then applying a Lomb-Scargle periodigram (Lomb 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Scargle 1982) analysis to this out-of-transit data to find a period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='631 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Next, we phase folded the data to the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='631 d period, binned the data by 100 points, and finally smoothed the binned data with scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='savgol_filter (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020) 7 that had a window set to 51 and order of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Our phase folded data and corresponding RR-Lyrae model can be seen in the second panel of Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We then sub- tracted (in magnitude space) the best RR-Lyrae light curve model from the TESS data, and reduced the cor- rected TESS data with the same procedures of the other TESS observations (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 7 We used an older version, ’savitzky_golay,’ which is the same algorithms before it was included in scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' HAT-South (Bakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2013) has public light curves for HATS-29, which we downloaded from the survey’s website 8, and is not contaminated by the background star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We compared the best fit Juliet transit with TESS against a fit with the HAT-South data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Upon confirming that the transit parameters - aside from tran- sit depth, which one would expect to differ due to the different photometric bands - were consistent with each other, we ran a joint Juliet fit with all the transit and RV data to obtain our final planetary parameters of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' See the bottom panel of Figure 6 for an overlay of the TESS and HAT-South data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' ATMOSPHERIC RETRIEVAL RESULTS Table 4 has the priors used for each model and the ∆ ln Z of each retrieval run are in Table 5 We thank the anonymous referee for helpful com- ments to the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' This work has been sup- ported by the National Aeronautics and Space Admin- istration’s Exoplanet Research Program via grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 8 hatsouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='org 12 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Summary of the photometric and RV data Transit data ASAS-SN RV data Star Sectors Transits Filter date range Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Spectrograph Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Source WASP-6 2, 29 13 V 2013-11-25 to 2018-11-26 883 CORALIE 35 Gillon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2009) g 2017-09-16 to 2022-04-18 2073 HARPS 55 Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2020) WASP-25 10 6 V 2012-01-24 to 2018-08-20 820 CORALIE 28 Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2012) g 2017-12-21 to 2022-04-18 2401 HARPS 31 Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2020) WASP-55 10, 37 8 V 2012-02-17 to 2018-08-19 961 CORALIE 20 Hellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2012) g 2017-12-16 to 2022-04-19 2156 HARPS 19 Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2020) WASP-96 2, 29 13 V 2014-04-30 to 2018-09-24 921 CORALIE 21 Hellier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2014)∗ g 2017-09-05 to 2022-02-10 2592 WASP-110 27 6 V 2014-04-29 to 2018-09-24 916 CORALIE 15 Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2014)∗ g 2017-09-05 to 2022-04-18 2286 WASP-124 1 8 V 2014-04-30 to 2018-09-19 1577 CORALIE 39 Maxted et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2016)∗ g 2017-09-07 to 2022-04-18 4217 HATS-29 13 6 V 2014-05-17 to 2018-09-24 989 HARPS 3 Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2016) (HAT-South g 2017-10-03 to 2022-04-18 2165 CYCLOPS 9 Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2016) data:) – 23 CORALIE 4 Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2016) Note—The HAT-South photometric data for HATS-29 (first columns, last row) was acquired from Bakos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Trifonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2020) reanalyzed archival HARPS data using SERVAL (Zechmeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The number of observations is denoted ’Obs.’, and is the unbinned/unclipped observations for the ASAS-SN data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' ∗data obtained through DACE Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Summary of TESS and ASAS-SN photometric monitoring Juliet fits Star TESS ASAS-SN Joint MAD Period Amplitude MADg MADv Period Amplitude Period Amplitude [ppm] [days] [ppm] [ppm] [ppm] [days] [ppm] [days] [ppm] WASP-6 348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='93+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='63 −6.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='07+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='96 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='85 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 Note—The MAD is for the binned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The amplitdues are obtained by using scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='minimize (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020) to fit a sine curve to the data phase folded on the Juliet best fit period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The subscripts g and v on the ASAS-SN MAD correspond to the g and V band filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 20-XRP20_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We thank N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Espinoza for provid- ing access to Exoretrievals, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Blanco-Cuaresma for continuous support using iSpec, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Shkolnik for help- ful discussion regarding GALEX data, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' DiTo- masso for discussion regarding analysis of RV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We also appreciate the support from the NSF Graduate Research Fellowship (GRFP), grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' DGE1745303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' RB and AJ acknowledge support from ANID – Millen- nium Science Initiative – ICN12_009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' AJ acknowledges additional support from FONDECYT project 1210718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' RB acknowledges support from FONDECYT project 11200751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Twin Planets 13 Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The priors for Exoretrievals and PLATON Exoretrievals PLATON parameter function bounds parameter function bounds reference pressure (P0, bars) log-uniform 8 to 3 reference pressure (Pclouds, Pa) log-uniform 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='99 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='99 planetary atmospheric uniform 600 to 1800K planetary atmospheric uniform 600 to 1800K temperature (Tp) temperature (Tp) stellar temperature uniform Teff-240 to Teff+240K stellar temperature gaussian µ=Teff, σ=150K (Tocc) (Tstar) stellar heterogeneities uniform Teff-3000 to Teff+3000K stellar heterogeneities uniform Teff-3000 to Teff+3000K temperature (Thet) temperature (Tspot) heterogeneity covering gaussian µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='041, σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='041 heterogeneity covering gaussian µ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='041, σ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='041 fraction (fhet) fraction (fspot) haze amplitude (a) log-uniform 30 to 30 scattering factor log-uniform 10 to 10 haze power law (γ) uniform 14 to 4 scattering slope (α) uniform 4 to 14 log cloud absorbing uniform 80 to 80 metallicity (Z/Z⊙) log-uniform 1 to 3 cross-section (σcloud) trace molecules’ log-uniform 30 to 0 C/O uniform 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='05 to 2 mixing ratios reference radius factor(f) uniform 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='8 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 1 bar, reference radius (R0) uniform Rp-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2Rp to Rp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2Rp Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' These priors were set to allow for a wide parameter space to be surveyed, but contained within physical regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Not all parameters were included in each model fit (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' We used 5000 live points for all runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For further description of the parameters of Exoretrievals, please refer to the Appendix D of Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Teff is the effective temperature of the host star, which is 5438K and 5392K for WASP-6 and WASP-110, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' γ is the exponent of the scattering slope power law, where −4 is a Rayleigh scattering slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' α is the wavelength dependence of scattering, with 4 being Rayleigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' f is a factor multiplied by the inputted planetary radius to produce the reference radius, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' R0 = fRp, Rp is the radius of the planet, corresponding to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='119Rj and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='177Rj for WASP-6b and WASP-110b, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Facilities: Magellan:Baade (IMACS), Smithso- nian Institution High Performance Cluster (SI/HPC), HST(STIS/WFC3), All-Sky Automated Survey for Su- pernovae (ASAS-SN), VLT(FORS2),TESS, ESO La Silla 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6m (HARPS), Swiss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2-metre Leonhard Eu- ler Telescope (CORALIE), Anglo-Australian Telescope (CYCLOPS), MPG/ESO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2(FEROS), and Gaia space- craft Software: Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2013), corner (Foreman-Mackey 2016), Matplotlib (Hunter 2007), NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020), Multinest (Feroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2009), PyMultiNest (Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2014), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2020), batman (Kreidberg 2015), george (Ambikasaran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2015) dynesty (Speagle 2020), Platon (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019), Juliet (Espinoza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2018), 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2013, PASP, 125, 154 14 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Top: The TESS SPOC of TIC 201604954, extracted from MAST 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='17909/fwdt-2x66, which was observed in sector 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' From visual inspection one can see that the RR-Lyrae oscillations dominate, with 43 complete oscillations observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Middle: The same data after removing the HATS-29b transits and phase folding on a period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='631343 days, which corresponds to the background RR-Lyrae’s period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Here the black are all phase folded observations, the blue is the data binned by 100, and the red is the savitzky_golay smoothed fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Bottom: The TESS data after modeling out the RR-Lyrae features, doing the same sigma clipping done for every other target (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Here cyan is the unbinned data and the blue is the same data binned by 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Overplotted is the HAT-South data, where the magenta and red are the unbinned and binned data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For both light curves we used a P and t0 of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='60588 and 2457031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='957, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 Relative Flux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='97 55 60 65 70 75 80 BJD - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='4586e6 (days)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='99 Mag Relative I 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='01 RR-Lyare fit 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='02 binned data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='50 Phase1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='015 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='005 Flux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='000 lative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='995 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='990 R TESS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='985 HATSouth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2011, arXiv e-prints, arXiv:1107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='5325 Mallonn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=', Köhler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=', Alexoudi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' 2019, A&A, 624, A62 16 McGruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' ∆ln Z for Exoretrievals (left) and PLATON (right) retrievals Exoretrievals PLATON Model: flat H2O Na K H2O + Na H2O + K + Na Model: WASP-6b: clear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='79 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='32 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='26 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='7 clear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 scatterers − − − 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='96 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='14 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='95 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='9 scattering 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='09 activity 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='21 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='85 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='03 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='53 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='07 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='44 activity 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='49 Both − − − 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='35 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='59 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='49 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='97 Both 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='91 WASP-110b: clear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='57 clear 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 haze+clouds − − − −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='21 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='09 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='12 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='46 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='6 scattering 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='93 activity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='39 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='73 activity 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='08 Both − − − −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='43 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='0 Both 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content='15 Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The ∆ln Z values are relative to a clear (and flat for Exoretrievals’s case) spectrum with the WASP-6b (top) spectrum that included the VLT/FORS2, HST/STIS, and HST/WFC3 data, and the WASP-110b (bottom) spectrum consisting of the VLT/FORS2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-6b the retrievals with water and sodium were heavily supported by Exoretrievals, where including potassium did not make a significant difference in ∆ln Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' The PLATON models that included scattering and activity were equally supported as the models with just activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' For WASP-110b the models with activity were supported, with Exoretrievals finding no significant contribution from atomic/molecular species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' Twin Planets 17 Marley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=', Ackerman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} +page_content=', Cuzzi, J.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE2T4oBgHgl3EQf3wh7/content/2301.04174v1.pdf'} diff --git a/z9AzT4oBgHgl3EQfRPsg/content/tmp_files/2301.01211v1.pdf.txt b/z9AzT4oBgHgl3EQfRPsg/content/tmp_files/2301.01211v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb7c728ceb8f62b4ca9a73a065e4e1a3d8490dea --- /dev/null +++ b/z9AzT4oBgHgl3EQfRPsg/content/tmp_files/2301.01211v1.pdf.txt @@ -0,0 +1,2172 @@ +Generative appearance replay for continual +unsupervised domain adaptation +Boqi Chen1,2*, Kevin Thandiackal1,2*, Pushpak Pati1 and Orcun Goksel2,3 +1IBM Research Europe, Zurich, Switzerland +2Computer-assisted Applications in Medicine, ETH Zurich, Zurich, Switzerland +3Department of Information Technology, Uppsala University, Uppsala, Sweden +Abstract +Deep learning models can achieve high accuracy when trained on large +amounts of labeled data. However, real-world scenarios often involve several +challenges: Training data may become available in installments, may origi- +nate from multiple different domains, and may not contain labels for training. +Certain settings, for instance medical applications, often involve further +restrictions that prohibit retention of previously seen data due to privacy +regulations. In this work, to address such challenges, we study unsupervised +segmentation in continual learning scenarios that involve domain shift. To +that end, we introduce GarDA (Generative Appearance Replay for continual +Domain Adaptation), a generative-replay based approach that can adapt a +segmentation model sequentially to new domains with unlabeled data. In +contrast to single-step unsupervised domain adaptation (UDA), continual +adaptation to a sequence of domains enables leveraging and consolidation of +information from multiple domains. Unlike previous approaches in incremental +UDA, our method does not require access to previously seen data, making it +applicable in many practical scenarios. We evaluate GarDA on two datasets +with different organs and modalities, where it substantially outperforms +existing techniques. +Keywords +Unsupervised domain adaptation, Continual learning, Optic disc segmen- +tation, Cardiac segmentation +1. Introduction +Deep Neural Networks (DNNs) have recently achieved +remarkable performance on various computer vision tasks with +natural images, such as classification [1], [2], [3] and semantic +segmentation [4], [5]. However, there exist several challenges +that hinder DNNs from achieving similar success in other +domains, e.g., healthcare. First, to achieve high-performance, +DNNs require large amounts of labeled training images, +which are challenging to obtain for medical applications, +since annotations can only be provided by medical experts. +Annotating medical images is therefore more costly compared +to annotating natural images. This is particularly critical for +applications requiring dense annotations, such as semantic +segmentation. Thus, there is a strong need for unsupervised +DNN approaches in healthcare. Second, medical datasets usu- +ally contain a relatively small number of images [6], [7], [8], +[9] compared to large-scale natural image datasets, such as +ImageNet [10]. Models trained on such small datasets often +*The authors contributed equally to this work. +do not generalize well to unseen domains [11]. Unsupervised +Domain Adaptation (UDA) [12], [13] aims to address both +aforementioned challenges, by first training a model on a +labeled source domain D0 and afterwards adapting it to an +unlabeled target domain D1, to alleviate annotation costs. +Real-world settings are often more complex than the above +scenario, and require long-term applicable solutions rather +than adapting to a single target domain alone. It is, for +example, typical that multiple datasets from different domains +D1, ..., DT become available at different time points. In such a +scenario, it is beneficial to keep a single model that is sequen- +tially adapted to these different domains, thereby consolidating +useful knowledge from more than one domain. Since patient +data is bound to strict privacy regulations, data sharing and +indefinite data storage are often not possible. Therefore such +medical scenarios are further complicated by the inability +to keep and access data from previously seen domains. In +the literature, this constrained learning problem is known as +continual learning (CL). In this work, we study a continual +UDA setting, where a model is adapted to a sequence of target +domains, the data of which is available only while training on +that respective domain for the first time. +The main challenges in continual UDA are twofold. First, +the model should not suffer from catastrophic forgetting [14], +[15], i.e., it should maintain its performance on the source +domain as well as all previously seen target domains. Second, +without access to the source or previous target domains, +the model should adapt accurately to new target domains. +Although strict continual UDA has been largely unexplored in +the literature, some recent techniques have studied a scenario +with looser constraints, which we herein term as incremental +UDA. In contrast to continual UDA, in incremental UDA the +data from the source domain is stored and reused during the +introduction of each subsequent target domain, as illustrated +in Fig. 1. +As an incremental UDA method, [16] proposed Multi-Head +Distillation (MuHDi), which performs knowledge distillation +using either the source images (which are kept at all times) or +the images from the current target domain. This is sub-optimal +since knowledge from previously seen target domains cannot +be preserved effectively. Another method, more closely related +to our solution, is ACE (Adapting to Changing Environ- +ments) [17], which builds on the idea of transferring the style +arXiv:2301.01211v1 [cs.CV] 3 Jan 2023 + +... +... +Incremental UDA +... +... +Continual UDA +Fig. 1: Overview of incremental (top) and continual (bottom) +UDA workflows. In the latter, the segmentation model M +sequentially adapts to new domains without storing data from +any previously seen domain, including D0. +of unlabeled target images to known labeled source images +(which then again have to be kept indefinitely). To address +forgetting, ACE generates images with the same content, i.e., +segmentation ground truth, as a given source image, but the +style of a target image. In addition, ACE employs a memory +unit to replay images in previously seen target styles to +the segmentation model. However, as for MuHDi, ACE also +violates a strict continual UDA constraint, since it relies on +future access to the source data. Furthermore, ACE can only +replay the previous target styles that have been stored in the +memory unit, hence lacking the ability to sample those for +replay from the entire distribution of previously seen target +styles. This in turn affects the knowledge preservation from +these intermediate targets. +In this work, we address the above-mentioned shortcomings +by proposing GarDA (Generative Appearance Replay for con- +tinual Domain Adaptation). We employ generative replay [18], +[19], [20], [21] to substitute images from any previously-seen +but currently-inaccessible domain with synthetic images that +are generated via a generative adversarial network (GAN) [22]. +This permits operation under a strict continual UDA setting, +i.e., even when the source and the intermediate target domains +become inaccessible after training on each. Importantly, for +replay we can arbitrarily sample images of any seen domain +via our generator, and are not limited by less expressive and +fixed memory buffers, as in ACE. Furthermore, thanks to our +powerful GAN-based approach for transferring the appearance +of target domain images onto source domain images, we can +generate more representative samples for replay, achieving +more effective knowledge preservation. +In summary, our main contributions are three-fold: (1) We +propose GarDA as the first segmentation technique that oper- +ates under a strict continual UDA setting, i.e., it mitigates +forgetting of the source and intermediate target domains +without storing any previously seen images. (2) We introduce +a novel combination of appearance transfer and generative +replay using a stochastic generator, which enables diverse +sampling from the entire distribution of all domains, and +is thus less prone to overfitting compared to methods em- +ploying separate memory buffers. (3) We achieve the new +state-of-the-art performance in continual UDA, demonstrated +herein through comprehensive benchmarking with two target +domains. To demonstrate the generalizability of GarDA, we +conduct experiments for two entirely different tasks: optic disc +segmentation in color fundus photography (CFP) images and +cardiac segmentation in magnetic resonance (MR) images. +2. Related work +Our work lies at the intersection of UDA and continual +learning, for which we provide below a brief overview with +relevant research. +2.1. Continual learning +Continual learning aims to simulate real-world data avail- +ability constraints in sequential learning, i.e., a model should +learn from a sequence of datasets without retaining any pre- +viously seen data and without forgetting previously acquired +knowledge [23], [15], [24], [25]. The research in continual +learning can be categorized into four major dimensions fo- +cusing on regularization [26], [27], [28], parameter isola- +tion [29], [30], [31], prototypical representations [32], [33], +[34], and generative replay [18], [19], [35], [36], [37], [38], +[20], [21]. Methods built on prototypical representations as +well as generative replay have been the most successful. The +former aims to store class-representative prototypes in feature +space to compensate any estimated drift when learning new +tasks. In the latter, a generative model learns to synthesize +images [18], [35], [36], [20], [21] or features [19], [37], [38] +of previously seen data, such that they can be replayed to +the classification/segmentation model when the original data +is not available anymore. While these methods have achieved +state-of-the-art performance on class- and task-incremental +benchmarks, they have not been employed in the context of +continual UDA, to the best of our knowledge. +2.2. UDA +Over the last years, a myriad of methods have been proposed +to tackle UDA for both classification and segmentation [12], +[13], with their common paradigm often being the domain +alignment. This aims to bring the source and target domain +representations closer, in terms of either the input images, +certain intermediate features, or the model outputs. If this is +done effectively, the classification/segmentation models trained +on the source domain are expected to generalize to the target +domain. [39] were the first to propose a Domain-Adversarial +Neural Network (DANN), where the main model’s feature +extractor uses the negative gradient of a domain classifier to +obtain domain-invariant features. Similarly, other approaches + +TABLE 1: Taxonomy of different domain adaptation settings. +Setting +Source +unavailable +after training +Multiple +target +domains +Unlabeled +target +domain(s) +Unsup. Domain Adaptation (UDA) + + + +Domain-Incremental Learning (DIL) + + + +Source-free UDA / UMA + + + +Incremental UDA + + + +Continual UDA + + + +have adopted GAN-inspired adversarial training for feature +alignment [40], [41], [42], which has been effective also for +the segmentation of medical images such as MR and computer +tomography (CT) images [43], [44]. Other methods such as +AdaptSegNet [45] or AdvEnt [46] also employ adversarial +losses, but these aim at aligning the model output spaces of +source and target domains. Alternatively, CycleGAN [47] aims +to align domains in the image space. Cycle-Consistent Ad- +versarial Domain Adaptation (CyCADA) [48] and Synergistic +Image and Feature Alignment (SIFA) [49] propose to unify +feature-level and image-level domain alignment for UDA. +All the aforementioned UDA methods were designed to +adapt models from a single source domain to a single target +domain, and they do not operate in continual (or incremental) +learning settings. In this work, our goal is to tackle continual +learning settings, for reasons motivated earlier including the +limitations of access to medical data. Continual UDA, with +a sequence of multiple target domains, introduces additional +challenges to single-domain UDA. +2.3. Sequential UDA +In the literature, UDA for a sequence of multiple target +domains has been studied under different settings. We herein +group them collectively under the umbrella of sequential UDA, +and propose the taxonomy presented in Table 1 to clarify +the differences among the individual settings. We denote the +problem of learning multiple labeled target domains without +retaining previously seen data as domain-incremental learning +(DIL) [50], [51], [52], [53], [54], [55], [56], [57]. Another +sub-category called source-free UDA – sometimes referred +to as unsupervised model adaptation (UMA) – has recently +gained increasing attention for both classification [58], [59], +[60], [61] and segmentation tasks [62], [63], [64], [65], [66]. +These methods perform UDA from a source to a single target +domain without using the actual source data. Note that in +the literature, source-free and continual UDA have sometimes +been used interchangeably. Instead, we propose to separate +these terms as shown in Table 1, since source-free UDA +aims for adaptation to one target domain only, therefore not +having to consider resilience during further target adaptations +in the future. Naming this setting as continual UDA would +be indeed inconsistent with the remaining continual learning +literature [15], [25] that deals with scenarios involving multiple +classes/tasks in a sequence. +In contrast to the above, incremental UDA aims at adapting +models to multiple target domains in an unsupervised manner. +For instance, MuHDi [16] employs domain adversarial training +for adaptation, together with multiple distillation losses to +mitigate forgetting. As mentioned earlier, knowledge distilla- +tion is sub-optimal when enforced using the source images +and/or the current target images, as these do not directly +prevent forgetting of the previously seen target domains. Wu +et al. [17] proposed ACE, which uses a deterministic encoder- +decoder architecture to create images in a target domain +style for replay. To tackle forgetting, ACE employs a replay +memory buffer containing style information of previously +seen target domains. Although this can mitigate forgetting +to some extent, the memory buffer can only hold the style +information of few samples, thus not representative of the +complete distribution of all previous target domains/images. In +addition, both MuHDi and ACE require access to source data +throughout the adaptation sequence, making them incremental +UDA in our taxonomy. +In this work, we study continual UDA, a more challenging +scenario compared to those aforementioned: The source data +is available only once at the beginning of the sequence for +training on the source domain. Afterwards, the model adapts +to a sequence of unlabeled target domains without storing data +from any previous domain. +3. Method +In this section, we present our proposed framework for +continual UDA. A segmentation model M is first trained +on images x0 ∈ X0 ⊆ RH×W ×C with segmentation label +maps y0 ∈ Y0 ⊆ {1, ..., L}H×W from the source domain +D0 = {(x0, y0)(i)}N0 +i=1, where N0, L, C, H, and W denote the +number of images, labels, image channels, image height, and +image width, respectively. Following training on the source, +D0 is afterwards not accessible anymore while adapting M +to a sequence of T ≥ 1 target domains D1:T = {X1, ..., XT } +which are lacking labels. In other words, while training on +the t-th target domain, we only have access to the unlabeled +images from the current domain, i.e., xt ∈ Dt, but we cannot +access samples from the source or previously encountered +target domains, i.e., D0:t−1. We substitute the missing D0:t−1 +with synthetic data produced by a GAN and replay this data +to M to prevent forgetting. As new target domains are added, +the GAN (consisting of a generator G and a discriminator +D) and the segmentation model M are trained alternately, as +illustrated in Figure 2. +First, M and the GAN are trained independently on the +source domain (Section 3.1). Next, G is trained to synthesize +images with the appearance characteristics of a new target +domain (Figure 2(a)) while remembering how to generate +images of previous domains (Figure 2(b)). This training phase +is described in Section 3.2 and Section 3.3, respectively. From +such a generator G, at any point we can create pairs of +synthetic samples with the same content but with separate +appearances: one of the source domain and one of a target + +(a) Adapt +(c) Adapt +trainable +frozen +Generator +Discriminator +Feature +extractor +Segmentation +model +Feature map +Adapted +feature map +(b) Distill +real/fake +real/fake +Train +Train +... +Adapt + +Distill +Adapt +... +Fig. 2: Overview of our method including (a) GAN adaptation (b) GAN distillation and (c) adaptation of the segmentation +model. +domain. We can then use these pairs to adapt the segmentation +model, by distilling the outputs of the previously trained model +Mp into the current model M, as seen in Figure 2(c). In +other words, the new M shall predict the same segmentation +map for a target-like image as the map predicted by Mp for +the corresponding source-like image. This step is described +in Section 3.4. The above process is repeated whenever data +from a new target domain becomes available. +3.1. Source training +Initially, we train a segmentation model M and a GAN (G, +D) separately on the source domain data. For an input image +x, M predicts a segmentation map. Specifically, we denote +the output logit of M for label l at pixel position (h, w) as +M (h,w,l)(x). The corresponding probability of belonging to +label l is computed via a softmax operation, i.e., +m(h,w,l)(x) = +eM (h,w,l)(x) +�L +j=1 eM (h,w,j)(x) . +(1) +Given a source image and segmentation label map (x, y) ∈ +D0, M is trained by minimizing the cross-entropy loss +LM = − +1 +HW +H +� +h=1 +W +� +w=1 +L +� +l=1 +1{y(h,w)=l}log +� +m(h,w,l)(x) +� +(2) +where 1{y(h,w)=l} denotes an indicator function that is 1 if +y(h,w)=l and 0 otherwise. +For the generative replay model, we employ a conditional +GAN consisting of a generator G and a projection discrimi- +nator D [67]. Given a randomly sampled noise vector z (that +aims to condition the image content) and a domain label τ +(for the corresponding appearance), G : (τ, z) �→ xτ|z learns +to generate synthetic samples xτ|z of domain Dτ. G is trained +by optimizing a non-saturating logistic loss [22] +LG = +E +z∼pz +� +a +� +− D +� +G(0, z) +� +, 0 +�� +, +(3) +where a(·) is the softplus operation. During training, D +counteracts G by trying to distinguish synthetic from real +images, by minimizing +LD = E +z∼pz +� +a +� +D +� +G(0, z), 0 +��� ++ +E +x∼D0 +� +a +� +− D +� +0, x +��� ++ λR1R1 , +(4) +where R1 is the gradient penalty term Ex∼D0 ∥∇xD(0, x)∥2 +2 +[68], which is only computed for real images and weighted +with λR1. + +3.2. GAN adaptation +The main challenge following the source training is the +unavailability of labeled target domain images, which prevents +the trivial solution of fine-tuning M on the target domain. +However, the challenge can be addressed, e.g., if a mechanism +could create (pseudo-)labeled target domain images. To this +end, we propose to adapt our GAN to the target domain such +that synthetic images can be sampled with content similar to +the source and appearance similar to any target. +As presented in Figure 2(a), for a new target domain Dt, we +freeze the previously trained generator G0:t−1 and instantiate +a trainable generator Gt. For simplicity, we denote Gt being +trained as G, and frozen G0:t−1 for previous domains as Gp. +We initialize G with the weights of Gp. +We employ a discriminator D that operates on feature repre- +sentations extracted using a frozen pretrained feature extractor +F, i.e., instead of distinguishing between real and fake images, +our D distinguishes their feature representations. Accordingly, +we first randomly sample a source-like image xp +0|z=Gp(0, z) +from Gp and a real image xt from the current dataset Dt. +These images are then passed through F, yielding feature +maps f p +0|z=F(xp +0|z), and ft=F(xt). We now aim to create a +target-like image xt|z=G(t, z) with features ft|z=F(xt|z) that +incorporates the content of xp +0|z and the appearance of xt. In +order to train G, we first need to define an objective, i.e., what +the synthesized features ft|z should look like. +Inspired by [69], [17], we leverage the idea behind adaptive +instance normalization (AdaIN) that the instance-level statis- +tics (e.g., mean and variance) of a feature map represent the +appearance of the corresponding image. Using AdaIN, we can +re-normalize the instance-level statistics of f p +0|z to those of ft +and obtain a new feature map ˜f p +t|z that encodes the content of +xp +0|z and the appearance of xt. The said re-normalization is +computed as +˜f p +t|z =AdaIN +� +f p +0|z, ft +� +=σ(ft) +� +� +f p +0|z − µ +� +f p +0|z +� +σ +� +f p +0|z +� +� +�+ µ(ft), +(5) +where µ(·) and σ(·) denote channel-wise mean and standard +deviation, respectively. During GAN training, discriminator D +is trained to identify ˜f p +t|z as real and ft|z as fake. To this end, +D optimizes the following objective: +Luda +D += E +z∼pz +� +a +� +D +� +t, ft|z +��� ++ +E +z∼pz +xt∼Dt +� +a +� +−D +� +t,˜f p +t|z +�� � ++ λR1R1 . +(6) +Meanwhile, G tries to confuse D while adapting to the new +domain by minimizing +Luda +G += +E +z∈pz +� +a +� +− D +� +t, ft|z +� �� ++ λconLcon , +(7) +where the first term encourages G to generate images with +realistic target appearance and the second term is a content +loss. The purpose of the content loss is for G to generate +two images with different appearance characteristics, but the +same content, when given the same z and two different domain +labels as input. That means that in the feature space, the only +difference between ft|z and f p +0|z should be the appearance +information, which we estimate through channel-wise mean +and variance. Thus, we define a content loss as the difference +between the synthesized target features after re-normalization +˜f0|z=AdaIN(ft|z, f p +0|z), and the source features f p +0|z, i.e., +Lcon = +H +� +h=1 +W +� +w=1 +C +� +h=c +��� f p (h,w,c) +0|z +− ˜f (h,w,c) +0|z +��� +2 +2 , +(8) +where H, W, and C are the height, width, and channels of +the feature maps. +3.3. GAN distillation +For continual UDA, in addition to adapting to a new target +domain, the generator G shall not forget any previously learned +knowledge, i.e., G should be able to generate data from +all previous domains D0:t−1. To counteract forgetting, we +introduce two distillation losses: Ldis +D and Ldis +G . As illustrated +in Figure 2(b), we sample uniformly from the previous do- +mains τ ∼ U{0, t − 1} and random noise vectors z ∼ pz, +and feed these both to the previous generator Gp and the +current generator G. For distillation, we treat the images +generated by the previous generator xp +τ|z = Gp(τ, z) as +real and the corresponding images generated by the current +generator xτ|z = G(τ, z) as fake. The distillation loss for the +discriminator is +Ldis +D = +E +z∼pz +τ∼U{0,t−1} +� +a +� +D +� +F +� +G(τ, z) +� +, τ +�� ++ a +� +−D +� +F +� +Gp(τ, z) +� +, τ +��� +(9) +and for the generator +Ldis +G = +Ladv +G +� +�� +� +E +z∼pz +τ∼U{0,t−1} +� +a +� +−D +� +F +� +G(τ, z) +� +, τ +��� ++ λimg +E +z∼pz +τ∼U{0,t−1} +� +∥G(τ, z) − Gp(τ, z)∥1 +� +� +�� +� +Limg +G +. +(10) +The first term Ladv +G +in Equation (10) is an adversarial (adv) +component evaluated at the feature level, while the second +term Limg +G +is an ℓ1 image (img) distillation loss at the image +level weighted by λimg. The combination of such image- and +feature-level distillation has been empirically demonstrated to +be effective for other continual learning tasks, e.g., class- +incremental learning [21]. +In practice, GAN adaptation and distillation (Figure 2(a&b)) +are optimized simultaneously by splitting a training batch into + +two halves: one for adaptation to a new domain, and the other +for distillation of previous domains. This training strategy is +more efficient as both the objectives are minimized in the same +forward pass. Then, the final objectives for D and G become +LD = Luda +D ++ Ldis +D +and +LG = Luda +G ++ Ldis +G . +(11) +3.4. Segmentation training +Subsequent to adapting the GAN, the segmentation model +M is trained to perform segmentation also in the new target +domain Dt (see Figure 2(c)). When adapting to Dt, we freeze +the previously trained model, called Mp hereafter, that has +seen D0:t−1 and instantiate a trainable model M initialized +with the weights of Mp. M is trained in a supervised manner +with images synthesized by G and corresponding pseudo- +labels predicted by Mp. To this end, we employ the concept of +knowledge distillation [70], i.e., the current model M is trained +to make the same predictions as the previous model Mp. Since +labeled data is only available during the source training, as +described in Section 3.1, we assume that the predictions of +Mp on the source images are the most accurate compared to +images from other target domains. For knowledge distillation, +we first pass a generated source image x0|z = G(0, z) to Mp +and obtain the pseudo segmentation map ˆm0 at each position +(h, w) as +ˆm(h,w,l) +0 += +eM (h,w,l) +p +(x0|z)/κ +�L +j=1 eM (h,w,j) +p +(x0|z)/κ , +(12) +where κ is a temperature parameter that we set to 2, as +commonly used in the knowledge distillation literature [70]. +This produces a softer probability distribution and encodes +more fine-grained information compared to one-hot labels. +For the same noise input z, a generated source image x0|z +and an image xτ|z = G(τ, z) of any seen domain Dτ∈{0,...,t} +are expected to have the same content, i.e., the same ground- +truth segmentation map. We can thus utilize ˆm0 as a (silver- +standard/approximate) ground truth for xτ|z. Similarly as +in Equation (2), we then train the segmentation model M +with the pseudo-labeled samples (xτ|z, ˆm0) by minimizing the +cross-entropy loss +LM = − +1 +HW +H +� +h=1 +W +� +w=1 +L +� +l=1 +ˆm(h,w,l) +0 +log +� +m(h,w,l) +τ +� +(13) +m(h,w,l) +τ += +eM (h,w,l)(xτ|z)/κ +�L +j=1 eM (h,w,j)(xτ|z)/κ . +(14) +Since the images xτ|z are sampled uniformly from all +domains including both the previous ones and the current one, +M is trained to generalize across all of them, i.e., it adapts to +the new target domain Dt while retaining performance on the +previously seen domains D0:t−1. +TABLE 2: Summary of the datasets used in our continual +UDA experiments. +Task +Domain +Device +vendor +Training +images +Test +images +ODS +D0 +Canon +640 +160 +D1 +Kowa +54 +27 +D2 +Nidek +278 +70 +CS +D0 +Philips +58 +16 +D1 +Siemens +76 +19 +D2 +Canon +40 +10 +4. Experiments +We evaluate GarDA on two different tasks, namely optic +disc segmentation in CFP images and cardiac segmentation in +MR images. Below, we first describe the employed datasets, +preprocessing steps, and implementation details in Secs. 4.1 +to 4.3. Then, we present comparisons to the state of the art +in Secs. 4.5 and 4.6, and discuss further results and ablation +studies in Sec. 4.7. +4.1. Optic disc segmentation (ODS) +The task of segmenting the optic disc is challenging and +crucial for the clinical detection of glaucoma [71]. To evaluate +our method on this task, we consider three public CFP +image datasets, i.e., the Retinal Fundus Glaucoma Challenge +REFUGE [72], the Indian Diabetic Retinopathy Image Dataset +IDRiD [6], and the Retinal Image database for Optic Nerve +Evaluation for Deep Learning RIM-ONE DL [7]. The datasets +were acquired from multiple centers and countries, using +different scanners. Therefore, they allow to create a realistic +and representative benchmark for continual UDA. +To construct a continual sequence of domains, we select +subsets of the aforementioned datasets where each domain +is represented by images acquired with a different camera. +We consider as labeled source domain D0 the set of images +from REFUGE that were acquired with a Canon CR-2 camera. +For the first unlabeled target domain D1, we use all images +from IDRiD, which were captured by a Kowa VX-10α digital +fundus camera. For a second unlabeled target domain D2, we +select the subset of images from RIM-ONE DL, which were +taken by a Nidek AFC-210 non-mydriatic fundus camera. The +samples from each domain/camera are then randomly split into +80% training images and 20% test images, as summarized +in Tab. 2. Note that we only employ a validation set (25% of +the training set) during source segmentation training, since for +the training on the target domains only generated images are +used. We follow the preprocessing pipeline employed in [73] +by first locating the optic disc center using a simple boundary +detection algorithm [74]. Then, the images are cropped to a +size of 640×640 pixels around the estimated optic disc center +before being further resized to 256 × 256 pixels. + +4.2. Cardiac segmentation (CS) +We also benchmark our method for CS in MR images, +where we segment three cardiac regions: the left ventricle (LV) +and the right ventricle (RV) cavities, and the left ventricle my- +ocardium (MYO). Note that the CS task provides an evaluation +setting different from ODS, as it contains higher dynamic- +range, grayscale images and aims at multi-class segmentation. +For CS, we use data from the Multi-Centre, Multi-Vendor +and Multi-Disease Cardiac Segmentation (M&Ms) challenge +[8]. It contains volumes from multiple hospitals in Spain and +Germany, acquired by 1.5T scanners of different vendors, +namely Philips, Siemens, and Canon. We select the labeled +subset of images obtained from the Philips scanner as the +source dataset, D0. The unlabeled target domains D1 and +D2 consist of all images acquired using Siemens and Canon +scanners, respectively. Similarly to ODS, we split the data +from each domain into 80% for training and 20% for testing, +where 25% of the training data is used as validation only +during the source segmentation training. The exact data distri- +bution is shown in Tab. 2, where “image” denotes a volume, +i.e., a set of multiple sequential 2D slices. Note that while +our model is trained on 2D slices, the evaluation is run in +3D. That is, at test time after 2D inference, the numbers of +correctly/falsely classified pixels from all 2D slices in a 3D +volume are aggregated to compute a 3D Dice score. +We preprocess the images by employing bias field cor- +rection [75] and min-max intensity normalization where the +minimum and maximum values are the 5th and 95th intensity +percentiles, respectively. In addition, we resample the 2D +slices to an in-plane resolution of 1.2 × 1.2 mm, before +cropping or padding them to a fixed size of 256 × 256 pixels. +4.3. Implementation details +We use a U-Net [76] segmentation model for GarDA and +all the competing methods. Both the encoder and the decoder +consist of five convolutional blocks. Skip connections are +added between each block of the encoder and the correspond- +ing block in the decoder. We employ the stochastic gradient +descent (SGD) optimizer with a learning rate of 0.0005. +Throughout all experimental settings, we use a batch size of +16 and a weight decay of 0.0005. Source domain training +is performed on all available real samples, whereas during +the introduction of target domains, arbitrarily many images +can be sampled from the generator for training. Therefore, +we measure the training duration for the source domain in +number of epochs (i.e., passes through the entire dataset) and +for adding target domains in number of iterations (i.e., batches +sampled from G). In particular, the model is trained for 100 +epochs on the source domain and for 500 iterations for every +target domain addition. +For the generative module, a conditional GAN with a +projection discriminator D [67] is used. The discriminator +incorporates a mini-batch discrimination layer [77] for bet- +ter sample diversity. Note that two types of discriminators +TABLE 3: Summary of hyperparameter optimization ranges +for each method. +Method +Parameter Values +Description +GarDA +λcon +{1, 10} +Content loss +λimg +{1, 10} +Image distillation +λR1 +{0.2, 2} +R1 regularization +AdaptSegNet +[45] +λcp +{1, 10, 20, 50} +Class prior loss +λseg +{1, 10, 20, 50} +Segmentation loss +AdvEnt +[46] +λcp +{1, 10, 20, 50} +Class prior loss +λseg +{1, 10, 20, 50} +Segmentation loss +MuHDi +[16] +λcp +{1, 10, 20, 50} +Class prior loss +λseg +{1, 10, 20, 50} +Segmentation loss +λFD +{0.01, 0.1, 1} +Feature distillation +λdd +{1, 10, 20, 50} +Distribution distillation +λprev +{0.1, 0.2, 0.5} +Distribution distillation +from previous model +ACE +[17] +λstyle +{0.1, 0.5, 1, 5} Style loss +λcont +{0.1, 1} +Content loss +λKL +{0.1, 1} +KL divergence loss +are utilized in GarDA. For the source domain training, the +discriminator distinguishes between real and fake images, +whereas during adaptation to target adaptation domains, D +discriminates features extracted by F. For the source training +on images, we use a D architecture with seven convolutional +layers. For target adaptation training, a smaller four-layer +backbone is used due to the lower dimensions of the feature +space. To extract these features, we follow previous work [69] +and utilize a frozen, pretrained VGG19 [78] architecture for +F. +Our generator G comprises two linear layers followed by +nine style-convolution layers [79] consisting of style modula- +tion, convolution, and noise injection. Besides, we compute an +exponential moving average (EMA) [80] of G’s parameters θ +over the training iterations n, i.e., θ(n) +EMA = βθ(n−1) +EMA + (1 − +β)θ(n) with θ(0) +EMA = θ(0) and β = 0.999. At the end of +the training, we construct a copy of G from these averaged +parameters, which is afterwards used to generate samples +for the training of the segmentation model. We found such +averaging to increase the robustness of G to avoid ending the +GAN training in a potentially poor local minimum. Both G +and D are trained with the Adam optimizer and an equalized +learning rate [77] of 0.0025. During training, images are +augmented with horizontal and vertical flips as well as random +rotations (90◦/180◦/270◦). A summary of the loss weights +used for the content loss, image distillation, as well as the +R1 regularization in the discriminator is shown in Tab. 3. +4.4. Comparisons with the state of the art +We compare GarDA to two established non-sequential UDA +methods, i.e., AdaptSegNet [45] and AdvEnt [46], and two +incremental UDA methods, i.e., ACE [17] and MuHDi [16]. +To enable a fair comparison, we adapt these methods to operate +under the continual UDA setting, i.e., they do not access real +data from the source domain when adapting to the target + +TABLE 4: Domain-wise test Dice scores (%) for optic disc segmentation (ODS) and cardiac segmentation (CS), at the end of +the continual sequence D0 (Canon)→ D1 (Kowa)→ D2 (Nidek) for ODS, and D0 (Philips)→ D1 (Siemens)→ D2 (Canon) for +CS. For the latter, the tabulated results are averages over Dice scores from the end-diastolic (ED) and end-systolic (ES) phases. +All results are given as mean ± standard deviation (std. dev.) over 3 random initializations. The best and the second-best +results per column (excluding LB and UB) are in bold and underlined, respectively. +Training +Method +ODS: Test Dice Scores (%) +CS (ES + ED): Test Dice Scores (%) +D0 +D1 +D2 +overall +D0 +D1 +D2 +overall +D0 +LB +95.5 +76.2 +44.6 +72.1 +84.6 +70.9 +58.3 +71.3 +D0 → D1 → D2 +AdaptSegNet [45] 95.4 ± 0.0 +72.4 ± 3.3 +47.3 ± 3.1 +71.7 ± 0.7 +78.8 ± 0.7 +62.0 ± 1.1 +48.4 ± 1.8 +63.1 ± 1.2 +AdvEnt [46] +95.9 ± 0.0 68.6 ± 5.6 +52.7 ± 3.4 +72.3 ± 2.9 +82.2 ± 0.1 +67.7 ± 1.7 +53.8 ± 1.3 +67.9 ± 1.0 +ACE [17] +87.8 ± 4.1 +82.9 ± 1.6 +57.4 ± 4.0 +76.0 ± 2.6 +82.0 ± 0.5 +73.6 ± 1.1 +58.6 ± 1.0 +71.4 ± 0.9 +MuHDi [16] +92.3 ± 3.9 +84.1 ± 0.5 +59.8 ± 5.9 +78.7 ± 1.3 +79.5 ± 1.2 +71.0 ± 2.7 +59.9 ± 5.9 +70.1 ± 3.3 +GarDA (ours) +94.6 ± 0.3 87.9 ± 0.1 65.2 ± 1.5 82.6 ± 0.4 +82.8 ± 0.1 76.0 ± 0.2 67.2 ± 0.1 75.3 ± 0.1 +D1 +SD-UB +− +92.3 +− +− +− +82.4 +− +− +D2 +SD-UB +− +− +93.1 +− +− +− +85.5 +− +D0 + D1 + D2 +MD-UB +96.2 +94.7 +90.5 +93.8 +84.1 +83.5 +84.5 +84.0 +domains. Inspired by [81], we therefore use the generator +of GarDA that is trained on the source domain, to provide +synthesized instead of real source domain images to the above +methods. Apart from this modification, we employ the meth- +ods as they were proposed originally. We implemented ACE +from scratch, and the other methods were based on the code +provided by MuHDi1 [16]. All models were implemented in +Pytorch [82] and trained on a single NVIDIA A100 GPU. The +hyperparameters for the methods are presented in Tab. 3. Since +source-free UDA methods only adapt to a single target domain +and DIL methods require labeled target domains, we do not +consider such methods for our experimental comparisons. +4.5. Experimental scenarios +Segmentation performances were evaluated using the Dice +score (aggregated over all foreground labels) achieved at the +end of the continual domain sequence D0→D1→D2, where +D0 denotes the labeled source domain and D1 & D2 represent +the unlabeled target domains. We present the domain-wise +Dice scores as well as an overall Dice score aggregated over all +domains, which enables the comparison of different methods +based on a single number. The domain-wise scores on the +target domains can be regarded as a performance measure for +adaptation. Comparisons between GarDA and the competing +methods are shown in Tab. 4 for ODS and CS. For each +method, we present the mean and standard deviation (std. +dev.) of Dice scores over three different random training +initializations. For reference, we also report lower bound (LB) +Dice scores for each domain, which are obtained by training +a segmentation model only on the source domain D0 and +then testing on all domains without any adaptation. We also +report single-domain upper bound (SD-UB) results, i.e., the +performance of a segmentation model that is trained in a +supervised manner (i.e., with labels) separately on either D1 +or D2, and tested on the same corresponding domain. Finally, +1https://github.com/valeoai/MuHDi +the multi-domain upper bound (MD-UB) shows the Dice score +achieved by a model that is trained jointly in a supervised +fashion using data from all three domains. +4.5.1. ODS performance analysis. The results in Tab. 4 show +a large gap between LB and SD-UB, i.e., 16.1 pp (percentage +points) on D1 and 48.5 pp on D2, indicating that the seg- +mentation model can benefit substantially from adaptation to +the target domains. GarDA is seen to surpass all competing +methods in terms of overall Dice score, as well as adaptation +performance, i.e., domain-wise scores for both target domains. +The non-sequential UDA method AdvEnt achieves the highest +Dice score on D0, but fails to adapt to the target domains. The +same observation can be made for AdaptSegNet. Among the +competing sequential UDA methods, MuHDi and ACE, the +former achieves better results on all domains, and overall the +second-best in adaptation after our method GarDA. Our ap- +proach outperforms MuHDi’s domain-wise mean Dice scores +on D0, D1, and D2 by 2.3 pp, 3.8 pp, and 5.4 pp, respectively. +This leads to an improvement of 3.9 pp in mean overall +Dice score – an improvement of almost 60% considering +that the previous state-of-the-art MuHDi is 6.6 pp above the +no-adaptation LB. Furthermore, such improvement brings the +state of the art in this task by over 25% closer to its upper +bound, given the initial gap of 15.1 pp from MuHDi to MD- +UB. It is also noteworthy to see that GarDA has by far +the smallest standard-deviations for adaptation, which shows +the robustness of our method to different initializations – a +very important quality for any deep-learning method and its +usability in practice. +4.5.2. CS performance analysis. In Tab. 4, we present the CS +results, which are averaged over the Dice scores computed on +images from the ED and the ES phases. Detailed results for +the individual ED and ES phases, as well as label-wise Dice +scores are reported in Appendix A. Similarly to ODS, CS +results in Tab. 4 also show a clear gap between LB and SD-UB, +i.e., 11.5 pp on D1 and 27.2 pp on D2. Compared to ODS, the + +0 +1 +2 +Step +60 +65 +70 +75 +80 +85 +90 +95 +100 +Overall Dice [%] +MD-UB +AdaptSegNet +AdvEnt +ACE +MuHDi +Ours +(a) ODS +0 +1 +2 +Step +60 +65 +70 +75 +80 +85 +90 +95 +100 +Overall Dice [%] +(b) CS +Fig. 3: Evolution of the mean overall Dice score over the seen +domains after each training step in the continual sequence for +(a) ODS and (b) CS. +lower Dice scores for SD-UB suggest the higher complexity of +CS, which is plausible since the labels occupy relatively small +part of the imaged field and the model aims to segment three +instead of one foreground label. GarDA is seen to outperform +all competing methods in all domain-wise and overall Dice +scores. On the source domain, AdvEnt performs comparably +to GarDA, but reaches significantly lower adaptation perfor- +mance in both target domains. Similarly, AdaptSegNet fails to +adapt successfully to D1 and D2. After adapting to the two +target domains, both AdvEnt and AdaptSegNet in fact perform +inferior to LB. In comparison to ACE and MuHDi, our method +improves domain-wise Dice scores on the target domains D1 +and D2 by, respectively, 2.4 pp and 7.3 pp or more. Indeed, +no competing method provides successful continual domain +adaptation, i.e., overall Dice scores better than LB, with only +ACE yielding 0.1 improvement, but with a standard-deviation +of 0.9; whereas GarDA achieves an overall Dice score that +is 3.9 pp higher than ACE’s – the second-best in this task. +Similarly to ODS, the standard deviations of GarDA are also +observed to be by far the lowest for any adapted domain +and overall, which shows the robustness of this method to +initialization. +4.5.3. Sequential performance analysis. In addition to the +Dice scores achieved at the end of the domain sequence, we +also analyze the evolution of overall segmentation performance +while the model adapts to each domain in Fig. 3. At each step +of the sequence, the overall Dice score is computed over all +domains seen up to that step. As seen in Fig. 3, the overall +scores decrease during the sequence as the models need to ad- +dress an increasing number of domains. It can be observed that +the non-sequential UDA baselines AdaptSegNet and AdvEnt +exhibit the largest drop in overall score compared to the initial +performance on D0. For ODS, both methods perform already +subpar compared to the other approaches after adapting to +D1, whereas for CS, AdaptSegNet still achieves a similar +overall score as ACE and our approach GarDA. However, +both AdaptSegNet and AdvEnt fail to adapt effectively to +D2 in the last step, which causes their final overall score to +be the lowest among all methods. Note that since MuHDi +is a continual extension of AdvEnt, their performances up +until D1 are identical. The benefits of MuHDi over AdvEnt +in a continual UDA setting are only seen in the last step, +by the gain in overall Dice score, i.e., 6.4 pp for ODS and +2.2 pp for CS. MuHDi outperforms ACE on ODS, but for CS, +attains slightly lower overall scores at the end of the sequence. +In comparison, our proposed method GarDA substantially +outperforms all other methods throughout the entire domain +sequence for both ODS and CS. +4.6. Qualitative results +In Fig. 4, we present test images for ODS and CS along with +the corresponding segmentation maps predicted by GarDA and +the competing methods. It can be observed that GarDA is +able to accurately segment the optic disc as well as cardiac +regions across different domains. As already highlighted in the +quantitative results, our method performs particularly better +on the last target domain in comparison to the other methods. +Although not perfect, GarDA is able to segment most of the +anatomical structures, whereas the other methods miss many +of the relevant parts; highlighted specifically by the results in +the last columns of D2. +To give an insight into how our GAN works, in Fig. 5 we +show examples of real images and images synthesized by our +generator after each adaptation step in the continual domain +sequence. It can be observed that the generator successfully +learns to create images with the appearance characteristics of +each domain. In particular, our generator captures the orange +tint of D1 as well as the darker appearance of D2 with +the more pronounced vessels in dark-red colors. Furthermore, +Fig. 5 shows that the content in the synthesized images stays +relatively consistent across domains, which is crucial for the +segmentation training to be effective. We can also see at step +t = 2 that, thanks to the GAN distillation losses, the generated +images of previously seen domains do not change significantly +compared to t = 0 or t = 1, when the domains were first +learned. Note that some of the generated images appear to +be rotated. It can be attributed to the employment of flipping +and random rotations during GAN training, which causes the +generator to reproduce such augmentations. However, this is +not an issue since these images are merely used to train the +segmentation model, which can accommodate any orientation. +Analogously, we show in Fig. 6 our generated images for CS +after each step in the continual domain sequence. Therefore, +we include reference rectangles to better emphasize appear- +ance differences between the domains around the anatomy of +interest for segmentation. In particular, images from D1 tend to +be darker and sometimes blurry, while images from D0 appear +to be the brightest overall. As MR images are in grayscale and +with high-dynamic range, the differences between the domains +are not as evident to the human eye as in the case of retinal +RGB images. Nevertheless, the results we presented earlier +indicate that such differences still significantly hinder gener- + +Fig. 4: Examples of real test images and our corresponding predicted segmentation maps. For ODS (top), the predicted optic +disc is colored in blue. For CS (bottom), the left ventricle (LV) is colored in orange, the right ventricle (RV) in purple, and +the myocardium (MYO) in blue. + +Image +Ground +truth +Ours + ACE +AdvEnt MuHDi +SegNet +Image +Ground +truth +Ours +ACE +MuHDi +AdvEnt +SegNet +AdaptFig. 5: Real sample images (left) from the ODS dataset domains, and (right) those sampled from GarDA’s generator at steps +t = 0, 1, 2. +Fig. 6: Real sample images (left) from the CS dataset domains, and (right) those sampled from GarDA’s generator at steps +t = 0, 1, 2. The pink rectangles serve as visual aid for the comparison of relevant anatomical structures between different +domains. + +01z +1/z +210. +217Xolz +2|z +117 +eaTABLE 5: Test Dice scores (%) for optic disc segmentation +(ODS) and cardiac segmentation (CS) on (a) D1 and (b) D2. +Results for CS are averaged over the end-diastolic (ED) and +end-systolic (ES) phases. Compared models are either adapted +to only a single domain (1st row) or continually adapted to +both target domains (2nd row). All results show the mean ± +std-dev over 3 random initializations. Superior results for the +column are shown in bold. +(a) D1 +Training +Test Dice Scores (%) on D1 +ODS +CS (ES + ED) +D0→D1 +85.1 ± 0.2 75.8 ± 0.0 +D0→D1→D2 87.9 ± 0.1 76.0 ± 0.2 +(b) D2 +Training +Test Dice Scores (%) on D2 +ODS +CS (ES + ED) +D0→D2 +63.9 ± 1.2 63.3 ± 0.2 +D0→D1→D2 65.2 ± 1.5 67.2 ± 0.1 +alization of models between domains, thereby necessitating +effective continual adaptation schemes. Similarly to ODS, our +generator is able to create CS images with different domain +characteristics for the same anatomical content. Additionally, +GarDA’s generator does not suffer from catastrophic forgetting +of previously seen domains. +4.7. Discussion +In the following subsections, we provide more insights into +our proposed method and the presented results. In particular, +we analyze the benefits of continual UDA over single-step +UDA in Sec. 4.7.1 and discuss the advantages of GarDA over +ACE in Sec. 4.7.2, which also employs a form of appearance +transfer. Finally, we examine the effectiveness of each of our +proposed contributions via ablation studies in Sec. 4.7.3. +4.7.1. Forward and backward transfer. An alternative to +continually adapting to new domains would be to only perform +single-step UDA, i.e., D0→D1 or D0→D2. We show in Tab. 5, +that these alternatives are in fact inferior to the model that is +continually trained on all domains D0→D1→D2. For both +ODS and CS, the continually trained model achieves better +performance on D1 compared to UDA directly to D1, which +suggests that the adaptation to a later target domain (D2) can +indeed provide useful information and improved performance +for a previous target domain. This can also be observed in the +evolution of the domain-wise Dice scores in Fig. 7, especially +for ODS. In the continual learning literature, this phenomenon +is known as backward transfer [83] of learned information. In +addition, we see in Tab. 5b that the intermediate adaptation +to D1 helps to improve the final Dice score on D2 for +both ODS and CS. This is known as forward transfer [83]. +Both transfers indicate that, when effectively incorporated, +information from additional domains can improve the accuracy +and generalizability of a model also on other domains. Owing +to this, we find continual UDA to overall perform superior to +single-step UDA. Furthermore, the continually trained model +is able to segment images from all involved domains, not only +two. This is generally beneficial, as one needs to keep only +a single model with accumulated information. For instance, +0 +1 +2 +Step +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Dice per domain [%] +(a) ODS +0 +1 +2 +Step +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Dice per domain [%] +0 +1 +2 +(b) CS +Fig. 7: Evolution of the mean domain-wise Dice scores on +D0, D1, and D2 after each step of training in the continual +sequence for (a) ODS and (b) CS. +when a care facility acquires a new imaging device, with +single-step UDA, they would need to find the right model +suitable for this device (or collect new data to adapt from +source) but with continual UDA, they can obtain the latest +continual model and expect it to perform well out-of-the-box +if it has seen similar images earlier. +4.7.2. Comparison to ACE. In order to generate images +from previous target domains, ACE first extracts feature maps +from a source image and re-normalizes them with appearance +statistics drawn from a memory buffer. The memory buffer +represents previously seen target domain images in the form of +mean and variance values of feature maps. The re-normalized +features are then fed to a decoder (an inverted VGG19 [78]) +that maps the features to the desired target images. There are +several shortcomings with this approach: First, ACE stores +a fixed number of mean and variance values in the memory +buffer, which cannot accurately represent the appearance infor- +mation of the entire distribution of target images. In contrast, +we employ a GAN trained to reproduce images from this +entire distribution. Indeed we illustrate in Fig. 8 the superior +quality of the images generated by our approach. While ACE’s +synthesized images exhibit severe checkerboard artifacts, the +images produced by our generator appear cleaner and closer +to real samples. Second, ACE always requires a source image +and a set of target domain statistics to generate a target image, +which limits the number and diversity of samples that can +be generated. On the other hand, our proposed generator is +able to produce arbitrarily many images from any domain +using a simple random noise vector input and a domain label. +Third, the inverted VGG decoder of ACE comprises more +than 140 million parameters, whereas our GAN (generator and +discriminator together) contains merely 23 million parameters. +Thus, the performance gain from our method GarDA also +comes at a lower budget. +4.7.3. Ablation study. We conduct ablation experiments for +each proposed component in GarDA to quantify their indi- + +Fig. 8: Examples of cardiac magnetic resonance (MR) images synthesized given the same content z by ACE (top) and our +proposed generator (bottom). +TABLE 6: Test Dice scores (%) of ablations on optic disc +segmentation (ODS), at the end of the continual sequence +D0 (Canon) →D1 (Kowa) →D2 (Nidek). For clarity, mean +results are shown without std. dev. The difference between our +method and its variations with ablated components is shown +in parentheses. +Training +Method +Test Dice Scores (%) +D0 +D1 +D2 +overall +D0 +LB +95.5 +76.2 +44.6 +72.1 +D0→D1→D2 +Ours − Lcon 85.9 (−8.8) +82.9 (−5.0) +72.0 (+6.8) +80.3 (−2.3) +Ours − Limg +G +82.2 (−12.5) 77.1 (−10.8) 52.9 (−12.3) 70.7 (−11.9) +Ours − Ladv 87.5 (−7.1) +76.8 (−11.1) 53.2 (−12.1) 72.5 (−10.1) +Ours +94.6 +87.9 +65.2 +82.6 +D1 +SD-UB +− +92.3 +− +− +D2 +SD-UB +− +− +93.1 +− +D0+D1+D2 +MD-UB +96.2 +94.7 +90.5 +93.8 +vidual impact on the final segmentation performance. This +evaluation is performed for ODS since the large gap between +lower and upper bound allows us to clearly demonstrate +performance gains. All results are summarized in Tab. 6. +The impact of the content loss can be seen in comparison +to the ablation “Ours−Lcon”. Omitting the content loss causes +the model to lose performance on previously seen domains, +exemplified by drops in domain-wise Dice scores by 8.8 pp and +5.0 pp on D0 and D1, respectively. Interestingly, the removal +of the content loss increases the Dice score on the last target +domain by 6.8 pp. This is likely because the generator is less +constrained to retain the same content across domains, which +allows it to better reproduce the appearance of the newer (last) +domain. However, this then comes at the cost of a larger drop +in performance on the earlier domains. +Our +proposed +GAN +distillation +losses, +consisting +of +Ldis +G =Ladv +G +λimgLimg +G +for the generator (Eq. (10)) and Ldis +D +for the discriminator (Eq. (9)), are designed to prevent +forgetting of the source and intermediate target domains. +We analyze the individual effects of adversarial and non- +adversarial components. The only non-adversarial component +is the image distillation loss, the ablation of which we +tabulate as “Ours−Limg +G ”. The results show that omitting +image distillation has a major – indeed the biggest – impact +on the segmentation performance. Namely, the Dice scores +drop for all domains by more than 10.0 pp. For ablating the +adversarial components Ladv=Ladv +G +Ldis +D , we tabulate the +results as “Ours−Ladv”. The impact is similar to the image +distillation, but does not affect the source domain performance +as much. In summary, our ablation study confirms that each of +the proposed components is essential to achieve our new state- +of-the-art segmentation performance for continual domain +adaptation. +5. Conclusions +In this work, we have proposed GarDA, a novel segmen- +tation method for continual UDA that can adapt to new +domains without forgetting previous ones. As GarDA employs +generative appearance replay, it does not require to store +previously seen data, neither from the source domain nor from +any intermediate target domains. To the best of our knowledge, +GarDA is the first segmentation method for such strictly +continual UDA. This makes our approach widely applicable in +practice, where data from previous domains cannot be retained +indefinitely. A potential limitation of our approach is the +duration of the GAN training, which may take several hours +depending on the desired quality of the domain appearance +adaptation. However, since new data from different domains +usually does not become available on a daily basis, this should +not be an issue in practice. +We evaluate the generalizability of our method by conduct- +ing comprehensive experiments for two very different tasks, +i.e., optic disc segmentation on color fundus photography im- +ages and cardiac segmentation on magnetic resonance images. +Our results demonstrate that GarDA substantially outperforms +all competing methods on both tasks. In addition, we provide +qualitative results showing that our proposed generator is able +to synthesize meaningful images from different domains that +enable the segmentation model to be trained continually. Our +ablation study highlights the impact of each proposed major + +ACE +Ourscomponent in GarDA, i.e., the content loss, image distillation, +and adversarial distillation. 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Ranzato, “Gradient episodic memory for continual +learning,” Adv. Neural Inform. Process. Syst. (NeurIPS), vol. 30, 2017. +Appendix A +Detailed cardiac segmentation results +Individual results for the end-systolic (ES) and end-diastolic +(ED) phases are presented in Tab. A1 and Tab. A2. For the +majority of labels and domains, it can be observed that our +approach achieves the best or the second-best results in terms +of label-wise and domain-wise Dice scores. + +TABLE A1: Label-wise test Dice scores (%) for cardiac segmentation (CS) in the end-systolic (ES) phase, at the end of the continual sequence D0 (Philips)→D1 +(Siemens)→D2 (Canon). All results are given as mean ± std. dev. over 3 random initializations. The best and the second-best results per column (excluding LB and +UB) are in bold and underlined, respectively. +Training +Method +Test Dice Scores ES (%) +D0 +D1 +D2 +overall +MYO +LV +RV +avg +MYO +LV +RV +avg +MYO +LV +RV +avg +D0 +LB +81.0 +82.0 +80.5 +81.2 +76.3 +68.1 +64.0 +69.5 +69.6 +66.9 +45.0 +60.5 +70.4 +D0→D1→ D2 +AdaptSegNet [45] 76.2 ± 1.0 +77.7 ± 1.0 +70.2 ± 0.7 +74.7 ± 0.8 +72.2 ± 0.6 +58.5 ± 1.2 +52.8 ± 1.5 +61.2 ± 1.0 +55.7 ± 3.1 +52.1 ± 2.2 +35.1 ± 1.6 +47.7 ± 2.2 +61.2 ± 1.1 +AdvEnt [46] +80.0 ± 0.7 81.0 ± 0.5 74.3 ± 1.5 +78.4 ± 0.5 +78.0 ± 0.7 +65.3 ± 1.3 +55.3 ± 2.5 +66.2 ± 1.1 +66.8 ± 1.5 +58.8 ± 2.1 +35.9 ± 2.6 +53.8 ± 0.9 +66.1 ± 0.8 +ACE [17] +79.5 ± 0.2 +79.1 ± 1.5 +76.8 ± 0.2 +78.5 ± 0.6 79.9 ± 0.3 68.8 ± 0.7 +65.9 ± 1.6 +71.6 ± 0.8 71.0 ± 0.4 64.8 ± 1.2 +41.6 ± 1.8 +59.1 ± 1.0 +69.7 ± 0.4 +MuHDi [16] +74.5 ± 2.0 +76.8 ± 1.3 +73.1 ± 5.2 +74.8 ± 2.0 +77.8 ± 1.9 +64.7 ± 3.2 +60.6 ± 4.0 +67.7 ± 2.8 +66.1 ± 3.0 +56.1 ± 7.1 +44.7 ± 5.9 +55.6 ± 5.1 +62.4 ± 2.7 +GarDA (ours) +78.0 ± 0.1 +79.5 ± 0.2 81.3 ± 0.0 79.6 ± 0.1 77.2 ± 0.3 72.0 ± 0.2 73.4 ± 0.1 74.2 ± 0.2 69.6 ± 0.2 72.5 ± 0.1 60.4 ± 0.2 67.5 ± 0.1 73.8 ± 0.1 +D1 +SD-UB +− +− +− +− +83.1 +76.7 +79.4 +79.7 +− +− +− +− +− +D2 +SD-UB +− +− +− +− +− +− +− +− +82.9 +84.5 +80.6 +82.7 +− +D0+D1+D2 +MD-UB +79.2 +79.7 +78.7 +79.2 +85.9 +76.3 +78.4 +80.2 +83.8 +82.9 +77.4 +81.3 +80.2 +TABLE A2: Label-wise test Dice scores (%) for cardiac segmentation (CS) in the end-diastolic (ED) phase, at the end of the continual sequence D0 (Philips)→D1 +(Siemens)→D2 (Canon). All results are given as mean ± std. dev. over 3 random initializations. The best and the second-best results per column (excluding LB and +UB) are in bold and underlined, respectively. +Training +Method +Test Dice Scores ED (%) +D0 +D1 +D2 +overall +MYO +LV +RV +avg +MYO +LV +RV +avg +MYO +LV +RV +avg +D0 +LB +94.5 +82.9 +86.9 +88.1 +84.5 +67.6 +64.6 +72.2 +77.7 +53.8 +37.0 +56.2 +72.2 +D0 → D1 → D2 +AdaptSegNet [45] 91.0 ± 0.7 +78.0 ± 0.9 +80.1 ± 0.6 +83.0 ± 0.6 +79.4 ± 1.2 +53.9 ± 1.3 +55.3 ± 2.1 +62.9 ± 1.2 +64.3 ± 2.3 +44.6 ± 0.8 +38.6 ± 2.2 +49.2 ± 1.3 +65.0 ± 0.9 +AdvEnt [46] +93.3 ± 0.7 81.4 ± 0.9 83.2 ± 1.0 +86.0 ± 0.3 +85.1 ± 0.4 +63.5 ± 1.7 +58.7 ± 5.1 +69.1 ± 2.1 +72.9 ± 3.2 +52.2 ± 7.1 +36.2 ± 7.8 +53.7 ± 1.7 +69.6 ± 1.2 +ACE [17] +93.1 ± 0.4 +79.7 ± 1.0 +83.8 ± 0.1 +85.5 ± 0.4 87.4 ± 0.5 69.0 ± 1.3 +70.4 ± 2.2 +75.6 ± 1.3 +78.6 ± 0.5 +54.0 ± 1.3 +41.5 ± 1.9 +58.1 ± 1.0 +73.1 ± 0.6 +MuHDi [16] +90.6 ± 1.5 +80.0 ± 0.5 +81.6 ± 0.0 +84.1 ± 0.5 +86.9 ± 1.3 +65.4 ± 2.8 +70.7 ± 4.5 +74.3 ± 2.6 +80.6 ± 4.0 +51.0 ± 8.0 60.9 ± 9.0 64.2 ± 6.8 +74.2 ± 3.1 +GarDA (ours) +92.6 ± 0.1 +79.3 ± 0.2 86.3 ± 0.1 86.1 ± 0.1 86.3 ± 0.4 69.5 ± 0.3 77.3 ± 0.1 77.7 ± 0.3 80.8 ± 0.2 59.4 ± 0.1 60.5 ± 0.1 66.9 ± 0.1 76.9 ± 0.1 +D1 +SD-UB +− +− +− +− +91.0 +79.2 +85.0 +85.1 +− +− +− +− +− +D2 +SD-UB +− +− +− +− +− +− +− +− +94.2 +82.2 +88.5 +88.3 +− +D0 + D1 + D2 +MD-UB +94.1 +83.3 +89.7 +89.0 +92.3 +79.9 +88.4 +86.9 +92.9 +82.0 +88.2 +87.7 +87.9 + diff --git a/z9AzT4oBgHgl3EQfRPsg/content/tmp_files/load_file.txt b/z9AzT4oBgHgl3EQfRPsg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2c04e2b9bd0e348c2cbb4350c85d27f34eb07dab --- /dev/null +++ b/z9AzT4oBgHgl3EQfRPsg/content/tmp_files/load_file.txt @@ -0,0 +1,2123 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf,len=2122 +page_content='Generative appearance replay for continual unsupervised domain adaptation Boqi Chen1,2*, Kevin Thandiackal1,2*, Pushpak Pati1 and Orcun Goksel2,3 1IBM Research Europe, Zurich, Switzerland 2Computer-assisted Applications in Medicine, ETH Zurich, Zurich, Switzerland 3Department of Information Technology, Uppsala University, Uppsala, Sweden Abstract Deep learning models can achieve high accuracy when trained on large amounts of labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, real-world scenarios often involve several challenges: Training data may become available in installments, may origi- nate from multiple different domains, and may not contain labels for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Certain settings, for instance medical applications, often involve further restrictions that prohibit retention of previously seen data due to privacy regulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In this work, to address such challenges, we study unsupervised segmentation in continual learning scenarios that involve domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To that end, we introduce GarDA (Generative Appearance Replay for continual Domain Adaptation), a generative-replay based approach that can adapt a segmentation model sequentially to new domains with unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In contrast to single-step unsupervised domain adaptation (UDA), continual adaptation to a sequence of domains enables leveraging and consolidation of information from multiple domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Unlike previous approaches in incremental UDA, our method does not require access to previously seen data, making it applicable in many practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We evaluate GarDA on two datasets with different organs and modalities, where it substantially outperforms existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Keywords Unsupervised domain adaptation, Continual learning, Optic disc segmen- tation, Cardiac segmentation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Introduction Deep Neural Networks (DNNs) have recently achieved remarkable performance on various computer vision tasks with natural images, such as classification [1], [2], [3] and semantic segmentation [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, there exist several challenges that hinder DNNs from achieving similar success in other domains, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' First, to achieve high-performance, DNNs require large amounts of labeled training images, which are challenging to obtain for medical applications, since annotations can only be provided by medical experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Annotating medical images is therefore more costly compared to annotating natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This is particularly critical for applications requiring dense annotations, such as semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Thus, there is a strong need for unsupervised DNN approaches in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Second, medical datasets usu- ally contain a relatively small number of images [6], [7], [8], [9] compared to large-scale natural image datasets, such as ImageNet [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Models trained on such small datasets often The authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' do not generalize well to unseen domains [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Unsupervised Domain Adaptation (UDA) [12], [13] aims to address both aforementioned challenges, by first training a model on a labeled source domain D0 and afterwards adapting it to an unlabeled target domain D1, to alleviate annotation costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Real-world settings are often more complex than the above scenario, and require long-term applicable solutions rather than adapting to a single target domain alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It is, for example, typical that multiple datasets from different domains D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', DT become available at different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In such a scenario, it is beneficial to keep a single model that is sequen- tially adapted to these different domains, thereby consolidating useful knowledge from more than one domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Since patient data is bound to strict privacy regulations, data sharing and indefinite data storage are often not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Therefore such medical scenarios are further complicated by the inability to keep and access data from previously seen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In the literature, this constrained learning problem is known as continual learning (CL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In this work, we study a continual UDA setting, where a model is adapted to a sequence of target domains, the data of which is available only while training on that respective domain for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The main challenges in continual UDA are twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' First, the model should not suffer from catastrophic forgetting [14], [15], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', it should maintain its performance on the source domain as well as all previously seen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Second, without access to the source or previous target domains, the model should adapt accurately to new target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Although strict continual UDA has been largely unexplored in the literature, some recent techniques have studied a scenario with looser constraints, which we herein term as incremental UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In contrast to continual UDA, in incremental UDA the data from the source domain is stored and reused during the introduction of each subsequent target domain, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As an incremental UDA method, [16] proposed Multi-Head Distillation (MuHDi), which performs knowledge distillation using either the source images (which are kept at all times) or the images from the current target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This is sub-optimal since knowledge from previously seen target domains cannot be preserved effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Another method, more closely related to our solution, is ACE (Adapting to Changing Environ- ments) [17], which builds on the idea of transferring the style arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='01211v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='CV] 3 Jan 2023 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Incremental UDA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Continual UDA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 1: Overview of incremental (top) and continual (bottom) UDA workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In the latter, the segmentation model M sequentially adapts to new domains without storing data from any previously seen domain, including D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' of unlabeled target images to known labeled source images (which then again have to be kept indefinitely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To address forgetting, ACE generates images with the same content, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', segmentation ground truth, as a given source image, but the style of a target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In addition, ACE employs a memory unit to replay images in previously seen target styles to the segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, as for MuHDi, ACE also violates a strict continual UDA constraint, since it relies on future access to the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Furthermore, ACE can only replay the previous target styles that have been stored in the memory unit, hence lacking the ability to sample those for replay from the entire distribution of previously seen target styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This in turn affects the knowledge preservation from these intermediate targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In this work, we address the above-mentioned shortcomings by proposing GarDA (Generative Appearance Replay for con- tinual Domain Adaptation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We employ generative replay [18], [19], [20], [21] to substitute images from any previously-seen but currently-inaccessible domain with synthetic images that are generated via a generative adversarial network (GAN) [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This permits operation under a strict continual UDA setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', even when the source and the intermediate target domains become inaccessible after training on each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Importantly, for replay we can arbitrarily sample images of any seen domain via our generator, and are not limited by less expressive and fixed memory buffers, as in ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Furthermore, thanks to our powerful GAN-based approach for transferring the appearance of target domain images onto source domain images, we can generate more representative samples for replay, achieving more effective knowledge preservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In summary, our main contributions are three-fold: (1) We propose GarDA as the first segmentation technique that oper- ates under a strict continual UDA setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', it mitigates forgetting of the source and intermediate target domains without storing any previously seen images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (2) We introduce a novel combination of appearance transfer and generative replay using a stochastic generator, which enables diverse sampling from the entire distribution of all domains, and is thus less prone to overfitting compared to methods em- ploying separate memory buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (3) We achieve the new state-of-the-art performance in continual UDA, demonstrated herein through comprehensive benchmarking with two target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To demonstrate the generalizability of GarDA, we conduct experiments for two entirely different tasks: optic disc segmentation in color fundus photography (CFP) images and cardiac segmentation in magnetic resonance (MR) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Related work Our work lies at the intersection of UDA and continual learning, for which we provide below a brief overview with relevant research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Continual learning Continual learning aims to simulate real-world data avail- ability constraints in sequential learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', a model should learn from a sequence of datasets without retaining any pre- viously seen data and without forgetting previously acquired knowledge [23], [15], [24], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The research in continual learning can be categorized into four major dimensions fo- cusing on regularization [26], [27], [28], parameter isola- tion [29], [30], [31], prototypical representations [32], [33], [34], and generative replay [18], [19], [35], [36], [37], [38], [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Methods built on prototypical representations as well as generative replay have been the most successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The former aims to store class-representative prototypes in feature space to compensate any estimated drift when learning new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In the latter, a generative model learns to synthesize images [18], [35], [36], [20], [21] or features [19], [37], [38] of previously seen data, such that they can be replayed to the classification/segmentation model when the original data is not available anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' While these methods have achieved state-of-the-art performance on class- and task-incremental benchmarks, they have not been employed in the context of continual UDA, to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' UDA Over the last years, a myriad of methods have been proposed to tackle UDA for both classification and segmentation [12], [13], with their common paradigm often being the domain alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This aims to bring the source and target domain representations closer, in terms of either the input images, certain intermediate features, or the model outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' If this is done effectively, the classification/segmentation models trained on the source domain are expected to generalize to the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' [39] were the first to propose a Domain-Adversarial Neural Network (DANN), where the main model’s feature extractor uses the negative gradient of a domain classifier to obtain domain-invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly, other approaches TABLE 1: Taxonomy of different domain adaptation settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Setting Source unavailable after training Multiple target domains Unlabeled target domain(s) Unsup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Domain Adaptation (UDA) \x17 \x17 \x13 Domain-Incremental Learning (DIL) \x13 \x13 \x17 Source-free UDA / UMA \x13 \x17 \x13 Incremental UDA \x17 \x13 \x13 Continual UDA \x13 \x13 \x13 have adopted GAN-inspired adversarial training for feature alignment [40], [41], [42], which has been effective also for the segmentation of medical images such as MR and computer tomography (CT) images [43], [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Other methods such as AdaptSegNet [45] or AdvEnt [46] also employ adversarial losses, but these aim at aligning the model output spaces of source and target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Alternatively, CycleGAN [47] aims to align domains in the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Cycle-Consistent Ad- versarial Domain Adaptation (CyCADA) [48] and Synergistic Image and Feature Alignment (SIFA) [49] propose to unify feature-level and image-level domain alignment for UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All the aforementioned UDA methods were designed to adapt models from a single source domain to a single target domain, and they do not operate in continual (or incremental) learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In this work, our goal is to tackle continual learning settings, for reasons motivated earlier including the limitations of access to medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Continual UDA, with a sequence of multiple target domains, introduces additional challenges to single-domain UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Sequential UDA In the literature, UDA for a sequence of multiple target domains has been studied under different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We herein group them collectively under the umbrella of sequential UDA, and propose the taxonomy presented in Table 1 to clarify the differences among the individual settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We denote the problem of learning multiple labeled target domains without retaining previously seen data as domain-incremental learning (DIL) [50], [51], [52], [53], [54], [55], [56], [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Another sub-category called source-free UDA – sometimes referred to as unsupervised model adaptation (UMA) – has recently gained increasing attention for both classification [58], [59], [60], [61] and segmentation tasks [62], [63], [64], [65], [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' These methods perform UDA from a source to a single target domain without using the actual source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that in the literature, source-free and continual UDA have sometimes been used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Instead, we propose to separate these terms as shown in Table 1, since source-free UDA aims for adaptation to one target domain only, therefore not having to consider resilience during further target adaptations in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Naming this setting as continual UDA would be indeed inconsistent with the remaining continual learning literature [15], [25] that deals with scenarios involving multiple classes/tasks in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In contrast to the above, incremental UDA aims at adapting models to multiple target domains in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For instance, MuHDi [16] employs domain adversarial training for adaptation, together with multiple distillation losses to mitigate forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As mentioned earlier, knowledge distilla- tion is sub-optimal when enforced using the source images and/or the current target images, as these do not directly prevent forgetting of the previously seen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' [17] proposed ACE, which uses a deterministic encoder- decoder architecture to create images in a target domain style for replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To tackle forgetting, ACE employs a replay memory buffer containing style information of previously seen target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Although this can mitigate forgetting to some extent, the memory buffer can only hold the style information of few samples, thus not representative of the complete distribution of all previous target domains/images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In addition, both MuHDi and ACE require access to source data throughout the adaptation sequence, making them incremental UDA in our taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In this work, we study continual UDA, a more challenging scenario compared to those aforementioned: The source data is available only once at the beginning of the sequence for training on the source domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Afterwards, the model adapts to a sequence of unlabeled target domains without storing data from any previous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Method In this section, we present our proposed framework for continual UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' A segmentation model M is first trained on images x0 ∈ X0 ⊆ RH×W ×C with segmentation label maps y0 ∈ Y0 ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', L}H×W from the source domain D0 = {(x0, y0)(i)}N0 i=1, where N0, L, C, H, and W denote the number of images, labels, image channels, image height, and image width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Following training on the source, D0 is afterwards not accessible anymore while adapting M to a sequence of T ≥ 1 target domains D1:T = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', XT } which are lacking labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In other words, while training on the t-th target domain, we only have access to the unlabeled images from the current domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', xt ∈ Dt, but we cannot access samples from the source or previously encountered target domains, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', D0:t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We substitute the missing D0:t−1 with synthetic data produced by a GAN and replay this data to M to prevent forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As new target domains are added, the GAN (consisting of a generator G and a discriminator D) and the segmentation model M are trained alternately, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' First, M and the GAN are trained independently on the source domain (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Next, G is trained to synthesize images with the appearance characteristics of a new target domain (Figure 2(a)) while remembering how to generate images of previous domains (Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This training phase is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' From such a generator G, at any point we can create pairs of synthetic samples with the same content but with separate appearances: one of the source domain and one of a target (a) Adapt (c) Adapt trainable frozen Generator Discriminator Feature extractor Segmentation model Feature map Adapted feature map (b) Distill real/fake real/fake Train Train .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Adapt + Distill Adapt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2: Overview of our method including (a) GAN adaptation (b) GAN distillation and (c) adaptation of the segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We can then use these pairs to adapt the segmentation model, by distilling the outputs of the previously trained model Mp into the current model M, as seen in Figure 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In other words, the new M shall predict the same segmentation map for a target-like image as the map predicted by Mp for the corresponding source-like image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This step is described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The above process is repeated whenever data from a new target domain becomes available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Source training Initially, we train a segmentation model M and a GAN (G, D) separately on the source domain data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For an input image x, M predicts a segmentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Specifically, we denote the output logit of M for label l at pixel position (h, w) as M (h,w,l)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The corresponding probability of belonging to label l is computed via a softmax operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', m(h,w,l)(x) = eM (h,w,l)(x) �L j=1 eM (h,w,j)(x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (1) Given a source image and segmentation label map (x, y) ∈ D0, M is trained by minimizing the cross-entropy loss LM = − 1 HW H � h=1 W � w=1 L � l=1 1{y(h,w)=l}log � m(h,w,l)(x) � (2) where 1{y(h,w)=l} denotes an indicator function that is 1 if y(h,w)=l and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the generative replay model, we employ a conditional GAN consisting of a generator G and a projection discrimi- nator D [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Given a randomly sampled noise vector z (that aims to condition the image content) and a domain label τ (for the corresponding appearance), G : (τ, z) �→ xτ|z learns to generate synthetic samples xτ|z of domain Dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' G is trained by optimizing a non-saturating logistic loss [22] LG = E z∼pz � a � − D � G(0, z) � , 0 �� , (3) where a(·) is the softplus operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' During training, D counteracts G by trying to distinguish synthetic from real images, by minimizing LD = E z∼pz � a � D � G(0, z), 0 ��� + E x∼D0 � a � − D � 0, x ��� + λR1R1 , (4) where R1 is the gradient penalty term Ex∼D0 ∥∇xD(0, x)∥2 2 [68], which is only computed for real images and weighted with λR1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' GAN adaptation The main challenge following the source training is the unavailability of labeled target domain images, which prevents the trivial solution of fine-tuning M on the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, the challenge can be addressed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', if a mechanism could create (pseudo-)labeled target domain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To this end, we propose to adapt our GAN to the target domain such that synthetic images can be sampled with content similar to the source and appearance similar to any target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As presented in Figure 2(a), for a new target domain Dt, we freeze the previously trained generator G0:t−1 and instantiate a trainable generator Gt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For simplicity, we denote Gt being trained as G, and frozen G0:t−1 for previous domains as Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We initialize G with the weights of Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We employ a discriminator D that operates on feature repre- sentations extracted using a frozen pretrained feature extractor F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', instead of distinguishing between real and fake images, our D distinguishes their feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Accordingly, we first randomly sample a source-like image xp 0|z=Gp(0, z) from Gp and a real image xt from the current dataset Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' These images are then passed through F, yielding feature maps f p 0|z=F(xp 0|z), and ft=F(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We now aim to create a target-like image xt|z=G(t, z) with features ft|z=F(xt|z) that incorporates the content of xp 0|z and the appearance of xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In order to train G, we first need to define an objective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', what the synthesized features ft|z should look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Inspired by [69], [17], we leverage the idea behind adaptive instance normalization (AdaIN) that the instance-level statis- tics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', mean and variance) of a feature map represent the appearance of the corresponding image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Using AdaIN, we can re-normalize the instance-level statistics of f p 0|z to those of ft and obtain a new feature map ˜f p t|z that encodes the content of xp 0|z and the appearance of xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The said re-normalization is computed as ˜f p t|z =AdaIN � f p 0|z, ft � =σ(ft) � � f p 0|z − µ � f p 0|z � σ � f p 0|z � � �+ µ(ft), (5) where µ(·) and σ(·) denote channel-wise mean and standard deviation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' During GAN training, discriminator D is trained to identify ˜f p t|z as real and ft|z as fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To this end, D optimizes the following objective: Luda D = E z∼pz � a � D � t, ft|z ��� + E z∼pz xt∼Dt � a � −D � t,˜f p t|z �� � + λR1R1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (6) Meanwhile, G tries to confuse D while adapting to the new domain by minimizing Luda G = E z∈pz � a � − D � t, ft|z � �� + λconLcon , (7) where the first term encourages G to generate images with realistic target appearance and the second term is a content loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The purpose of the content loss is for G to generate two images with different appearance characteristics, but the same content, when given the same z and two different domain labels as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' That means that in the feature space, the only difference between ft|z and f p 0|z should be the appearance information, which we estimate through channel-wise mean and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Thus, we define a content loss as the difference between the synthesized target features after re-normalization ˜f0|z=AdaIN(ft|z, f p 0|z), and the source features f p 0|z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', Lcon = H � h=1 W � w=1 C � h=c ��� f p (h,w,c) 0|z − ˜f (h,w,c) 0|z ��� 2 2 , (8) where H, W, and C are the height, width, and channels of the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' GAN distillation For continual UDA, in addition to adapting to a new target domain, the generator G shall not forget any previously learned knowledge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', G should be able to generate data from all previous domains D0:t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To counteract forgetting, we introduce two distillation losses: Ldis D and Ldis G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As illustrated in Figure 2(b), we sample uniformly from the previous do- mains τ ∼ U{0, t − 1} and random noise vectors z ∼ pz, and feed these both to the previous generator Gp and the current generator G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For distillation, we treat the images generated by the previous generator xp τ|z = Gp(τ, z) as real and the corresponding images generated by the current generator xτ|z = G(τ, z) as fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The distillation loss for the discriminator is Ldis D = E z∼pz τ∼U{0,t−1} � a � D � F � G(τ, z) � , τ �� + a � −D � F � Gp(τ, z) � , τ ��� (9) and for the generator Ldis G = Ladv G � �� � E z∼pz τ∼U{0,t−1} � a � −D � F � G(τ, z) � , τ ��� + λimg E z∼pz τ∼U{0,t−1} � ∥G(τ, z) − Gp(τ, z)∥1 � � �� � Limg G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (10) The first term Ladv G in Equation (10) is an adversarial (adv) component evaluated at the feature level, while the second term Limg G is an ℓ1 image (img) distillation loss at the image level weighted by λimg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The combination of such image- and feature-level distillation has been empirically demonstrated to be effective for other continual learning tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', class- incremental learning [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In practice, GAN adaptation and distillation (Figure 2(a&b)) are optimized simultaneously by splitting a training batch into two halves: one for adaptation to a new domain, and the other for distillation of previous domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This training strategy is more efficient as both the objectives are minimized in the same forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Then, the final objectives for D and G become LD = Luda D + Ldis D and LG = Luda G + Ldis G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (11) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Segmentation training Subsequent to adapting the GAN, the segmentation model M is trained to perform segmentation also in the new target domain Dt (see Figure 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' When adapting to Dt, we freeze the previously trained model, called Mp hereafter, that has seen D0:t−1 and instantiate a trainable model M initialized with the weights of Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' M is trained in a supervised manner with images synthesized by G and corresponding pseudo- labels predicted by Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To this end, we employ the concept of knowledge distillation [70], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', the current model M is trained to make the same predictions as the previous model Mp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Since labeled data is only available during the source training, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1, we assume that the predictions of Mp on the source images are the most accurate compared to images from other target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For knowledge distillation, we first pass a generated source image x0|z = G(0, z) to Mp and obtain the pseudo segmentation map ˆm0 at each position (h, w) as ˆm(h,w,l) 0 = eM (h,w,l) p (x0|z)/κ �L j=1 eM (h,w,j) p (x0|z)/κ , (12) where κ is a temperature parameter that we set to 2, as commonly used in the knowledge distillation literature [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This produces a softer probability distribution and encodes more fine-grained information compared to one-hot labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the same noise input z, a generated source image x0|z and an image xτ|z = G(τ, z) of any seen domain Dτ∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=',t} are expected to have the same content, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', the same ground- truth segmentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We can thus utilize ˆm0 as a (silver- standard/approximate) ground truth for xτ|z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly as in Equation (2), we then train the segmentation model M with the pseudo-labeled samples (xτ|z, ˆm0) by minimizing the cross-entropy loss LM = − 1 HW H � h=1 W � w=1 L � l=1 ˆm(h,w,l) 0 log � m(h,w,l) τ � (13) m(h,w,l) τ = eM (h,w,l)(xτ|z)/κ �L j=1 eM (h,w,j)(xτ|z)/κ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (14) Since the images xτ|z are sampled uniformly from all domains including both the previous ones and the current one, M is trained to generalize across all of them, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', it adapts to the new target domain Dt while retaining performance on the previously seen domains D0:t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' TABLE 2: Summary of the datasets used in our continual UDA experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Task Domain Device vendor Training images Test images ODS D0 Canon 640 160 D1 Kowa 54 27 D2 Nidek 278 70 CS D0 Philips 58 16 D1 Siemens 76 19 D2 Canon 40 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Experiments We evaluate GarDA on two different tasks, namely optic disc segmentation in CFP images and cardiac segmentation in MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Below, we first describe the employed datasets, preprocessing steps, and implementation details in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Then, we present comparisons to the state of the art in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6, and discuss further results and ablation studies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Optic disc segmentation (ODS) The task of segmenting the optic disc is challenging and crucial for the clinical detection of glaucoma [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To evaluate our method on this task, we consider three public CFP image datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', the Retinal Fundus Glaucoma Challenge REFUGE [72], the Indian Diabetic Retinopathy Image Dataset IDRiD [6], and the Retinal Image database for Optic Nerve Evaluation for Deep Learning RIM-ONE DL [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The datasets were acquired from multiple centers and countries, using different scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Therefore, they allow to create a realistic and representative benchmark for continual UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To construct a continual sequence of domains, we select subsets of the aforementioned datasets where each domain is represented by images acquired with a different camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We consider as labeled source domain D0 the set of images from REFUGE that were acquired with a Canon CR-2 camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the first unlabeled target domain D1, we use all images from IDRiD, which were captured by a Kowa VX-10α digital fundus camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For a second unlabeled target domain D2, we select the subset of images from RIM-ONE DL, which were taken by a Nidek AFC-210 non-mydriatic fundus camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The samples from each domain/camera are then randomly split into 80% training images and 20% test images, as summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that we only employ a validation set (25% of the training set) during source segmentation training, since for the training on the target domains only generated images are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We follow the preprocessing pipeline employed in [73] by first locating the optic disc center using a simple boundary detection algorithm [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Then, the images are cropped to a size of 640×640 pixels around the estimated optic disc center before being further resized to 256 × 256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Cardiac segmentation (CS) We also benchmark our method for CS in MR images, where we segment three cardiac regions: the left ventricle (LV) and the right ventricle (RV) cavities, and the left ventricle my- ocardium (MYO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that the CS task provides an evaluation setting different from ODS, as it contains higher dynamic- range, grayscale images and aims at multi-class segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For CS, we use data from the Multi-Centre, Multi-Vendor and Multi-Disease Cardiac Segmentation (M&Ms) challenge [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It contains volumes from multiple hospitals in Spain and Germany, acquired by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5T scanners of different vendors, namely Philips, Siemens, and Canon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We select the labeled subset of images obtained from the Philips scanner as the source dataset, D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The unlabeled target domains D1 and D2 consist of all images acquired using Siemens and Canon scanners, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly to ODS, we split the data from each domain into 80% for training and 20% for testing, where 25% of the training data is used as validation only during the source segmentation training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The exact data distri- bution is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 2, where “image” denotes a volume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', a set of multiple sequential 2D slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that while our model is trained on 2D slices, the evaluation is run in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' That is, at test time after 2D inference, the numbers of correctly/falsely classified pixels from all 2D slices in a 3D volume are aggregated to compute a 3D Dice score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We preprocess the images by employing bias field cor- rection [75] and min-max intensity normalization where the minimum and maximum values are the 5th and 95th intensity percentiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In addition, we resample the 2D slices to an in-plane resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 mm, before cropping or padding them to a fixed size of 256 × 256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Implementation details We use a U-Net [76] segmentation model for GarDA and all the competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Both the encoder and the decoder consist of five convolutional blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Skip connections are added between each block of the encoder and the correspond- ing block in the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We employ the stochastic gradient descent (SGD) optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Throughout all experimental settings, we use a batch size of 16 and a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Source domain training is performed on all available real samples, whereas during the introduction of target domains, arbitrarily many images can be sampled from the generator for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Therefore, we measure the training duration for the source domain in number of epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', passes through the entire dataset) and for adding target domains in number of iterations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', batches sampled from G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In particular, the model is trained for 100 epochs on the source domain and for 500 iterations for every target domain addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the generative module, a conditional GAN with a projection discriminator D [67] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The discriminator incorporates a mini-batch discrimination layer [77] for bet- ter sample diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that two types of discriminators TABLE 3: Summary of hyperparameter optimization ranges for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Method Parameter Values Description GarDA λcon {1, 10} Content loss λimg {1, 10} Image distillation λR1 {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2, 2} R1 regularization AdaptSegNet [45] λcp {1, 10, 20, 50} Class prior loss λseg {1, 10, 20, 50} Segmentation loss AdvEnt [46] λcp {1, 10, 20, 50} Class prior loss λseg {1, 10, 20, 50} Segmentation loss MuHDi [16] λcp {1, 10, 20, 50} Class prior loss λseg {1, 10, 20, 50} Segmentation loss λFD {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1, 1} Feature distillation λdd {1, 10, 20, 50} Distribution distillation λprev {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5} Distribution distillation from previous model ACE [17] λstyle {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5, 1, 5} Style loss λcont {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1, 1} Content loss λKL {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1, 1} KL divergence loss are utilized in GarDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the source domain training, the discriminator distinguishes between real and fake images, whereas during adaptation to target adaptation domains, D discriminates features extracted by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the source training on images, we use a D architecture with seven convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For target adaptation training, a smaller four-layer backbone is used due to the lower dimensions of the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To extract these features, we follow previous work [69] and utilize a frozen, pretrained VGG19 [78] architecture for F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Our generator G comprises two linear layers followed by nine style-convolution layers [79] consisting of style modula- tion, convolution, and noise injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Besides, we compute an exponential moving average (EMA) [80] of G’s parameters θ over the training iterations n, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', θ(n) EMA = βθ(n−1) EMA + (1 − β)θ(n) with θ(0) EMA = θ(0) and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' At the end of the training, we construct a copy of G from these averaged parameters, which is afterwards used to generate samples for the training of the segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We found such averaging to increase the robustness of G to avoid ending the GAN training in a potentially poor local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Both G and D are trained with the Adam optimizer and an equalized learning rate [77] of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' During training, images are augmented with horizontal and vertical flips as well as random rotations (90◦/180◦/270◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' A summary of the loss weights used for the content loss, image distillation, as well as the R1 regularization in the discriminator is shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Comparisons with the state of the art We compare GarDA to two established non-sequential UDA methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', AdaptSegNet [45] and AdvEnt [46], and two incremental UDA methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', ACE [17] and MuHDi [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To enable a fair comparison, we adapt these methods to operate under the continual UDA setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', they do not access real data from the source domain when adapting to the target TABLE 4: Domain-wise test Dice scores (%) for optic disc segmentation (ODS) and cardiac segmentation (CS), at the end of the continual sequence D0 (Canon)→ D1 (Kowa)→ D2 (Nidek) for ODS, and D0 (Philips)→ D1 (Siemens)→ D2 (Canon) for CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the latter, the tabulated results are averages over Dice scores from the end-diastolic (ED) and end-systolic (ES) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All results are given as mean ± standard deviation (std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=') over 3 random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The best and the second-best results per column (excluding LB and UB) are in bold and underlined, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Training Method ODS: Test Dice Scores (%) CS (ES + ED): Test Dice Scores (%) D0 D1 D2 overall D0 D1 D2 overall D0 LB 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 D0 → D1 → D2 AdaptSegNet [45] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 AdvEnt [46] 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ACE [17] 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 MuHDi [16] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 GarDA (ours) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 D1 SD-UB − 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 − − − 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 − − D2 SD-UB − − 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 − − − 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 − D0 + D1 + D2 MD-UB 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Inspired by [81], we therefore use the generator of GarDA that is trained on the source domain, to provide synthesized instead of real source domain images to the above methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Apart from this modification, we employ the meth- ods as they were proposed originally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We implemented ACE from scratch, and the other methods were based on the code provided by MuHDi1 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All models were implemented in Pytorch [82] and trained on a single NVIDIA A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The hyperparameters for the methods are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Since source-free UDA methods only adapt to a single target domain and DIL methods require labeled target domains, we do not consider such methods for our experimental comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Experimental scenarios Segmentation performances were evaluated using the Dice score (aggregated over all foreground labels) achieved at the end of the continual domain sequence D0→D1→D2, where D0 denotes the labeled source domain and D1 & D2 represent the unlabeled target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We present the domain-wise Dice scores as well as an overall Dice score aggregated over all domains, which enables the comparison of different methods based on a single number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The domain-wise scores on the target domains can be regarded as a performance measure for adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Comparisons between GarDA and the competing methods are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4 for ODS and CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For each method, we present the mean and standard deviation (std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=') of Dice scores over three different random training initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For reference, we also report lower bound (LB) Dice scores for each domain, which are obtained by training a segmentation model only on the source domain D0 and then testing on all domains without any adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We also report single-domain upper bound (SD-UB) results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', the performance of a segmentation model that is trained in a supervised manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', with labels) separately on either D1 or D2, and tested on the same corresponding domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Finally, 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='com/valeoai/MuHDi the multi-domain upper bound (MD-UB) shows the Dice score achieved by a model that is trained jointly in a supervised fashion using data from all three domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' ODS performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4 show a large gap between LB and SD-UB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 pp (percentage points) on D1 and 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 pp on D2, indicating that the seg- mentation model can benefit substantially from adaptation to the target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' GarDA is seen to surpass all competing methods in terms of overall Dice score, as well as adaptation performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', domain-wise scores for both target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The non-sequential UDA method AdvEnt achieves the highest Dice score on D0, but fails to adapt to the target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The same observation can be made for AdaptSegNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Among the competing sequential UDA methods, MuHDi and ACE, the former achieves better results on all domains, and overall the second-best in adaptation after our method GarDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Our ap- proach outperforms MuHDi’s domain-wise mean Dice scores on D0, D1, and D2 by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 pp, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 pp, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 pp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This leads to an improvement of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 pp in mean overall Dice score – an improvement of almost 60% considering that the previous state-of-the-art MuHDi is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 pp above the no-adaptation LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Furthermore, such improvement brings the state of the art in this task by over 25% closer to its upper bound, given the initial gap of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 pp from MuHDi to MD- UB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It is also noteworthy to see that GarDA has by far the smallest standard-deviations for adaptation, which shows the robustness of our method to different initializations – a very important quality for any deep-learning method and its usability in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' CS performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4, we present the CS results, which are averaged over the Dice scores computed on images from the ED and the ES phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Detailed results for the individual ED and ES phases, as well as label-wise Dice scores are reported in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly to ODS, CS results in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4 also show a clear gap between LB and SD-UB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 pp on D1 and 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 pp on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Compared to ODS, the 0 1 2 Step 60 65 70 75 80 85 90 95 100 Overall Dice [%] MD-UB AdaptSegNet AdvEnt ACE MuHDi Ours (a) ODS 0 1 2 Step 60 65 70 75 80 85 90 95 100 Overall Dice [%] (b) CS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3: Evolution of the mean overall Dice score over the seen domains after each training step in the continual sequence for (a) ODS and (b) CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' lower Dice scores for SD-UB suggest the higher complexity of CS, which is plausible since the labels occupy relatively small part of the imaged field and the model aims to segment three instead of one foreground label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' GarDA is seen to outperform all competing methods in all domain-wise and overall Dice scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' On the source domain, AdvEnt performs comparably to GarDA, but reaches significantly lower adaptation perfor- mance in both target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly, AdaptSegNet fails to adapt successfully to D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' After adapting to the two target domains, both AdvEnt and AdaptSegNet in fact perform inferior to LB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In comparison to ACE and MuHDi, our method improves domain-wise Dice scores on the target domains D1 and D2 by, respectively, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 pp and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 pp or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Indeed, no competing method provides successful continual domain adaptation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', overall Dice scores better than LB, with only ACE yielding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 improvement, but with a standard-deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' whereas GarDA achieves an overall Dice score that is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 pp higher than ACE’s – the second-best in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly to ODS, the standard deviations of GarDA are also observed to be by far the lowest for any adapted domain and overall, which shows the robustness of this method to initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Sequential performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In addition to the Dice scores achieved at the end of the domain sequence, we also analyze the evolution of overall segmentation performance while the model adapts to each domain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' At each step of the sequence, the overall Dice score is computed over all domains seen up to that step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 3, the overall scores decrease during the sequence as the models need to ad- dress an increasing number of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It can be observed that the non-sequential UDA baselines AdaptSegNet and AdvEnt exhibit the largest drop in overall score compared to the initial performance on D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For ODS, both methods perform already subpar compared to the other approaches after adapting to D1, whereas for CS, AdaptSegNet still achieves a similar overall score as ACE and our approach GarDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, both AdaptSegNet and AdvEnt fail to adapt effectively to D2 in the last step, which causes their final overall score to be the lowest among all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that since MuHDi is a continual extension of AdvEnt, their performances up until D1 are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The benefits of MuHDi over AdvEnt in a continual UDA setting are only seen in the last step, by the gain in overall Dice score, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 pp for ODS and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 pp for CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' MuHDi outperforms ACE on ODS, but for CS, attains slightly lower overall scores at the end of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In comparison, our proposed method GarDA substantially outperforms all other methods throughout the entire domain sequence for both ODS and CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Qualitative results In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4, we present test images for ODS and CS along with the corresponding segmentation maps predicted by GarDA and the competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It can be observed that GarDA is able to accurately segment the optic disc as well as cardiac regions across different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As already highlighted in the quantitative results, our method performs particularly better on the last target domain in comparison to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Although not perfect, GarDA is able to segment most of the anatomical structures, whereas the other methods miss many of the relevant parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' highlighted specifically by the results in the last columns of D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To give an insight into how our GAN works, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 5 we show examples of real images and images synthesized by our generator after each adaptation step in the continual domain sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It can be observed that the generator successfully learns to create images with the appearance characteristics of each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In particular, our generator captures the orange tint of D1 as well as the darker appearance of D2 with the more pronounced vessels in dark-red colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 5 shows that the content in the synthesized images stays relatively consistent across domains, which is crucial for the segmentation training to be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We can also see at step t = 2 that, thanks to the GAN distillation losses, the generated images of previously seen domains do not change significantly compared to t = 0 or t = 1, when the domains were first learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Note that some of the generated images appear to be rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' It can be attributed to the employment of flipping and random rotations during GAN training, which causes the generator to reproduce such augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, this is not an issue since these images are merely used to train the segmentation model, which can accommodate any orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Analogously, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 6 our generated images for CS after each step in the continual domain sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Therefore, we include reference rectangles to better emphasize appear- ance differences between the domains around the anatomy of interest for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In particular, images from D1 tend to be darker and sometimes blurry, while images from D0 appear to be the brightest overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As MR images are in grayscale and with high-dynamic range, the differences between the domains are not as evident to the human eye as in the case of retinal RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Nevertheless, the results we presented earlier indicate that such differences still significantly hinder gener- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4: Examples of real test images and our corresponding predicted segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For ODS (top), the predicted optic disc is colored in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For CS (bottom), the left ventricle (LV) is colored in orange, the right ventricle (RV) in purple, and the myocardium (MYO) in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Image Ground truth Ours ACE AdvEnt MuHDi SegNet Image Ground truth Ours ACE MuHDi AdvEnt SegNet AdaptFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 5: Real sample images (left) from the ODS dataset domains, and (right) those sampled from GarDA’s generator at steps t = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 6: Real sample images (left) from the CS dataset domains, and (right) those sampled from GarDA’s generator at steps t = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The pink rectangles serve as visual aid for the comparison of relevant anatomical structures between different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 01z 1/z 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 217Xolz 2|z 117 eaTABLE 5: Test Dice scores (%) for optic disc segmentation (ODS) and cardiac segmentation (CS) on (a) D1 and (b) D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Results for CS are averaged over the end-diastolic (ED) and end-systolic (ES) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Compared models are either adapted to only a single domain (1st row) or continually adapted to both target domains (2nd row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All results show the mean ± std-dev over 3 random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Superior results for the column are shown in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (a) D1 Training Test Dice Scores (%) on D1 ODS CS (ES + ED) D0→D1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 D0→D1→D2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 (b) D2 Training Test Dice Scores (%) on D2 ODS CS (ES + ED) D0→D2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 D0→D1→D2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 alization of models between domains, thereby necessitating effective continual adaptation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Similarly to ODS, our generator is able to create CS images with different domain characteristics for the same anatomical content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Additionally, GarDA’s generator does not suffer from catastrophic forgetting of previously seen domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Discussion In the following subsections, we provide more insights into our proposed method and the presented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In particular, we analyze the benefits of continual UDA over single-step UDA in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 and discuss the advantages of GarDA over ACE in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2, which also employs a form of appearance transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Finally, we examine the effectiveness of each of our proposed contributions via ablation studies in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Forward and backward transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' An alternative to continually adapting to new domains would be to only perform single-step UDA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', D0→D1 or D0→D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We show in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 5, that these alternatives are in fact inferior to the model that is continually trained on all domains D0→D1→D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For both ODS and CS, the continually trained model achieves better performance on D1 compared to UDA directly to D1, which suggests that the adaptation to a later target domain (D2) can indeed provide useful information and improved performance for a previous target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This can also be observed in the evolution of the domain-wise Dice scores in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 7, especially for ODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In the continual learning literature, this phenomenon is known as backward transfer [83] of learned information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In addition, we see in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 5b that the intermediate adaptation to D1 helps to improve the final Dice score on D2 for both ODS and CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This is known as forward transfer [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Both transfers indicate that, when effectively incorporated, information from additional domains can improve the accuracy and generalizability of a model also on other domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Owing to this, we find continual UDA to overall perform superior to single-step UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Furthermore, the continually trained model is able to segment images from all involved domains, not only two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This is generally beneficial, as one needs to keep only a single model with accumulated information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For instance, 0 1 2 Step 50 55 60 65 70 75 80 85 90 95 100 Dice per domain [%] (a) ODS 0 1 2 Step 50 55 60 65 70 75 80 85 90 95 100 Dice per domain [%] 0 1 2 (b) CS Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 7: Evolution of the mean domain-wise Dice scores on D0, D1, and D2 after each step of training in the continual sequence for (a) ODS and (b) CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' when a care facility acquires a new imaging device, with single-step UDA, they would need to find the right model suitable for this device (or collect new data to adapt from source) but with continual UDA, they can obtain the latest continual model and expect it to perform well out-of-the-box if it has seen similar images earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Comparison to ACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In order to generate images from previous target domains, ACE first extracts feature maps from a source image and re-normalizes them with appearance statistics drawn from a memory buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The memory buffer represents previously seen target domain images in the form of mean and variance values of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The re-normalized features are then fed to a decoder (an inverted VGG19 [78]) that maps the features to the desired target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' There are several shortcomings with this approach: First, ACE stores a fixed number of mean and variance values in the memory buffer, which cannot accurately represent the appearance infor- mation of the entire distribution of target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In contrast, we employ a GAN trained to reproduce images from this entire distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Indeed we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 8 the superior quality of the images generated by our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' While ACE’s synthesized images exhibit severe checkerboard artifacts, the images produced by our generator appear cleaner and closer to real samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Second, ACE always requires a source image and a set of target domain statistics to generate a target image, which limits the number and diversity of samples that can be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' On the other hand, our proposed generator is able to produce arbitrarily many images from any domain using a simple random noise vector input and a domain label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Third, the inverted VGG decoder of ACE comprises more than 140 million parameters, whereas our GAN (generator and discriminator together) contains merely 23 million parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Thus, the performance gain from our method GarDA also comes at a lower budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We conduct ablation experiments for each proposed component in GarDA to quantify their indi- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 8: Examples of cardiac magnetic resonance (MR) images synthesized given the same content z by ACE (top) and our proposed generator (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' TABLE 6: Test Dice scores (%) of ablations on optic disc segmentation (ODS), at the end of the continual sequence D0 (Canon) →D1 (Kowa) →D2 (Nidek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For clarity, mean results are shown without std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The difference between our method and its variations with ablated components is shown in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Training Method Test Dice Scores (%) D0 D1 D2 overall D0 LB 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 D0→D1→D2 Ours − Lcon 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 (−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 (−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 (+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 (−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3) Ours − Limg G 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 (−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5) 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 (−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 (−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3) 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 (−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9) Ours − Ladv 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 (−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 (−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 (−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1) 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 (−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1) Ours 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 D1 SD-UB − 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 − − D2 SD-UB − − 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 − D0+D1+D2 MD-UB 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 vidual impact on the final segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This evaluation is performed for ODS since the large gap between lower and upper bound allows us to clearly demonstrate performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All results are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The impact of the content loss can be seen in comparison to the ablation “Ours−Lcon”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Omitting the content loss causes the model to lose performance on previously seen domains, exemplified by drops in domain-wise Dice scores by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 pp and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 pp on D0 and D1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Interestingly, the removal of the content loss increases the Dice score on the last target domain by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This is likely because the generator is less constrained to retain the same content across domains, which allows it to better reproduce the appearance of the newer (last) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, this then comes at the cost of a larger drop in performance on the earlier domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Our proposed GAN distillation losses, consisting of Ldis G =Ladv G +λimgLimg G for the generator (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (10)) and Ldis D for the discriminator (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (9)), are designed to prevent forgetting of the source and intermediate target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We analyze the individual effects of adversarial and non- adversarial components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The only non-adversarial component is the image distillation loss, the ablation of which we tabulate as “Ours−Limg G ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The results show that omitting image distillation has a major – indeed the biggest – impact on the segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Namely, the Dice scores drop for all domains by more than 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For ablating the adversarial components Ladv=Ladv G +Ldis D , we tabulate the results as “Ours−Ladv”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The impact is similar to the image distillation, but does not affect the source domain performance as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In summary, our ablation study confirms that each of the proposed components is essential to achieve our new state- of-the-art segmentation performance for continual domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Conclusions In this work, we have proposed GarDA, a novel segmen- tation method for continual UDA that can adapt to new domains without forgetting previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' As GarDA employs generative appearance replay, it does not require to store previously seen data, neither from the source domain nor from any intermediate target domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' To the best of our knowledge, GarDA is the first segmentation method for such strictly continual UDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' This makes our approach widely applicable in practice, where data from previous domains cannot be retained indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' A potential limitation of our approach is the duration of the GAN training, which may take several hours depending on the desired quality of the domain appearance adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' However, since new data from different domains usually does not become available on a daily basis, this should not be an issue in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' We evaluate the generalizability of our method by conduct- ing comprehensive experiments for two very different tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', optic disc segmentation on color fundus photography im- ages and cardiac segmentation on magnetic resonance images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Our results demonstrate that GarDA substantially outperforms all competing methods on both tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In addition, we provide qualitative results showing that our proposed generator is able to synthesize meaningful images from different domains that enable the segmentation model to be trained continually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Our ablation study highlights the impact of each proposed major ACE Ourscomponent in GarDA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=', the content loss, image distillation, and adversarial distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Overall, the trends observed in our experiments indicate GarDA to potentially yield better results for increasingly longer domain sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' In the future, when new datasets from different domains become available, longer continual sequences with more domains shall be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Acknowledgments We would like to thank Krishna Chaitanya and Neerav Karani for insightful discussions as well as their valuable advice on 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Chanan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Killeen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Lin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Gimelshein, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Antiga, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Desmaison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' K¨opf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' DeVito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Raison, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Tejani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Chilamkurthy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Steiner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Fang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Bai, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Chintala, “PyTorch: An Imperative Style, High- Performance Deep Learning Library,” in Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Neural Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (NeurIPS), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' [83] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Lopez-Paz and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Ranzato, “Gradient episodic memory for continual learning,” Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Neural Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' (NeurIPS), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Appendix A Detailed cardiac segmentation results Individual results for the end-systolic (ES) and end-diastolic (ED) phases are presented in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' A1 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' For the majority of labels and domains, it can be observed that our approach achieves the best or the second-best results in terms of label-wise and domain-wise Dice scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' TABLE A1: Label-wise test Dice scores (%) for cardiac segmentation (CS) in the end-systolic (ES) phase, at the end of the continual sequence D0 (Philips)→D1 (Siemens)→D2 (Canon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All results are given as mean ± std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' over 3 random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The best and the second-best results per column (excluding LB and UB) are in bold and underlined, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Training Method Test Dice Scores ES (%) D0 D1 D2 overall MYO LV RV avg MYO LV RV avg MYO LV RV avg D0 LB 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 D0→D1→ D2 AdaptSegNet [45] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 AdvEnt [46] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ACE [17] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 MuHDi [16] 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 GarDA (ours) 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 D1 SD-UB − − − − 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 − − − − − D2 SD-UB − − − − − − − − 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 − D0+D1+D2 MD-UB 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 TABLE A2: Label-wise test Dice scores (%) for cardiac segmentation (CS) in the end-diastolic (ED) phase, at the end of the continual sequence D0 (Philips)→D1 (Siemens)→D2 (Canon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' All results are given as mean ± std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' over 3 random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' The best and the second-best results per column (excluding LB and UB) are in bold and underlined, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content=' Training Method Test Dice Scores ED (%) D0 D1 D2 overall MYO LV RV avg MYO LV RV avg MYO LV RV avg D0 LB 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 D0 → D1 → D2 AdaptSegNet [45] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 AdvEnt [46] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 ACE [17] 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 MuHDi [16] 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 65.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 D1 SD-UB − − − − 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 − − − − − D2 SD-UB − − − − − − − − 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 − D0 + D1 + D2 MD-UB 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='0 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} +page_content='9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQfRPsg/content/2301.01211v1.pdf'} diff --git a/ztE2T4oBgHgl3EQfiAcw/content/tmp_files/2301.03953v1.pdf.txt b/ztE2T4oBgHgl3EQfiAcw/content/tmp_files/2301.03953v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4918f028aff3776ddb0c4d9b43ea571b7911ed0 --- /dev/null +++ b/ztE2T4oBgHgl3EQfiAcw/content/tmp_files/2301.03953v1.pdf.txt @@ -0,0 +1,2542 @@ +1 +Channel-aware Decoupling Network for Multi-turn +Dialogue Comprehension +Zhuosheng Zhang, Hai Zhao, Longxiang Liu +Abstract—Training machines to understand natural language +and interact with humans is one of the major goals of artificial +intelligence. Recent years have witnessed an evolution from +matching networks to pre-trained language models (PrLMs). +In contrast to the plain-text modeling as the focus of the +PrLMs, dialogue texts involve multiple speakers and reflect +special characteristics such as topic transitions and structure +dependencies between distant utterances. However, the related +PrLM models commonly represent dialogues sequentially by +processing the pairwise dialogue history as a whole. Thus +the hierarchical information on either utterance interrelation +or speaker roles coupled in such representations is not well +addressed. In this work, we propose compositional learning for +holistic interaction across the utterances beyond the sequential +contextualization from PrLMs, in order to capture the utterance- +aware and speaker-aware representations entailed in a dialogue +history. We decouple the contextualized word representations +by masking mechanisms in Transformer-based PrLM, making +each word only focus on the words in current utterance, other +utterances, and two speaker roles (i.e., utterances of sender and +utterances of the receiver), respectively. In addition, we employ +domain-adaptive training strategies to help the model adapt to +the dialogue domains. Experimental results show that our method +substantially boosts the strong PrLM baselines in four public +benchmark datasets, achieving new state-of-the-art performance +over previous methods. +Index Terms—Dialogue Modeling, Open Domain Conversation +System, Natural Language Generation, Deep Neural Networks. +I. INTRODUCTION +L +Anguage is not only an effective medium for people +to communicate with each other but also a natural +interface between humans and machines. However, building +an intelligent dialogue system that can understand human +Z. Zhang and H. Zhao are with the Department of Computer Science and +Engineering, Shanghai Jiao Tong University, and also with Key Laboratory +of Shanghai Education Commission for Intelligent Interaction and Cognitive +Engineering, Shanghai Jiao Tong University, and also with MoE Key Lab of +Artificial Intelligence, AI Institute, Shanghai Jiao Tong University. L. Liu +is with Key Laboratory of Intelligent Information Processing Institute of +Computing Technology, Chinese Academy of Sciences (ICT/CAS). This work +was conducted when L. Liu was with the Department of Computer Science and +Engineering, Shanghai Jiao Tong University. E-mail: zhangzs@sjtu.edu.cn, +zhaohai@cs.sjtu.edu.cn, liulongxiang21s@ict.ac.cn. Z. Zhang and L. Liu +contribute equally to this work. (Corresponding author: Hai Zhao) +This work was partially supported by Key Projects of National Natural +Science Foundation of China (U1836222 and 61733011). +Part of this study has been accepted as ”Filling the Gap of Utterance-aware +and Speaker-aware Representation for Multi-turn Dialogue” [1] in the Thirty- +Fifth AAAI Conference on Artificial Intelligence (AAAI 2021), with partially +material overlapped. This article extends the conference version by studying +channel-aware decoupling in a broader view of multi-turn dialogue modeling. +Towards this goal, we extend the descriptions in Introduction, Related Work, +Model, Experiments, and Analysis correspondingly. For the techniques, this +work extends the proposed model with domain-adaptive strategies, more +baselines, and comprehensive analyses with new conclusions. +TABLE I +AN EXAMPLE OF RESPONSE-SELECTION FOR MULTI-TURN DIALOGUE IN +MUTUAL DATASET. F AND M DENOTE DIFFERENT SPEAKERS. +Utterance (Context) +F: Excuse me, sir. This is a non smoking area. +M: Oh, sorry. I will move to the smoking area. +F: I’m afraid no table in the smoking area is available now. +Response Candidates +A. Sorry. I won’t smoke in the hospital again.  +B. OK. I won’t smoke. Could you please give me a menu?  +C. Could you please tell the customer over there not to smoke? +We can’t stand the smell  +D. Sorry. I will smoke when I get off the bus.  +conversations and give logically correct, fluent responses is +one of the eternal goals of artificial intelligence. It has been +drawing increasing interest from both academia and industry +areas. The methods of building a chatbot that is capable of +performing multi-turn dialogue can be categorized into two +lines: generation-based methods and retrieval-based methods. +Generation-based methods [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] directly +generate a response using an encoder-decoder framework, +which tends to be short and lacks diversity. Retrieval-based +methods [12, 13, 14, 15, 16, 17, 18, 19] retrieve a list of +response candidates, then use a model to rank the candidates +and select the best one as a reply. Since the responses of +retrieval-based are generally more natural, fluent, and syn- +tactically correct, retrieval-based methods are more mature +for producing multi-turn dialogue systems both in academia +and industry [20, 21, 22], which is our major focus in this +work. Table I shows an example from Multi-Turn Dialogue +Reasoning dataset (MuTual) [23]. In order to choose the right +answer, the machine is required to understand and infer from +the meaning of ”table” and its coreference, indicating the +requirement of reasoning ability instead of simple matching. +Early studies concerning dialogue comprehension mainly +focus on the matching networks that calculate similarity scores +between the pairwise sequence of dialogue context and candi- +date response at different granularities. The matching matrices +will be fused to get a feature vector, then the sequence of +feature vectors will be further integrated by RNNs to get the +final representation for scoring. However, these methods have +two sides of disadvantages. First, interactions mainly involve +each utterance and response, ignoring the global interactions +between utterances. Second, the relative positions between the +response and different utterances are not taken into consider- +ation, lacking the sequential information of context-response +pairs. +Copyright © 20XX IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including +reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any +copyrighted component of this work in other works. +arXiv:2301.03953v1 [cs.CL] 10 Jan 2023 + +2 +Recently, pre-trained language models (PrLMs), such as +BERT [24], RoBERTa [25], and ELECTRA [26], have +achieved impressive performance in a wide range of NLP +tasks [27, 28, 29, 30, 31, 32, 33]. The word embeddings +derived by these language models are pre-trained on large +corpora and are then fine-tuned in task-specific datasets. In +the multi-turn dialogue scope, there are also some efforts +using PrLMs to yield promising performances [34, 35]. The +mainstream is employing PrLMs as encoders by concatenating +the context and candidate responses directly, as a linear +sequence of successive tokens and implicitly capturing the +contextualized representations of those tokens through self- +attention [36, 37, 38, 39]. However, simply embedding the +token to high-dimensional space cannot faithfully model the +dialogue-related information, such as positional or turn order +information [40]; In addition, the mechanism of self-attention +runs through the whole dialogue, resulting in entangled infor- +mation that originally belongs to different parts. +Multi-turn dialogue modeling has critical challenges of tran- +sition and inherency, which the existing multi-turn dialogue +methods rarely consider to our best knowledge. +1) There exists speaker role switching in multi-turn dia- +logue. Each speaker has different thinking habits and speak- +ing purposes, which lead to the unique speaking style of +each speaker’s role and fuzziness of utterance coreference. +Therefore, it is the essential difference between dialogue and +passage. Although Speaker-Aware BERT [38] has considered +role transitions, adding the embedding of speaker-id into the +inputs of PrLMs is not effective enough. +2) Utterances have their own inherent meaning and con- +textual meaning. A clear understanding of local and global +meaning can reduce the negative influence of irrelevant content +in a long-distance context. Some studies on neural machine +translation have taken this into account and achieved gratifying +results[41], which inspire this work. +3) The widely adopted PrLMs are pre-trained on general +corpora, which would be less effective to directly fine-tune +these models on downstream tasks if there is a domain shift. +A recent trend is to post-train PrLMs on dialogue corpus and +achieve state-of-the-art results [42, 38, 39]. We also employ +this post-training method in this work and investigate whether +it can be enhanced by modeling the dialogue properties +discussed above. +In this work, beyond the sequential contextualization from +PrLMs, we proposed a novel end-to-end Channel-aware De- +coupling Network (CDN) to enhance dialogue comprehension +with compositional utterance interaction and thus fill the +obvious gap of utterance-aware and speaker-aware represen- +tations. In detail, the contextualized word representation of +dialogue text is decomposed into four parts that contain dif- +ferent information and then fused after sufficient interactions. +More specifically, the PrLM receives concatenated context and +response and outputs the contextualized representation of each +word. For each word, we use a masking mechanism inside a +self-attention network to limit the focus of each word only +to the words in current utterance, other utterances, utterances +of the sender, and utterances of the receiver respectively. To +avoid ambiguity, we call the first two complementary parts the +utterance-aware channel and the last two the speaker-aware +channel. We then fuse the information inside each channel via +a gating mechanism to control the information reservation. +The word-level information will be further aggregated to the +utterance level. For utterance-level representations, BiGRU is +adopted to get dialogue-level representations. Inspired by Tao +et al. [13], we put the information fusion of two channels at the +end to get the final dialogue representation for classification. +In addition, we employ domain-adaptive training strategies to +help the model adapt to the dialogue domains. Experimental +results on four public benchmark datasets show that the +proposed model outperforms the baseline models substantially +on all the evaluation metrics and achieves new state-of-the-art +results. +In summary, our contributions are mainly in three folds: +1) A novel end-to-end Mask-based Decoupling-Fusion Net- +work is proposed to fill the gap between utterance- +aware and speaker-aware representations for multi-turn +dialogue. +2) An effective domain-adaptive post-training method with +intra-utterance and inter-utterance objectives to better +adapt PrLMs to our dialogue comprehension tasks. +3) Experimental results on public datasets show the su- +periority of our method and verify the effectiveness +of our model beyond strong PrLMs. We have also +achieved various state-of-the-art performances compared +with existing methods. +II. BACKGROUND AND RELATED WORK +A. Multi-turn Dialogue Comprehension +As an active research topic in the NLP field, multi-turn +dialogue comprehension has attracted great attention from both +academia and industry whose aim is to teach machines to read +dialogue contexts and solve typical tasks such as response +selection [43, 12, 15, 23, 44]. However, selecting a coherent +and informative response for a given dialogue context remains +a challenge. The multi-turn dialogue typically involves two or +more speakers that engage in various conversation topics and +intentions. Thus the utterances are rich in interactions, e.g., +with criss-cross discourse structures [45, 46, 47]. A critical +challenge is learning rich and robust context representations +and interactive relationships of dialogue utterances so that the +resulting model can adequately capture the semantics of each +utterance and the relationships among all the utterances inside +the dialogue. Previous studies can be classified into two lines: +matching models and pre-trained language models. +B. Matching Models +Matching models aim to match the response with contexts +and calculate the matching score. They can be divided into +two categories: single-turn and multi-turn matching models. +Earlier research mainly considered the context utterances as +one single utterance, using the dual encoder framework to +encode the whole context and the response, respectively. The +encoder varies in LSTM [48], CNNs [49], Attentive-LSTM +[50]. + +3 +Encoder +... +... +U1/S1 +U2/S2 +R/S1 +Gated Fusing +Gated Fusing +GRU +GRU +GRU +GRU +GRU +GRU +Classifier +U1 +U2 +R +U1 +U2 +R +U1 +U2 +R +U1 +U2 +R +S1 +S2 +S1 +S1 +S2 +S1 +Current Utterance (C1) +Other Utterance (C2) +S1 +S2 +S1 +S1 +S2 +S1 +Utterance of Receiver (C3) Utterance of Sender (C4) +Encoding +Decoupling +Interaction +Fig. 1. Overall framework of CDN. Here for simplicity, inputs contain two utterances and one response spoken by two speakers with total six words. A more +detailed figure and description on the decoupling block will be shown below. The decoupling block is composed of four independent self-attention blocks +with the same inputs and different masks. The same word in different blocks attends to different scope of information. +As for multi-turn matching, Sequential Matching Network +(SMN) [12] is the representative work, where a word-word +and a sequence-sequence similarity matrix will be calculated, +forming a matching image for each utterance-response pair +which will be further integrated by CNN and RNN. Several +related works like Deep Utterance Aggregation (DUA) [15], +Deep Attention Matching Network (DAM) [16], Interaction- +over-Interaction (IoI) [14] are extensions of SMN from differ- +ent perspectives. DUA [15] points out that the last utterance +should be considered explicitly and use self-attention to get a +better segment matching matrix. DAM [16] uses hierarchi- +cally stacked layers of self-attention to represent matching +from different levels. IoI [14] mainly deepens the layers of +response-utterance interaction blocks and combines final states +at different depths. +C. Pre-trained Language Models +Recently, inspired by the impressive performance of PrLMs, +the mainstream is employing PrLMs to handle the whole +pairwise texts as a linear sequence of successive tokens and +implicitly capture the contextualized representations of those +tokens through self-attention [36, 37, 38, 39]. The word +embeddings derived by these language models are pre-trained +on large corpora and are then utilized as either distributed +word embeddings [51] or fine-tuned according to the specific +task needs [24]. Most of the pre-trained Language Models +(PrLMs) are based on Transformer, among which Bidirectional +Encoder Representations from Transformers (BERT) [24] is +one of the most representative works. BERT uses multiple lay- +ers of stacked Transformer Encoder to obtain contextualized +representations of the language at different levels. BERT has +achieved unprecedented performances in many downstream +tasks of NLP. Several subsequent variants have been proposed +to further enhance the capacity of PrLMs, such as RoBERTa +[25], ALBERT [52], and ELECTRA [26]. To evaluate our +methods on both sides of the fast inference speed and strong +performance, we will adopt BERT and ELECTRA as our +backbone models in this work. +III. CHANNEL-AWARE DECOUPLING NETWORK +Our proposed model, named Channel-aware Decoupling +Network (CDN), consists of six parts: Encoding, Decoupling, +Fusing, Words Aggregating, Utterances Integrating, and Scor- +ing. In this section, we will formulate the problem and then +introduce each sub-module in detail. +A. Problem Formulation +Suppose that we have a dataset D = {(yi, ci, ri)}N +i=1, +where ci = {ui,1, ..., ui,ni} represents the dialogue context +with {ui,k}ni +k=1 as utterances. ri is a response candidate, and +yi ∈ {0, 1} denotes a label. The goal is to learn a discriminator +g(·, ·) from D, and at the inference phase, given any new +context c and response r, we use the discriminator to calculate +g(c, r) as their matching score. To select several best responses +as return to human inputs, we rank matching scores or set a +threshold among the list of candidates. +B. Model Overview +Figure 1 shows the overall framework of CDN. CDN first +encodes each word in the concatenated context and response to +contextualized representation using the PrLMs. In the decou- +pling module, a self-attention mechanism with different masks +forces each word to only focus on the information in spe- +cific scopes, decoupling to utterance-aware and speaker-aware +channels. Note that now the information flow inside different +channels is independent. In each channel, complementary +information will be fused by a gate. Then word representations +belonging to the same utterance will be aggregated to represent +the utterance. The sequence of utterance representations will + +4 +be delivered to Bidirectional Gated Recurrent Unit (BiGRU) to +produce the channel-aware dialogue-level representation. After +that, the information from different channels is fused to get the +final dialogue representation. The classifier will then output a +score of the dialogue to reflect the matching degree between +context and candidate response. +CDN is superior to existing methods in the following +ways. First, compared to SA-BERT [38], the speaker-aware +information is built upon semantically rich contextualized +representations instead of a simple additional embedding term, +making its influence on dialogue scoring preserved better. +Second, compared to the work of designing a complicated +matching network, our structure is simpler and can represent +information at different levels in a more unified manner. Third, +compared to pure PrLMs, the additional layer of CDN can +decouple information to different channels, making full use of +the contextualized representations. Besides, since the number +of utterances is far less than that of words, the top BiGRU +layers can keep their memory capacity, alleviating the defects +of the Transformer in reflecting relative position information. +C. Encoding Context and Response +We employ a pre-trained language model such as ELEC- +TRA to obtain the initial word representations. The tokens +in utterances and response are concatenated successively and +then fed into the encoder. Following Devlin et al. [24], we add +“[CLS]” token in the front of the sequence and insert “[SEP]” +between adjacent utterances. Let Ui = [ui,0, ui,1, ..., ui,ni], +R = [r0, r1, ..., rnr], where ni and nr is respectively the +length of i-th utterances and response, then the output is +E = [e1, e2, ..., en0+...+nk+nr] where k is the maximum +number of utterances, and ei ∈ Rd, d is hidden size. +D. Channel-aware Information Decoupling +We first introduce the multi-head self-attention (MHSA) +with mask, which can be formulated as: +Attention(Q, K, V, M) = softmax(QKT +√dk ++ M)V, +headi = Attention(EW Q +i , EW K +i , EW V +i , M), +MHSA(E, M) = Concat(head1, ..., headh)W O, +(1) +where W Q +i +∈ Rdmodel×dq, W K +i +∈ Rdmodel×dk, W V +i +∈ +Rdmodel×dv, W O +i +∈ Rhdv×dmodel are parameter matrices, dq, +dk, dv denote the dimension of Query vectors, Key vectors +and Value vectors, h denotes the number of heads. M denotes +the mask. +We have four masks {Mk}4 +k=1 ∈ Rl×l defined as: +M1[i, j] = +� +0, +if Ti = Tj +−∞, +otherwise +M2[i, j] = +� +0, +if Ti ̸= Tj +−∞, +otherwise +M3[i, j] = +� +0, +if Si = Sj +−∞, +otherwise +M4[i, j] = +� +0, +if Si ̸= Sj +−∞, +otherwise +(2) +where i, j is the word position in the whole dialogue, Ti is +the index of utterance the i-th word is located in, and Si is the +speaker that the i-th word is spoken by. And M1, M2, M3, M4 +only attend to the word in current utterance, other utterances, +utterances of sender, and utterances of receiver. We call the +first two utterance-aware channels and the last two speaker- +aware channels. +The decoupled channel-aware information {Ck}4 +k=1 ∈ Rl×d +are derived by multi-head self-attention with mask. +Ci = MHSA(E, Mi), i ∈ {1, 2, 3, 4}, +(3) +where E ∈ Rl×d is the output of PrLMs stated in Section +III-C. +E. Complementary Information Fusing +To fuse the complementary information inside each channel, +we use a gate to fuse them. Inspired by [53], we use a gate +to calculate the ratio of information preservation based on +matching heuristics considering the “Siamese” architecture as +information from two parts, the element-wise product and +difference as “similarity” or “closeness” measures. Let E +denote the original representations output from PrLMs. For +utterance-aware channel, ¯E = C1 and ˆE = C2. For speaker- +aware channel, ¯E = C3 and ˆE = C4. Ci is defined in Equation +(3). The gate is formulated as: +˜ +E1 = ReLU(FC([E, ¯E, E − ¯E, E ⊙ ¯E])), +˜ +E2 = ReLU(FC([E, ˆE, E − ˆE, E ⊙ ˆE])), +P = Sigmoid(FC(([ ˜ +E1, ˜ +E2]))), +G(E, ¯E, ˆE) = P, +(4) +where Sigmoid, ReLU [54] are activation functions, FC is +fully-connected layer, [·, ·] means concatenation, and ⊙ is +element-wise multiplication. +Using two parametric-independent gates, the channel-aware +information can be fused, which is defined as: +P1 = G1(E, C1, C2), +P2 = G2(E, C3, C4), +Cu = P1 ⊙ C1 + (1 − P1) ⊙ C2, +Cs = P2 ⊙ C3 + (1 − P2) ⊙ C4, +(5) +where the calculation of {Gi}2 +i=1 is defined in Equation 4, Cu +and Cs is the fused utterance-aware and speaker-aware word +representations, respectively. +F. Utterance Representations +For each channel, word representations will be aggregated +by simple max-pooling over words in the same utterance to +get the utterance representations. Let Lu and Ls be the output +in this part. Then they are defined as: +Lu[i, :] = MaxPooling +Tj=i +(Cu[j, :]) ∈ Rd, +Ls[i, :] = MaxPooling +Tj=i +(Cs[j, :]) ∈ Rd. +(6) + +5 +G. Dialogue Representation +To get the channel-aware dialogue representation, the se- +quence of utterance representations will be delivered to Bi- +GRU. Suppose that the hidden states of the BiGRU are +(h1, . . . , hk), then ∀j, 1 ≤ j ≤ k, hj ∈ R2d is given by +←− +h j = ←−− +GRU(←− +h j−1, ←− +L[j]), +−→ +h j = −−→ +GRU(−→ +h j−1, −→ +L[j]), +hj = [←− +h j; −→ +h j]. +(7) +where L[j] denotes the sentence representation derived from +Eq. (6). For each channel, we take the hidden state of BiGRU +at the last step as channel-aware dialogue representation. Let +the two vectors be v1 and v2. Then two channels can be easily +fused to get the final dialogue representation. +v = Tanh(W[v1; v2] + b), +(8) +where W ∈ Rd×4d, b ∈ Rd are trainable parameters. Tanh is +the activation function. +H. Scoring +The dialogue vector will be fed into a classifier with a +fully connected and softmax layer. We learn model g(·, ·) by +minimizing cross-entropy loss with dataset D. Let Θ denote +the parameters of CDN. For binary classification like ECD, +Douban, and Ubuntu, the objective function L(D, Θ) can be +formulated as: +− +N +� +i=1 +[yilog(g(ci, ri)) + (1 − yi)log(1 − g(ci, ri))]. +(9) +where N is the number of training data. For multiple-choice +tasks like MuTual, the loss function is: +− +N +� +i=1 +C +� +k=1 +yi,klog(g(ci, ri,k)). +(10) +where C is the number of candidate response options for each +input context. +IV. DOMAIN-ADAPTIVE POST-TRAINING +Although the PrLMs demonstrate superior performance due +to their strong representation ability from self-supervised pre- +training, it is still challenging to effectively adapt task-related +knowledge during the detailed task-specific training, which is +usually in the way of fine-tuning [55]. Generally, those PrLMs +handle the whole input text as a linear sequence of successive +tokens and implicitly capture the contextualized representa- +tions of those tokens through self-attention. The such fine- +tuning paradigm of exploiting PrLMs would be suboptimal to +model dialogue tasks that hold exclusive text features that plain +text for PrLM training may hardly embody. In this following +section, we first describe the standard general-purpose training +in BERT, and then present our extensions to the concerned +dialogue domains. +A. General-Purpose Training +As the standard pre-training procedure, PrLMs are pre- +trained on large-scale domain-free texts and then used for fine- +tuning according to the specific task needs. There are token- +level and sentence-level objectives used in the general-purpose +pre-training. The most widely-used PrLM for domain-adaption +in the dialogue field is BERT [24], whose pre-training is based +on two loss functions: (1) a masked language model (MLM) +loss, and (2) a next sentence prediction (NSP) loss. MLM +first masks out some tokens from the input sentences and then +trains the model to predict them by the rest of the tokens. +NSP is another widely used pre-training objective, which trains +the model to distinguish whether two input sentences are +continuous segments from the training corpus. +B. Domain-Adaptive Training +The original PrLMs are trained on a large text corpus to +learn general language representations. To incorporate specific +in-domain knowledge, adaptation on in-domain corpora, also +known as domain-aware pre-training, is designed, which di- +rectly employs the original PrLMs as mentioned in the general- +purpose paragraph above, using the dialogue-domain corpus. +In this work, we extend the MLM and NSP as intra-utterance +and inter-utterance objectives to coordinate with our dialogue +comprehension tasks. +• Intra-utterance Objective To better adapt PrLMs to dia- +logue scenarios, we conduct the post-training with three levels +of masks for MLM, including subword, word, and span-level +masks. Subword is the original method used in BERT [24], +and word means the whole-word-masking (WWM) where the +whole word will be masked during the data preprocessing. The +Span masking mechanism follows [56], which masks contigu- +ous random spans from a geometric distribution rather than +random tokens. Compared with subword masks, both span and +WWM mechanisms would potentially improve the model’s +ability to capture high-level topic information in dialogue texts. +The training objective is then formed as follows: +Lintra = − +N +� +k=1 +[mi log ˆmi] , +(11) +where ˆmi denotes the predicted token id and mi is the ground- +truth. N is the number of examples. +• Inter-utterance Objective For a dialogue context with +multiple utterances c = {u1, ..., un}, we form a context text +c′ = {u1, ..., uk−1}, k ∈ [1, n] for each utterance u+ +k . We +then randomly sample an utterance u− +k from the corpus as the +negative example. The goal is to identify utterance uk as the +true utterance against the negative one by feeding sequence +“[CLS] c′ [SEP] uk [SEP]” into BERT. Following the setting +proposed in BERT, we take the hidden representation of [CLS] +through a full-connected layer followed by a sigmoid function +for classification. Here, we minimize the cross entropy loss in +post-training. +Linter = − +N +� +k=1 +[si log ˆsi] , +(12) + +6 +where ˆsi denotes the predicted label and si is the ground-truth. +N is the number of examples. +During post-training, we combine both of the intra- and +inter- utterance objectives: +L = Lintra + Linter. +(13) +V. EXPERIMENTS +A. Datasets +We tested our model on two English datasets: Ubuntu Dia- +logue Corpus (Ubuntu) and Multi-Turn Dialogue Reasoning +(MuTual), and two Chinese datasets: Douban Conversation +Corpus (Douban) and E-commerce Dialogue Corpus (ECD). +1) Ubuntu Dialogue Corpus: Ubuntu [43] consists of En- +glish multi-turn conversations about technical support col- +lected from chat logs of the Ubuntu forum. The dataset +contains 1 million context-response pairs, 0.5 million for +validation, and 0.5 million for testing. In the training set, each +context has one positive response generated by humans and +one negative response sampled randomly. In the validation and +test sets, for each context, there are 9 negative responses and +1 positive response. +2) Douban Conversation Corpus: Douban [12] is different +from Ubuntu in the following ways. First, it is an open domain +task where dialogues are extracted from the Douban Group. +Second, Response candidates on the test set are collected +by using the last turn as the query to retrieve 10 response +candidates and labeled by humans. Third, there could be more +than one correct response for a context. Response candidates +on the test set are collected by a standard search engine +Apache Lucene1, other than negative sampling without human +judgment on Ubuntu Dialogue Corpus. +3) E-commerce Dialogue Corpus: ECD [15] dataset is +extracted from conversations between customers and service +staff on Taobao. It contains over 5 types of conversations based +on over 20 commodities. There are also 1 million context- +response pairs in the training set, 0.5 million in the validation +set, and 0.5 million in the test set. +4) Multi-Turn Dialogue Reasoning: MuTual [23] consists +of 8,860 manually annotated dialogues based on Chinese stu- +dent English listening comprehension exams. For each context, +there is one positive response and three negative responses. +The difference compared to the above three datasets is that +only MuTual is reasoning-based. There are more than 6 types +of reasoning abilities reflected in MuTual. +B. Setup +For the sake of computational efficiency, the maximum +number of utterances is specialized as 20. The concatenated +context, response, ”[CLS]” and ”[SEP]” in one sample are +truncated according to the ”longest first” rule or padded to a +certain length, which is 256 for MuTual and 384 for the other +three datasets. Our model is implemented using Pytorch and +based on the Transformer Library. We use ELECTRA [26] as +our underlying model. AdamW [68] is used as our optimizer. +1http://lucene.apache.org/ +The batch size is 24 for MuTual, and 64 for others. The initial +learning rate is 4 × 10−6 for MuTual and 3 × 10−6 for others. +We run up to 3 epochs for MuTual and 2 epochs for others and +select the model that achieves the best result in the validation +process. +Our domain adaptive post-training for the corresponding +response selection tasks is based on the three large-scale +dialogue corpus, including Ubuntu, Douban, and ECD, respec- +tively. The data is the same as the training data in fine-tuning +but only used in unsupervised pre-training. Because there is +no appropriate domain data for the small-scale Mutual dataset, +we only report the fine-tuning results without post-training. +For the English tasks, we use the pre-trained weights bert- +base-uncased and electra-large-discriminator for fine-tuning; +for the Chinese tasks, the weights are from bert-base-chinese +and hfl/chinese-electra-large-discriminator.2 +C. Baseline Models +The following models are our baselines for comparison: +• Single-turn matching models: Single-turn Matching +Models consider the context utterances as one single utterance, +using the dual encoder framework to encode the whole context +and the response, respectively. The variants include CNN, +LSTM, BiLSTM [59], MV-LSTM [60], and Match-LSTM [62] +construct the dialog context by concatenating utterances as a +long document, and match the dialog context with a candidate +response. +• Multi-turn matching models: Multi-turn matching mod- +els focus on the utterance-level interaction with responses. +The major difference lies in the different types of match- +ing, including cross-attention, self-attention, etc. The typical +studies include Sequential Matching Network (SMN) [12], +Deep Attention Matching Network (DAM) [16], Deep Ut- +terance Aggregation (DUA) [15], Interaction-over-Interaction +(IoI) [14] which have been stated in Section II-B. Besides, +Multi-Representation Fusion Network (MRFN) [13] matches +context and response with multiple types of representations. +Multi-hop Selector Network (MSN) [17] utilizes a multi-hop +selector to filter necessary utterances and match among them. +• PrLMs-based +models: PrLMs-based models apply +PrLMs to encode the dialogue history and candidate response +as a whole. BERT [24], RoBERTa [25], ALBERT [52], +ELECTRA [26] have been stated in Section II-B. Besides, +Option Comparison Network (OCN) [69] is involved, which +compares the options before matching response and contexts.3 +D. Evaluation Metrics +Following the previous work [43, 12], we calculate the +proportion of true positive responses among the top-k selected +responses from the list of n available candidates for one +context, denoted as Rn@k. Besides, additional conventional +metrics of information retrieval are employed on Douban: +Mean Average Precision (MAP) [70], Mean Reciprocal Rank +(MRR) [71], and precision at position 1 (P@1). +2Those weights are available in the Transformers repo: https://github.com/ +huggingface/transformers/. +3On the leaderboard of MuTual, since some work is not publicly available, +we will not introduce here. + +7 +TABLE II +TEST RESULTS ON UBUNTU, DOUBAN, AND E-COMMERCE DATASETS. † DENOTES THE METHODS WITH DOMAIN-ADAPTIVE POST-TRAINING. THE +EVALUATION RESULTS ARE COLLECTED FROM PUBLISHED LITERATURE [57, 34, 39, 58]. +Models +Ubuntu +Douban +E-commerce +R10@1 +R10@2 +R10@5 +MAP +MRR +P@1 +R10@1 +R10@2 +R10@5 +R10@1 +R10@2 +R10@5 +Single-turn models with concatenated matching +CNN [59] +0.549 +0.684 +0.896 +0.417 +0.440 +0.226 +0.121 +0.252 +0.647 +0.328 +0.515 +0.792 +LSTM [59] +0.638 +0.784 +0.949 +0.485 +0.537 +0.320 +0.187 +0.343 +0.720 +0.365 +0.536 +0.828 +BiLSTM [59] +0.630 +0.780 +0.944 +0.479 +0.514 +0.313 +0.184 +0.330 +0.716 +0.365 +0.536 +0.825 +MV-LSTM [60] +0.653 +0.804 +0.946 +0.498 +0.538 +0.348 +0.202 +0.351 +0.710 +0.412 +0.591 +0.857 +Match-LSTM [61] +0.653 +0.799 +0.944 +0.500 +0.537 +0.345 +0.202 +0.348 +0.720 +0.410 +0.590 +0.858 +Multi-turn matching network with separate interaction +Multi-View [62] +0.662 +0.801 +0.951 +0.505 +0.543 +0.342 +0.202 +0.350 +0.729 +0.421 +0.601 +0.861 +DL2R [63] +0.626 +0.783 +0.944 +0.488 +0.527 +0.330 +0.193 +0.342 +0.705 +0.399 +0.571 +0.842 +SMN [12] +0.726 +0.847 +0.961 +0.529 +0.569 +0.397 +0.233 +0.396 +0.724 +0.453 +0.654 +0.886 +DUA [15] +0.752 +0.868 +0.962 +0.551 +0.599 +0.421 +0.243 +0.421 +0.780 +0.501 +0.700 +0.921 +DAM [16] +0.767 +0.874 +0.969 +0.550 +0.601 +0.427 +0.254 +0.410 +0.757 +0.526 +0.727 +0.933 +MRFN [13] +0.786 +0.886 +0.976 +0.571 +0.617 +0.448 +0.276 +0.435 +0.783 +- +- +- +IMN [64] +0.794 +0.889 +0.974 +0.570 +0.615 +0.433 +0.262 +0.452 +0.789 +0.621 +0.797 +0.964 +IoI [14] +0.796 +0.894 +0.974 +0.573 +0.621 +0.444 +0.269 +0.451 +0.786 +0.563 +0.768 +0.950 +MSN [17] +0.800 +0.899 +0.978 +0.587 +0.632 +0.470 +0.295 +0.452 +0.788 +0.606 +0.770 +0.937 +G-MSN [58] +0.812 +0.911 +0.987 +0.599 +0.645 +0.476 +0.308 +0.468 +0.826 +0.613 +0.786 +0.964 +PrLM-based methods for fine-tuning +BERT-SS-DA [65] +0.813 +0.901 +0.977 +0.602 +0.643 +0.458 +0.280 +0.491 +0.843 +0.648 +0.843 +0.980 +TADAM [66] +0.821 +0.906 +0.978 +0.594 +0.633 +0.453 +0.282 +0.472 +0.828 +0.660 +0.834 +0.975 +PoDS [57] +0.828 +0.912 +0.981 +0.598 +0.636 +0.460 +0.287 +0.468 +0.845 +0.633 +0.810 +0.967 +ELECTRA [1] +0.845 +0.919 +0.979 +0.599 +0.643 +0.471 +0.287 +0.474 +0.831 +0.607 +0.813 +0.960 +SA-BERT† [38] +0.855 +0.928 +0.983 +0.619 +0.659 +0.496 +0.313 +0.481 +0.847 +0.704 +0.879 +0.985 +PoDS† [57] +0.856 +0.929 +0.985 +0.599 +0.637 +0.460 +0.287 +0.469 +0.839 +0.671 +0.842 +0.973 +BERT-VFT† [42] +0.858 +0.931 +0.985 +- +- +- +- +- +- +- +- +- +DCM† [18] +0.868 +0.936 +0.987 +0.611 +0.649 +- +0.294 +0.498 +0.842 +0.685 +0.864 +0.982 +UMSELECTRA[67] +0.854 +0.929 +0.984 +0.608 +0.650 +0.472 +0.291 +0.488 +0.845 +0.648 +0.831 +0.974 +UMSELECTRA† [67] +0.875 +0.941 +0.988 +0.623 +0.663 +0.492 +0.307 +0.501 +0.851 +0.707 +0.853 +0.974 +Our Implementation +ELECTRA +0.845 +0.919 +0.979 +0.599 +0.643 +0.471 +0.287 +0.474 +0.831 +0.607 +0.813 +0.960 ++ CDN +0.866 +0.932 +0.984 +0.624 +0.663 +0.498 +0.325 +0.511 +0.855 +0.639 +0.829 +0.971 +ELECTRA† +0.890 +0.947 +0.989 +0.625 +0.663 +0.483 +0.301 +0.513 +0.865 +0.657 +0.834 +0.977 ++ CDN† +0.914 +0.961 +0.993 +0.634 +0.674 +0.496 +0.312 +0.540 +0.868 +0.673 +0.857 +0.979 +E. Results +Tables II-III show the test results on four datasets. +1) Generally, the previous models based on multi-turn +matching networks perform worse than simple PrLMs-based +ones, illustrating the power of contextualized representations +in context-sensitive dialogue modeling. +2) PrLM can perform even better when equipped with CDN, +verifying the effectiveness of our model, where utterance- +aware and speaker-aware information can be better exploited. +3) It is observed that post-training on dialogue data can sub- +stantially boost the baseline, i.e., by 4.5% R10@1 on Ubuntu. +Even on the backbone baseline with domain-adaptive post- +training, CDN can still yield consistent performance gains, +which indicates that the CDN architecture is generally effec- +tive even working with stronger backbone PrLM encoders after +post-training. A similar phenomenon is observed in existing +studies [16, 13, 64, 14] that post-training on dialogue data +brings more improvements than modifying attention architec- +tures in Transformers. However, post-training is less efficient, +which requires costly training on an additional domain-specific +corpus. +4) Our model outperforms other models in most metrics. +CDN surpasses the previous SOTA SA-BERT model on +most datasets, which is augmented by extra domain adap- +tation strategies to conduct language model pre-training on +in-domain dialogue corpus before fine-tuning on tasks. In +addition, CDN also ranks the best on the MuTual leaderboard.4 +VI. ANALYSIS +A. Ablation Study +Since our model first decouples information to utterance- +aware and speaker-aware channels and then uses BiGRU +to learn a channel-aware dialogue representation from the +sequence of utterance representations, we wonder about the +effect of two channels and whether BiGRU can be replaced +by simple pooling. First, we visualize the attention distri- +bution in C1, C2, C3 and C4 as shown in Figure 2. The +illustration shows that channel-aware decoupling operations +distinguish the four parts of utterance-aware and speaker- +aware representations. +Then, we perform an ablation study +on MuTual dev sets as shown in Table IV.5 Results show that +each part is necessary. The most important part is speaker- +aware information since speaker role transition is an essential +feature in multi-party dialogues. Then it comes to BiGRU, +4https://nealcly.github.io/MuTual-leaderboard/ +5We conduct the ablation studies on MuTual because MuTual is of the +highest quality, which is a manually annotated dialogues based on English +listening comprehension exams. + +8 +a) C1 +b) C2 +c) C3 +d) C4 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +. +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +. +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +. +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +[CLS] +f +: +why +are +you +so +worried +, +peter +? +the +exam +? +[SEP] +m +: +no +, +mom +. +i +finished +preparing +for +my +exam +three +weeks +ago +. +i +' +m +worried +about +... +[SEP] +f +: +just +keep +calm +and +keep +working +hard +. +[SEP] +Fig. 2. Visualization of attention distribution in C1, C2, C3 and C4, which indicates that channel-aware decoupling operations distinguish the four parts of +utterance-aware and speaker-aware representations. The example context is from the MuTual validation set. The text is “[CLS] f : why are you so worried , +peter ? the exam ? [SEP] m : no , mom . i finished preparing for my exam three weeks ago . i ’ m worried about my paper . you know , the deadline is +coming up . but i have n ’ t collected enough information . if i ca n ’ t finish it on time , my teacher will not be happy with me . [SEP] f : just keep calm +and keep working hard . [SEP]”. The special tokens, i.e., [CLS] and [SEP] are bold in blue, which indicate the utterance boundary. +TABLE III +RESULTS ON THE MUTUAL TEST SET. THE UPPER AND MIDDLE BLOCKS +PRESENT THE PUBLIC MODELS W/O AND W/ PRLMS, RESPECTIVELY. THE +RESULTS ARE COLLECTED FROM PUBLISHED LITERATURE [43, 1, 72]. +THE LOWER BLOCK SHOWS OUR IMPLEMENTATIONS ON THE +REPRODUCED BERT AND ELECTRA BASELINES FOLLOWING THE +OFFICIAL IMPLEMENTATIONS IN THE MUTUAL DATASET PAPER [23]. +Model +R4@1 +R4@2 +MRR +TF-IDF [43] +0.279 +0.536 +0.542 +Dual LSTM [43] +0.260 +0.491 +0.743 +SMN [12] +0.299 +0.585 +0.595 +DAM [16] +0.241 +0.465 +0.518 +GPT-2 [73] +0.332 +0.602 +0.584 +GPT-2-FT [73] +0.392 +0.670 +0.629 +BERT [24] +0.648 +0.847 +0.795 +RoBERTa [25] +0.713 +0.892 +0.836 +RoBERTa + OCN [69] +0.867 +0.958 +0.926 +ALBERT [74] +0.847 +0.962 +0.916 +GRN-v2 [74] +0.915 +0.983 +0.954 +BERT +0.656 +0.866 +0.803 ++ CDN +0.692 +0.881 +0.823 +ELECTRA +0.900 +0.979 +0.946 ++ CDN +0.916 +0.984 +0.956 +revealing the strength of BiGRU in modeling the sentence- +level transition between the turns of utterances. +B. Number of Decoupling Layers +Inspired by the stacked manner of Transformer [75], we +have also performed a study on the effects of stacked decou- +pling layers. The results are shown in Figure 3, where we +can see only one layer is enough and better than deeper. This +can be explained that a deeper decoupling module is harder +to learn, and only one interaction is enough. +C. Number of BiGRU Layers +To see whether a deeper BiGRU will be beneficial to the +modeling of a dialogue representation from the sequence of +utterance representations, we have conducted experiments on +the number of BiGRU layers. As shown in Figure 4, a deeper +BiGRU causes a big drop on R@1 metric, showing that the +TABLE IV +ABLATION STUDY ON MUTUAL DEV SETS. “UA” MEANS +UTTERANCE-AWARE, AND “SA” MEANS SPEAKER-AWARE. +Model +R4@1 +R4@2 +MRR +ELECTRA + CDN +0.923 +0.979 +0.958 +w/o SA-Mask +0.909 +0.973 +0.949 +w/o UA-Mask +0.913 +0.977 +0.953 +w/o BiGRU (w/ Max-Pool) +0.909 +0.982 +0.951 +w/o BiGRU (w/ Mean-Pool) +0.911 +0.980 +0.952 +TABLE V +INFLUENCE OF FUSING METHODS ON THE MUTUAL DEV SET. +Fusing Method +R4@1 +R4@2 +MRR +ELECTRA + CDN +0.923 +0.979 +0.958 +-Gate +0.914 +0.975 +0.952 +-Original Info +0.918 +0.980 +0.955 +-Gate +0.910 +0.981 +0.951 +shallow BiGRU is effective for capturing the information flow +in the dialogue context. +D. Effects of Fusing Methods +As described in Section III-E, we use a gate to fuse the +complementary information inside each channel to accumulate +the information from the four decoupled channels. A simple +fully-connected layer can be applied as a more simple alterna- +tive. To explore whether it is necessary to use those gates, we +perform a comparative study as results are presented in Table +V. We can see the gate mechanism yields better performance.6 +In addition, we find using the information from the original +sequence representation E is also beneficial, which could serve +as the reference for calculating the gating ratio to measure +how similar is the decoupled representation with regards to +the original representation. +6It is observed that “-Gate” yields better results towards the R4@2 metric. +Since the results on this metric are quite high (0.979-0.981), they are only +listed for completeness. Following the existing literature [67], we focus on +the more distinguishable metrics R1@2 and MRR. + +9 +1 +2 +3 +4 +5 +0.9 +0.95 +1 +Decoupling Block +Accuracy(%) +R@1 +R@2 +MRR +Fig. 3. +Influence of the number of Decoupling Blocks. +TABLE VI +INFLUENCE OF THE AGGREGATING METHODS ON THE MUTUAL DEV SET. +CNN USES A FILTER OF SIZE 3, AND CNN-MULTI COMBINES FILTERS OF +SIZE 2, 3, 4, WHICH IS SIMILAR TO [76]. +Aggregating Method +R@1 +R@2 +MRR +Max-Pooling +0.923 +0.979 +0.958 +Mean-Pooling +0.911 +0.975 +0.951 +CNN +0.916 +0.974 +0.953 +CNN-Multi +0.902 +0.974 +0.946 +E. Effects of Aggregating Methods +To aggregate the word representations in one utterance, +we can use simple global pooling. And another widely used +method is Convolution Neural Network (CNN). Deeper CNN +can capture a wide scope of the receptive field, making it +successful in Computer Vision [77] and Text Classification +[76]. We have also compared different aggregating methods on +the MuTual dev set as shown in Table VI. We can see that max- +pooling is better than mean-pooling since it can preserve some +activated signals, removing the disturbance of less important +signals. However, the two CNN-based methods perform worse +than max-pooling, especially those with multiple filter sizes. +This can be explained that shared filters between different +sentences are not flexible enough and cannot generalize well. +F. Effects of Underlying Pre-trained Models +To test the generality of the benefits of our CDN to other +PrLMs, we alter the underlying PrLMs to other variants in +different sizes or types. As shown in Table VII, we see that +our CDN is generally effective for the widely-used PrLMs. +G. Influence of Domain-adaptive Post-Training +We are interested in the influence of the post-training ob- +jectives (MLM and NSP) on overall performance. Firstly, we +compare the three kinds of masking strategies, i.e., subword, +span, and word level masks. Then, we evaluate the impact of +the NSP objective to see if it works in the dialogue scenarios. +For the analysis, we use the Ubuntu dataset instead of the +MuTual dataset because MuTual is a very small-scale dataset +(only 8K context-response pairs) even with high quality, which +is not appropriate for evaluating post-training. Therefore, we +1 +2 +3 +4 +5 +0.9 +0.95 +1 +BiGRU Layer +Accuracy(%) +R@1 +R@2 +MRR +Fig. 4. +Influence of the number of BiGRU layers on MuTual. +TABLE VII +PERFORMANCES OF SINGLE PRLM BASELINE AND WITH CDN ON THE +MUTUAL DEV SET. +PrLMs +R@1 +R@2 +MRR +Baseline +CDN Baseline +CDN Baseline +CDN +BERTbase +0.653 +0.684 +0.860 +0.871 +0.800 +0.818 +RoBERTabase +0.709 +0.731 +0.886 +0.898 +0.833 +0.846 +ELECTRAbase +0.762 +0.813 +0.916 +0.928 +0.865 +0.893 +BERTlarge +0.691 +0.726 +0.879 +0.901 +0.822 +0.844 +RoBERTalarge +0.834 +0.845 +0.952 +0.953 +0.908 +0.914 +ELECTRAlarge +0.906 +0.923 +0.977 +0.979 +0.949 +0.958 +used Ubuntu which is widely used by existing studies (those +reported in Table II), for assessing post-training. +Table VIII shows the results. We see that the contribution +of the masking methods does not vary too much, and the +NSP objective boosts the performance remarkably. Although +NSP has been shown trivial in RoBERTa [25] during general- +purpose pre-training, it yields surprising gains in dialogue sce- +narios. The most plausible reason is that dialogue emphasizes +the relevance between dialogue context and the subsequent +response, which shares a similar goal with NSP. The findings +indicate that the utterance-level language modeling would be +effective for dialogue-related tasks, which would be a possible +research topic in future studies. +As pre-trained weights from the free-domain corpus are +widely-used as the initialization, we are also interested in +whether the pre-trained weights are dispensable when us- +ing our domain-adaptive training. The comparison between +general pre-training and domain-adaptive training without the +weights from the general pre-training is shown in Table IX. We +observe that training with the domain-aware corpus is much +more efficient in our dialogue task. With much less training +data and computation cost, our method outperforms all the +baselines without large-scale pre-training on the domain-free +corpus in Table II, and even achieves better performance than +various pre-trained methods such as BERT and TADAM. +H. Application Potentials +In terms of applications, our model can be used to build the +retrieval-based chatbot, and the model latency can be tolerated. +For our CDN model, the model latency mainly depends on the +underlying PrLMs and batch size. Taking the MuTual dataset + +10 +TABLE VIII +ABLATION STUDY OF DOMAIN-ADAPTIVE POST-TRAINING ON UBUNTU +(ELECTRA-BASED CDN). +Model +R10@1 +R10@2 +R10@5 +Subword +0.884 +0.945 +0.987 +Span +0.887 +0.946 +0.988 +WWM +0.889 +0.947 +0.989 +Subword + NSP +0.912 +0.959 +0.992 +Span + NSP +0.912 +0.960 +0.992 +WWM + NSP +0.914 +0.961 +0.993 +TABLE IX +COMPARISON OF OF GENERAL PRE-TRAINING AND DOMAIN-ADAPTIVE +TRAINING (WITHOUT THE WEIGHTS FROM THE GENERAL PRE-TRAINING). +Model +Data +Steps +Batch +Device +R10@1 +R10@2 +R10@5 +Vanilla +160GB +1.75M +TPU +2048 +0.845 +0.919 +0.979 +Domain +340M +470K +64 +GPU +0.821 +0.907 +0.979 +as an example, the inference phase is 2 min/epoch using our +settings, where 886 instances are included. Such velocity is +supposed to be tolerated in real-world applications. For real- +world scenarios, our model can be applied to build a chatbot +if responses can be first selected coarsely and then ranked by +our model. +Besides the retrieval-based chatbots, we have two ways to +generalize CDN to generation-based dialogue systems as (1) +re-ranking module (2) the encoder in the traditional encoder- +decoder framework. A generation-based model can generate +some candidate responses, then to choose the better response, +our model can be used to rank the candidates. Another way is +using the major parts of CDN (without the final classification +layer) as an encoder to learn the utterance-aware and speaker- +aware representations. Then a decoder of advanced generation- +based models can be utilized upon CDN to generate better +responses. +VII. CONCLUSION +In this paper, we propose a novel and simple Channel-aware +Decoupling Network (CDN), which decouples the utterance- +aware and speaker-aware information, tackling the problem of +role transition and noisy distant texts. Experiments on four +retrieval-based multi-turn dialogue datasets show the superior- +ity over existing methods. The ablation study of different sub- +modules explains their effectiveness and relative importance. +We find that domain-adaptive post-training also enhances the +model performance substantially. Even on the backbone with +post-training, CDN still yields consistent gains, showing that +CDN is generally effective no matter how strong the backbone +model is. Our work reveals a way to make better use of +the semantically rich contextualized representations from pre- +trained language models and gives insights on how to combine +the traditional RNN models with powerful transformer-based +models. 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Zisserman, “Very deep convolu- +tional networks for large-scale image recognition,” in 3rd +International Conference on Learning Representations, +ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Confer- +ence Track Proceedings, Y. Bengio and Y. LeCun, Eds., +2015. +Zhuosheng Zhang received his Bachelor’s degree in +internet of things from Wuhan University in 2016, +his M.S. degree in computer science from Shanghai +Jiao Tong University in 2020. He is working towards +the Ph.D. degree in computer science with the Center +for Brain-like Computing and Machine Intelligence +of Shanghai Jiao Tong University. He was an intern- +ship research fellow at NICT from 2019-2020. His +research interests include natural language process- +ing, machine reading comprehension, and dialogue +systems. +Hai Zhao received the BEng degree in sensor +and instrument engineering, and the MPhil degree +in control theory and engineering from Yanshan +University in 1999 and 2000, respectively, and the +PhD degree in computer science from Shanghai Jiao +Tong University, China in 2005. He is currently a +full professor at department of computer science and +engineering, Shanghai Jiao Tong University after he +joined the university in 2009. He was a research +fellow at the City University of Hong Kong from +2006 to 2009, a visiting scholar in Microsoft Re- +search Asia in 2011, a visiting expert in NICT, Japan in 2012. He is an ACM +professional member, and served as area co-chair in ACL 2017 on Tagging, +Chunking, Syntax and Parsing, (senior) area chairs in ACL 2018, 2019 +on Phonology, Morphology and Word Segmentation. His research interests +include natural language processing and related machine learning, data mining +and artificial intelligence. +Longxiang Liu received Bachelor’s degree in the +department of computer science and engineering +from Shanghai Jiao Tong University in 2021. He is +now a graduate student studying in the University +of Chinese Academy of Sciences. He is working to- +wards the M.S. degree in computer science with the +Natural Language Processing Group of Key Labora- +tory of Intelligent Information Processing of Institute +of Computing Technology in Chinese Academy of +Sciences. He has done researches in Center for +Brain-like Computing and Machine Intelligence of +Shanghai Jiao Tong University. His research interests include natural language +processing, machine reading comprehension and dialogue systems. + diff --git a/ztE2T4oBgHgl3EQfiAcw/content/tmp_files/load_file.txt b/ztE2T4oBgHgl3EQfiAcw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec7b8ce8caf7b61a28d63aae6d99eadca5f536a1 --- /dev/null +++ b/ztE2T4oBgHgl3EQfiAcw/content/tmp_files/load_file.txt @@ -0,0 +1,1690 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf,len=1689 +page_content='1 Channel-aware Decoupling Network for Multi-turn Dialogue Comprehension Zhuosheng Zhang, Hai Zhao, Longxiang Liu Abstract—Training machines to understand natural language and interact with humans is one of the major goals of artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Recent years have witnessed an evolution from matching networks to pre-trained language models (PrLMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In contrast to the plain-text modeling as the focus of the PrLMs, dialogue texts involve multiple speakers and reflect special characteristics such as topic transitions and structure dependencies between distant utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, the related PrLM models commonly represent dialogues sequentially by processing the pairwise dialogue history as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Thus the hierarchical information on either utterance interrelation or speaker roles coupled in such representations is not well addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In this work, we propose compositional learning for holistic interaction across the utterances beyond the sequential contextualization from PrLMs, in order to capture the utterance- aware and speaker-aware representations entailed in a dialogue history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We decouple the contextualized word representations by masking mechanisms in Transformer-based PrLM, making each word only focus on the words in current utterance, other utterances, and two speaker roles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', utterances of sender and utterances of the receiver), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In addition, we employ domain-adaptive training strategies to help the model adapt to the dialogue domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Experimental results show that our method substantially boosts the strong PrLM baselines in four public benchmark datasets, achieving new state-of-the-art performance over previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Index Terms—Dialogue Modeling, Open Domain Conversation System, Natural Language Generation, Deep Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' INTRODUCTION L Anguage is not only an effective medium for people to communicate with each other but also a natural interface between humans and machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, building an intelligent dialogue system that can understand human Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Zhang and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Zhao are with the Department of Computer Science and Engineering, Shanghai Jiao Tong University, and also with Key Laboratory of Shanghai Education Commission for Intelligent Interaction and Cognitive Engineering, Shanghai Jiao Tong University, and also with MoE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Liu is with Key Laboratory of Intelligent Information Processing Institute of Computing Technology, Chinese Academy of Sciences (ICT/CAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' This work was conducted when L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Liu was with the Department of Computer Science and Engineering, Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' E-mail: zhangzs@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='cn, zhaohai@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='cn, liulongxiang21s@ict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Zhang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Liu contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (Corresponding author: Hai Zhao) This work was partially supported by Key Projects of National Natural Science Foundation of China (U1836222 and 61733011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Part of this study has been accepted as ”Filling the Gap of Utterance-aware and Speaker-aware Representation for Multi-turn Dialogue” [1] in the Thirty- Fifth AAAI Conference on Artificial Intelligence (AAAI 2021), with partially material overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' This article extends the conference version by studying channel-aware decoupling in a broader view of multi-turn dialogue modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Towards this goal, we extend the descriptions in Introduction, Related Work, Model, Experiments, and Analysis correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For the techniques, this work extends the proposed model with domain-adaptive strategies, more baselines, and comprehensive analyses with new conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' TABLE I AN EXAMPLE OF RESPONSE-SELECTION FOR MULTI-TURN DIALOGUE IN MUTUAL DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' F AND M DENOTE DIFFERENT SPEAKERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Utterance (Context) F: Excuse me, sir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' This is a non smoking area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' M: Oh, sorry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' I will move to the smoking area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' F: I’m afraid no table in the smoking area is available now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Response Candidates A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Sorry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' I won’t smoke in the hospital again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' \x17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' I won’t smoke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Could you please give me a menu?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' \x13 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Could you please tell the customer over there not to smoke?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We can’t stand the smell \x17 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Sorry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' I will smoke when I get off the bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' \x17 conversations and give logically correct, fluent responses is one of the eternal goals of artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' It has been drawing increasing interest from both academia and industry areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The methods of building a chatbot that is capable of performing multi-turn dialogue can be categorized into two lines: generation-based methods and retrieval-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Generation-based methods [2, 3, 4, 5, 6, 7, 8, 9, 10, 11] directly generate a response using an encoder-decoder framework, which tends to be short and lacks diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Retrieval-based methods [12, 13, 14, 15, 16, 17, 18, 19] retrieve a list of response candidates, then use a model to rank the candidates and select the best one as a reply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Since the responses of retrieval-based are generally more natural, fluent, and syn- tactically correct, retrieval-based methods are more mature for producing multi-turn dialogue systems both in academia and industry [20, 21, 22], which is our major focus in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Table I shows an example from Multi-Turn Dialogue Reasoning dataset (MuTual) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In order to choose the right answer, the machine is required to understand and infer from the meaning of ”table” and its coreference, indicating the requirement of reasoning ability instead of simple matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Early studies concerning dialogue comprehension mainly focus on the matching networks that calculate similarity scores between the pairwise sequence of dialogue context and candi- date response at different granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The matching matrices will be fused to get a feature vector, then the sequence of feature vectors will be further integrated by RNNs to get the final representation for scoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, these methods have two sides of disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' First, interactions mainly involve each utterance and response, ignoring the global interactions between utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Second, the relative positions between the response and different utterances are not taken into consider- ation, lacking the sequential information of context-response pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Copyright © 20XX IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='03953v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='CL] 10 Jan 2023 2 Recently, pre-trained language models (PrLMs), such as BERT [24], RoBERTa [25], and ELECTRA [26], have achieved impressive performance in a wide range of NLP tasks [27, 28, 29, 30, 31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The word embeddings derived by these language models are pre-trained on large corpora and are then fine-tuned in task-specific datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In the multi-turn dialogue scope, there are also some efforts using PrLMs to yield promising performances [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The mainstream is employing PrLMs as encoders by concatenating the context and candidate responses directly, as a linear sequence of successive tokens and implicitly capturing the contextualized representations of those tokens through self- attention [36, 37, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, simply embedding the token to high-dimensional space cannot faithfully model the dialogue-related information, such as positional or turn order information [40];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In addition, the mechanism of self-attention runs through the whole dialogue, resulting in entangled infor- mation that originally belongs to different parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Multi-turn dialogue modeling has critical challenges of tran- sition and inherency, which the existing multi-turn dialogue methods rarely consider to our best knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 1) There exists speaker role switching in multi-turn dia- logue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Each speaker has different thinking habits and speak- ing purposes, which lead to the unique speaking style of each speaker’s role and fuzziness of utterance coreference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Therefore, it is the essential difference between dialogue and passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Although Speaker-Aware BERT [38] has considered role transitions, adding the embedding of speaker-id into the inputs of PrLMs is not effective enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 2) Utterances have their own inherent meaning and con- textual meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A clear understanding of local and global meaning can reduce the negative influence of irrelevant content in a long-distance context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Some studies on neural machine translation have taken this into account and achieved gratifying results[41], which inspire this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3) The widely adopted PrLMs are pre-trained on general corpora, which would be less effective to directly fine-tune these models on downstream tasks if there is a domain shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A recent trend is to post-train PrLMs on dialogue corpus and achieve state-of-the-art results [42, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We also employ this post-training method in this work and investigate whether it can be enhanced by modeling the dialogue properties discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In this work, beyond the sequential contextualization from PrLMs, we proposed a novel end-to-end Channel-aware De- coupling Network (CDN) to enhance dialogue comprehension with compositional utterance interaction and thus fill the obvious gap of utterance-aware and speaker-aware represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In detail, the contextualized word representation of dialogue text is decomposed into four parts that contain dif- ferent information and then fused after sufficient interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' More specifically, the PrLM receives concatenated context and response and outputs the contextualized representation of each word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For each word, we use a masking mechanism inside a self-attention network to limit the focus of each word only to the words in current utterance, other utterances, utterances of the sender, and utterances of the receiver respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' To avoid ambiguity, we call the first two complementary parts the utterance-aware channel and the last two the speaker-aware channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We then fuse the information inside each channel via a gating mechanism to control the information reservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The word-level information will be further aggregated to the utterance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For utterance-level representations, BiGRU is adopted to get dialogue-level representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Inspired by Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [13], we put the information fusion of two channels at the end to get the final dialogue representation for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In addition, we employ domain-adaptive training strategies to help the model adapt to the dialogue domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Experimental results on four public benchmark datasets show that the proposed model outperforms the baseline models substantially on all the evaluation metrics and achieves new state-of-the-art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In summary, our contributions are mainly in three folds: 1) A novel end-to-end Mask-based Decoupling-Fusion Net- work is proposed to fill the gap between utterance- aware and speaker-aware representations for multi-turn dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 2) An effective domain-adaptive post-training method with intra-utterance and inter-utterance objectives to better adapt PrLMs to our dialogue comprehension tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3) Experimental results on public datasets show the su- periority of our method and verify the effectiveness of our model beyond strong PrLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We have also achieved various state-of-the-art performances compared with existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Multi-turn Dialogue Comprehension As an active research topic in the NLP field, multi-turn dialogue comprehension has attracted great attention from both academia and industry whose aim is to teach machines to read dialogue contexts and solve typical tasks such as response selection [43, 12, 15, 23, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, selecting a coherent and informative response for a given dialogue context remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The multi-turn dialogue typically involves two or more speakers that engage in various conversation topics and intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Thus the utterances are rich in interactions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', with criss-cross discourse structures [45, 46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A critical challenge is learning rich and robust context representations and interactive relationships of dialogue utterances so that the resulting model can adequately capture the semantics of each utterance and the relationships among all the utterances inside the dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Previous studies can be classified into two lines: matching models and pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Matching Models Matching models aim to match the response with contexts and calculate the matching score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' They can be divided into two categories: single-turn and multi-turn matching models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Earlier research mainly considered the context utterances as one single utterance, using the dual encoder framework to encode the whole context and the response, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The encoder varies in LSTM [48], CNNs [49], Attentive-LSTM [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3 Encoder .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' U1/S1 U2/S2 R/S1 Gated Fusing Gated Fusing GRU GRU GRU GRU GRU GRU Classifier U1 U2 R U1 U2 R U1 U2 R U1 U2 R S1 S2 S1 S1 S2 S1 Current Utterance (C1) Other Utterance (C2) S1 S2 S1 S1 S2 S1 Utterance of Receiver (C3) Utterance of Sender (C4) Encoding Decoupling Interaction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Overall framework of CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Here for simplicity, inputs contain two utterances and one response spoken by two speakers with total six words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A more detailed figure and description on the decoupling block will be shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The decoupling block is composed of four independent self-attention blocks with the same inputs and different masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The same word in different blocks attends to different scope of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' As for multi-turn matching, Sequential Matching Network (SMN) [12] is the representative work, where a word-word and a sequence-sequence similarity matrix will be calculated, forming a matching image for each utterance-response pair which will be further integrated by CNN and RNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Several related works like Deep Utterance Aggregation (DUA) [15], Deep Attention Matching Network (DAM) [16], Interaction- over-Interaction (IoI) [14] are extensions of SMN from differ- ent perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' DUA [15] points out that the last utterance should be considered explicitly and use self-attention to get a better segment matching matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' DAM [16] uses hierarchi- cally stacked layers of self-attention to represent matching from different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' IoI [14] mainly deepens the layers of response-utterance interaction blocks and combines final states at different depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Pre-trained Language Models Recently, inspired by the impressive performance of PrLMs, the mainstream is employing PrLMs to handle the whole pairwise texts as a linear sequence of successive tokens and implicitly capture the contextualized representations of those tokens through self-attention [36, 37, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The word embeddings derived by these language models are pre-trained on large corpora and are then utilized as either distributed word embeddings [51] or fine-tuned according to the specific task needs [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Most of the pre-trained Language Models (PrLMs) are based on Transformer, among which Bidirectional Encoder Representations from Transformers (BERT) [24] is one of the most representative works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' BERT uses multiple lay- ers of stacked Transformer Encoder to obtain contextualized representations of the language at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' BERT has achieved unprecedented performances in many downstream tasks of NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Several subsequent variants have been proposed to further enhance the capacity of PrLMs, such as RoBERTa [25], ALBERT [52], and ELECTRA [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' To evaluate our methods on both sides of the fast inference speed and strong performance, we will adopt BERT and ELECTRA as our backbone models in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' CHANNEL-AWARE DECOUPLING NETWORK Our proposed model, named Channel-aware Decoupling Network (CDN), consists of six parts: Encoding, Decoupling, Fusing, Words Aggregating, Utterances Integrating, and Scor- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In this section, we will formulate the problem and then introduce each sub-module in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Problem Formulation Suppose that we have a dataset D = {(yi, ci, ri)}N i=1, where ci = {ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', ui,ni} represents the dialogue context with {ui,k}ni k=1 as utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' ri is a response candidate, and yi ∈ {0, 1} denotes a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The goal is to learn a discriminator g(·, ·) from D, and at the inference phase, given any new context c and response r, we use the discriminator to calculate g(c, r) as their matching score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' To select several best responses as return to human inputs, we rank matching scores or set a threshold among the list of candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Model Overview Figure 1 shows the overall framework of CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' CDN first encodes each word in the concatenated context and response to contextualized representation using the PrLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In the decou- pling module, a self-attention mechanism with different masks forces each word to only focus on the information in spe- cific scopes, decoupling to utterance-aware and speaker-aware channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Note that now the information flow inside different channels is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In each channel, complementary information will be fused by a gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then word representations belonging to the same utterance will be aggregated to represent the utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The sequence of utterance representations will 4 be delivered to Bidirectional Gated Recurrent Unit (BiGRU) to produce the channel-aware dialogue-level representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' After that, the information from different channels is fused to get the final dialogue representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The classifier will then output a score of the dialogue to reflect the matching degree between context and candidate response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' CDN is superior to existing methods in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' First, compared to SA-BERT [38], the speaker-aware information is built upon semantically rich contextualized representations instead of a simple additional embedding term, making its influence on dialogue scoring preserved better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Second, compared to the work of designing a complicated matching network, our structure is simpler and can represent information at different levels in a more unified manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Third, compared to pure PrLMs, the additional layer of CDN can decouple information to different channels, making full use of the contextualized representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Besides, since the number of utterances is far less than that of words, the top BiGRU layers can keep their memory capacity, alleviating the defects of the Transformer in reflecting relative position information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Encoding Context and Response We employ a pre-trained language model such as ELEC- TRA to obtain the initial word representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The tokens in utterances and response are concatenated successively and then fed into the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Following Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [24], we add “[CLS]” token in the front of the sequence and insert “[SEP]” between adjacent utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Let Ui = [ui,0, ui,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', ui,ni], R = [r0, r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', rnr], where ni and nr is respectively the length of i-th utterances and response, then the output is E = [e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', en0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='+nk+nr] where k is the maximum number of utterances, and ei ∈ Rd, d is hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Channel-aware Information Decoupling We first introduce the multi-head self-attention (MHSA) with mask, which can be formulated as: Attention(Q, K, V, M) = softmax(QKT √dk + M)V, headi = Attention(EW Q i , EW K i , EW V i , M), MHSA(E, M) = Concat(head1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', headh)W O, (1) where W Q i ∈ Rdmodel×dq, W K i ∈ Rdmodel×dk, W V i ∈ Rdmodel×dv, W O i ∈ Rhdv×dmodel are parameter matrices, dq, dk, dv denote the dimension of Query vectors, Key vectors and Value vectors, h denotes the number of heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' M denotes the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We have four masks {Mk}4 k=1 ∈ Rl×l defined as: M1[i, j] = � 0, if Ti = Tj −∞, otherwise M2[i, j] = � 0, if Ti ̸= Tj −∞, otherwise M3[i, j] = � 0, if Si = Sj −∞, otherwise M4[i, j] = � 0, if Si ̸= Sj −∞, otherwise (2) where i, j is the word position in the whole dialogue, Ti is the index of utterance the i-th word is located in, and Si is the speaker that the i-th word is spoken by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' And M1, M2, M3, M4 only attend to the word in current utterance, other utterances, utterances of sender, and utterances of receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We call the first two utterance-aware channels and the last two speaker- aware channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The decoupled channel-aware information {Ck}4 k=1 ∈ Rl×d are derived by multi-head self-attention with mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Ci = MHSA(E, Mi), i ∈ {1, 2, 3, 4}, (3) where E ∈ Rl×d is the output of PrLMs stated in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Complementary Information Fusing To fuse the complementary information inside each channel, we use a gate to fuse them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Inspired by [53], we use a gate to calculate the ratio of information preservation based on matching heuristics considering the “Siamese” architecture as information from two parts, the element-wise product and difference as “similarity” or “closeness” measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Let E denote the original representations output from PrLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For utterance-aware channel, ¯E = C1 and ˆE = C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For speaker- aware channel, ¯E = C3 and ˆE = C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Ci is defined in Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The gate is formulated as: ˜ E1 = ReLU(FC([E, ¯E, E − ¯E, E ⊙ ¯E])), ˜ E2 = ReLU(FC([E, ˆE, E − ˆE, E ⊙ ˆE])), P = Sigmoid(FC(([ ˜ E1, ˜ E2]))), G(E, ¯E, ˆE) = P, (4) where Sigmoid, ReLU [54] are activation functions, FC is fully-connected layer, [·, ·] means concatenation, and ⊙ is element-wise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Using two parametric-independent gates, the channel-aware information can be fused, which is defined as: P1 = G1(E, C1, C2), P2 = G2(E, C3, C4), Cu = P1 ⊙ C1 + (1 − P1) ⊙ C2, Cs = P2 ⊙ C3 + (1 − P2) ⊙ C4, (5) where the calculation of {Gi}2 i=1 is defined in Equation 4, Cu and Cs is the fused utterance-aware and speaker-aware word representations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Utterance Representations For each channel, word representations will be aggregated by simple max-pooling over words in the same utterance to get the utterance representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Let Lu and Ls be the output in this part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then they are defined as: Lu[i, :] = MaxPooling Tj=i (Cu[j, :]) ∈ Rd, Ls[i, :] = MaxPooling Tj=i (Cs[j, :]) ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (6) 5 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Dialogue Representation To get the channel-aware dialogue representation, the se- quence of utterance representations will be delivered to Bi- GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Suppose that the hidden states of the BiGRU are (h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' , hk), then ∀j, 1 ≤ j ≤ k, hj ∈ R2d is given by ←− h j = ←−− GRU(←− h j−1, ←− L[j]), −→ h j = −−→ GRU(−→ h j−1, −→ L[j]), hj = [←− h j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' −→ h j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (7) where L[j] denotes the sentence representation derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For each channel, we take the hidden state of BiGRU at the last step as channel-aware dialogue representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Let the two vectors be v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then two channels can be easily fused to get the final dialogue representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' v = Tanh(W[v1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' v2] + b), (8) where W ∈ Rd×4d, b ∈ Rd are trainable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Tanh is the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Scoring The dialogue vector will be fed into a classifier with a fully connected and softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We learn model g(·, ·) by minimizing cross-entropy loss with dataset D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Let Θ denote the parameters of CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For binary classification like ECD, Douban, and Ubuntu, the objective function L(D, Θ) can be formulated as: − N � i=1 [yilog(g(ci, ri)) + (1 − yi)log(1 − g(ci, ri))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (9) where N is the number of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For multiple-choice tasks like MuTual, the loss function is: − N � i=1 C � k=1 yi,klog(g(ci, ri,k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (10) where C is the number of candidate response options for each input context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' DOMAIN-ADAPTIVE POST-TRAINING Although the PrLMs demonstrate superior performance due to their strong representation ability from self-supervised pre- training, it is still challenging to effectively adapt task-related knowledge during the detailed task-specific training, which is usually in the way of fine-tuning [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Generally, those PrLMs handle the whole input text as a linear sequence of successive tokens and implicitly capture the contextualized representa- tions of those tokens through self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The such fine- tuning paradigm of exploiting PrLMs would be suboptimal to model dialogue tasks that hold exclusive text features that plain text for PrLM training may hardly embody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In this following section, we first describe the standard general-purpose training in BERT, and then present our extensions to the concerned dialogue domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' General-Purpose Training As the standard pre-training procedure, PrLMs are pre- trained on large-scale domain-free texts and then used for fine- tuning according to the specific task needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' There are token- level and sentence-level objectives used in the general-purpose pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The most widely-used PrLM for domain-adaption in the dialogue field is BERT [24], whose pre-training is based on two loss functions: (1) a masked language model (MLM) loss, and (2) a next sentence prediction (NSP) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' MLM first masks out some tokens from the input sentences and then trains the model to predict them by the rest of the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' NSP is another widely used pre-training objective, which trains the model to distinguish whether two input sentences are continuous segments from the training corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Domain-Adaptive Training The original PrLMs are trained on a large text corpus to learn general language representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' To incorporate specific in-domain knowledge, adaptation on in-domain corpora, also known as domain-aware pre-training, is designed, which di- rectly employs the original PrLMs as mentioned in the general- purpose paragraph above, using the dialogue-domain corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In this work, we extend the MLM and NSP as intra-utterance and inter-utterance objectives to coordinate with our dialogue comprehension tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Intra-utterance Objective To better adapt PrLMs to dia- logue scenarios, we conduct the post-training with three levels of masks for MLM, including subword, word, and span-level masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Subword is the original method used in BERT [24], and word means the whole-word-masking (WWM) where the whole word will be masked during the data preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The Span masking mechanism follows [56], which masks contigu- ous random spans from a geometric distribution rather than random tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Compared with subword masks, both span and WWM mechanisms would potentially improve the model’s ability to capture high-level topic information in dialogue texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The training objective is then formed as follows: Lintra = − N � k=1 [mi log ˆmi] , (11) where ˆmi denotes the predicted token id and mi is the ground- truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' N is the number of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Inter-utterance Objective For a dialogue context with multiple utterances c = {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', un}, we form a context text c′ = {u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', uk−1}, k ∈ [1, n] for each utterance u+ k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We then randomly sample an utterance u− k from the corpus as the negative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The goal is to identify utterance uk as the true utterance against the negative one by feeding sequence “[CLS] c′ [SEP] uk [SEP]” into BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Following the setting proposed in BERT, we take the hidden representation of [CLS] through a full-connected layer followed by a sigmoid function for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Here, we minimize the cross entropy loss in post-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Linter = − N � k=1 [si log ˆsi] , (12) 6 where ˆsi denotes the predicted label and si is the ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' N is the number of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' During post-training, we combine both of the intra- and inter- utterance objectives: L = Lintra + Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' (13) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' EXPERIMENTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Datasets We tested our model on two English datasets: Ubuntu Dia- logue Corpus (Ubuntu) and Multi-Turn Dialogue Reasoning (MuTual), and two Chinese datasets: Douban Conversation Corpus (Douban) and E-commerce Dialogue Corpus (ECD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 1) Ubuntu Dialogue Corpus: Ubuntu [43] consists of En- glish multi-turn conversations about technical support col- lected from chat logs of the Ubuntu forum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The dataset contains 1 million context-response pairs, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='5 million for validation, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='5 million for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In the training set, each context has one positive response generated by humans and one negative response sampled randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In the validation and test sets, for each context, there are 9 negative responses and 1 positive response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 2) Douban Conversation Corpus: Douban [12] is different from Ubuntu in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' First, it is an open domain task where dialogues are extracted from the Douban Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Second, Response candidates on the test set are collected by using the last turn as the query to retrieve 10 response candidates and labeled by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Third, there could be more than one correct response for a context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Response candidates on the test set are collected by a standard search engine Apache Lucene1, other than negative sampling without human judgment on Ubuntu Dialogue Corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3) E-commerce Dialogue Corpus: ECD [15] dataset is extracted from conversations between customers and service staff on Taobao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' It contains over 5 types of conversations based on over 20 commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' There are also 1 million context- response pairs in the training set, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='5 million in the validation set, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='5 million in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 4) Multi-Turn Dialogue Reasoning: MuTual [23] consists of 8,860 manually annotated dialogues based on Chinese stu- dent English listening comprehension exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For each context, there is one positive response and three negative responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The difference compared to the above three datasets is that only MuTual is reasoning-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' There are more than 6 types of reasoning abilities reflected in MuTual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Setup For the sake of computational efficiency, the maximum number of utterances is specialized as 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The concatenated context, response, ”[CLS]” and ”[SEP]” in one sample are truncated according to the ”longest first” rule or padded to a certain length, which is 256 for MuTual and 384 for the other three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Our model is implemented using Pytorch and based on the Transformer Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We use ELECTRA [26] as our underlying model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' AdamW [68] is used as our optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 1http://lucene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='org/ The batch size is 24 for MuTual, and 64 for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The initial learning rate is 4 × 10−6 for MuTual and 3 × 10−6 for others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We run up to 3 epochs for MuTual and 2 epochs for others and select the model that achieves the best result in the validation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Our domain adaptive post-training for the corresponding response selection tasks is based on the three large-scale dialogue corpus, including Ubuntu, Douban, and ECD, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The data is the same as the training data in fine-tuning but only used in unsupervised pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Because there is no appropriate domain data for the small-scale Mutual dataset, we only report the fine-tuning results without post-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For the English tasks, we use the pre-trained weights bert- base-uncased and electra-large-discriminator for fine-tuning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' for the Chinese tasks, the weights are from bert-base-chinese and hfl/chinese-electra-large-discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Baseline Models The following models are our baselines for comparison: Single-turn matching models: Single-turn Matching Models consider the context utterances as one single utterance, using the dual encoder framework to encode the whole context and the response, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The variants include CNN, LSTM, BiLSTM [59], MV-LSTM [60], and Match-LSTM [62] construct the dialog context by concatenating utterances as a long document, and match the dialog context with a candidate response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Multi-turn matching models: Multi-turn matching mod- els focus on the utterance-level interaction with responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The major difference lies in the different types of match- ing, including cross-attention, self-attention, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The typical studies include Sequential Matching Network (SMN) [12], Deep Attention Matching Network (DAM) [16], Deep Ut- terance Aggregation (DUA) [15], Interaction-over-Interaction (IoI) [14] which have been stated in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Besides, Multi-Representation Fusion Network (MRFN) [13] matches context and response with multiple types of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Multi-hop Selector Network (MSN) [17] utilizes a multi-hop selector to filter necessary utterances and match among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' PrLMs-based models: PrLMs-based models apply PrLMs to encode the dialogue history and candidate response as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' BERT [24], RoBERTa [25], ALBERT [52], ELECTRA [26] have been stated in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Besides, Option Comparison Network (OCN) [69] is involved, which compares the options before matching response and contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Evaluation Metrics Following the previous work [43, 12], we calculate the proportion of true positive responses among the top-k selected responses from the list of n available candidates for one context, denoted as Rn@k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Besides, additional conventional metrics of information retrieval are employed on Douban: Mean Average Precision (MAP) [70], Mean Reciprocal Rank (MRR) [71], and precision at position 1 (P@1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 2Those weights are available in the Transformers repo: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='com/ huggingface/transformers/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3On the leaderboard of MuTual, since some work is not publicly available, we will not introduce here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 7 TABLE II TEST RESULTS ON UBUNTU, DOUBAN, AND E-COMMERCE DATASETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' † DENOTES THE METHODS WITH DOMAIN-ADAPTIVE POST-TRAINING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' THE EVALUATION RESULTS ARE COLLECTED FROM PUBLISHED LITERATURE [57, 34, 39, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Models Ubuntu Douban E-commerce R10@1 R10@2 R10@5 MAP MRR P@1 R10@1 R10@2 R10@5 R10@1 R10@2 R10@5 Single-turn models with concatenated matching CNN [59] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='684 0.' metadata={'source': 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+page_content=' 1) Generally, the previous models based on multi-turn matching networks perform worse than simple PrLMs-based ones, illustrating the power of contextualized representations in context-sensitive dialogue modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 2) PrLM can perform even better when equipped with CDN, verifying the effectiveness of our model, where utterance- aware and speaker-aware information can be better exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3) It is observed that post-training on dialogue data can sub- stantially boost the baseline, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='5% R10@1 on Ubuntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Even on the backbone baseline with domain-adaptive post- training, CDN can still yield consistent performance gains, which indicates that the CDN architecture is generally effec- tive even working with stronger backbone PrLM encoders after post-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A similar phenomenon is observed in existing studies [16, 13, 64, 14] that post-training on dialogue data brings more improvements than modifying attention architec- tures in Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, post-training is less efficient, which requires costly training on an additional domain-specific corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 4) Our model outperforms other models in most metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' CDN surpasses the previous SOTA SA-BERT model on most datasets, which is augmented by extra domain adap- tation strategies to conduct language model pre-training on in-domain dialogue corpus before fine-tuning on tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In addition, CDN also ranks the best on the MuTual leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='4 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Ablation Study Since our model first decouples information to utterance- aware and speaker-aware channels and then uses BiGRU to learn a channel-aware dialogue representation from the sequence of utterance representations, we wonder about the effect of two channels and whether BiGRU can be replaced by simple pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' First, we visualize the attention distri- bution in C1, C2, C3 and C4 as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The illustration shows that channel-aware decoupling operations distinguish the four parts of utterance-aware and speaker- aware representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then, we perform an ablation study on MuTual dev sets as shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='5 Results show that each part is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The most important part is speaker- aware information since speaker role transition is an essential feature in multi-party dialogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then it comes to BiGRU, 4https://nealcly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='io/MuTual-leaderboard/ 5We conduct the ablation studies on MuTual because MuTual is of the highest quality, which is a manually annotated dialogues based on English listening comprehension exams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 8 a) C1 b) C2 c) C3 d) C4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='000 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' i finished preparing for my exam three weeks ago .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=" i ' m worried about ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] f : just keep calm and keep working hard .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] [CLS] f : why are you so worried , peter ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' the exam ?' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' the exam ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] m : no , mom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' i finished preparing for my exam three weeks ago .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=" i ' m worried about ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] f : just keep calm and keep working hard .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] [CLS] f : why are you so worried , peter ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' the exam ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] m : no , mom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' i finished preparing for my exam three weeks ago .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=" i ' m worried about ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] f : just keep calm and keep working hard .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] [CLS] f : why are you so worried , peter ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' the exam ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] m : no , mom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' i finished preparing for my exam three weeks ago .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=" i ' m worried about ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] f : just keep calm and keep working hard .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Visualization of attention distribution in C1, C2, C3 and C4, which indicates that channel-aware decoupling operations distinguish the four parts of utterance-aware and speaker-aware representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The example context is from the MuTual validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The text is “[CLS] f : why are you so worried , peter ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' the exam ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] m : no , mom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' i finished preparing for my exam three weeks ago .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' i ’ m worried about my paper .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' you know , the deadline is coming up .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' but i have n ’ t collected enough information .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' if i ca n ’ t finish it on time , my teacher will not be happy with me .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP] f : just keep calm and keep working hard .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' [SEP]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The special tokens, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', [CLS] and [SEP] are bold in blue, which indicate the utterance boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' TABLE III RESULTS ON THE MUTUAL TEST SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' THE UPPER AND MIDDLE BLOCKS PRESENT THE PUBLIC MODELS W/O AND W/ PRLMS, RESPECTIVELY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' THE RESULTS ARE COLLECTED FROM PUBLISHED LITERATURE [43, 1, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' THE LOWER BLOCK SHOWS OUR IMPLEMENTATIONS ON THE REPRODUCED BERT AND ELECTRA BASELINES FOLLOWING THE OFFICIAL IMPLEMENTATIONS IN THE MUTUAL DATASET PAPER [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Model R4@1 R4@2 MRR TF-IDF [43] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='542 Dual LSTM [43] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='260 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='743 SMN [12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='595 DAM [16] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='518 GPT-2 [73] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='584 GPT-2-FT [73] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='629 BERT [24] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='795 RoBERTa [25] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='836 RoBERTa + OCN [69] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='926 ALBERT [74] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='916 GRN-v2 [74] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='954 BERT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='803 + CDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='823 ELECTRA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='946 + CDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='956 revealing the strength of BiGRU in modeling the sentence- level transition between the turns of utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Number of Decoupling Layers Inspired by the stacked manner of Transformer [75], we have also performed a study on the effects of stacked decou- pling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The results are shown in Figure 3, where we can see only one layer is enough and better than deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' This can be explained that a deeper decoupling module is harder to learn, and only one interaction is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Number of BiGRU Layers To see whether a deeper BiGRU will be beneficial to the modeling of a dialogue representation from the sequence of utterance representations, we have conducted experiments on the number of BiGRU layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' As shown in Figure 4, a deeper BiGRU causes a big drop on R@1 metric, showing that the TABLE IV ABLATION STUDY ON MUTUAL DEV SETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' “UA” MEANS UTTERANCE-AWARE, AND “SA” MEANS SPEAKER-AWARE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Model R4@1 R4@2 MRR ELECTRA + CDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='958 w/o SA-Mask 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='949 w/o UA-Mask 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='953 w/o BiGRU (w/ Max-Pool) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='951 w/o BiGRU (w/ Mean-Pool) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='952 TABLE V INFLUENCE OF FUSING METHODS ON THE MUTUAL DEV SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Fusing Method R4@1 R4@2 MRR ELECTRA + CDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='958 Gate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='952 Original Info 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='955 Gate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='910 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='951 shallow BiGRU is effective for capturing the information flow in the dialogue context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Effects of Fusing Methods As described in Section III-E, we use a gate to fuse the complementary information inside each channel to accumulate the information from the four decoupled channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A simple fully-connected layer can be applied as a more simple alterna- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' To explore whether it is necessary to use those gates, we perform a comparative study as results are presented in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We can see the gate mechanism yields better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='6 In addition, we find using the information from the original sequence representation E is also beneficial, which could serve as the reference for calculating the gating ratio to measure how similar is the decoupled representation with regards to the original representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 6It is observed that “-Gate” yields better results towards the R4@2 metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Since the results on this metric are quite high (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='981), they are only listed for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Following the existing literature [67], we focus on the more distinguishable metrics R1@2 and MRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 9 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='95 1 Decoupling Block Accuracy(%) R@1 R@2 MRR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Influence of the number of Decoupling Blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' TABLE VI INFLUENCE OF THE AGGREGATING METHODS ON THE MUTUAL DEV SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' CNN USES A FILTER OF SIZE 3, AND CNN-MULTI COMBINES FILTERS OF SIZE 2, 3, 4, WHICH IS SIMILAR TO [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Aggregating Method R@1 R@2 MRR Max-Pooling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='958 Mean-Pooling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='911 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='951 CNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='953 CNN-Multi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='946 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Effects of Aggregating Methods To aggregate the word representations in one utterance, we can use simple global pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' And another widely used method is Convolution Neural Network (CNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Deeper CNN can capture a wide scope of the receptive field, making it successful in Computer Vision [77] and Text Classification [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We have also compared different aggregating methods on the MuTual dev set as shown in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We can see that max- pooling is better than mean-pooling since it can preserve some activated signals, removing the disturbance of less important signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' However, the two CNN-based methods perform worse than max-pooling, especially those with multiple filter sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' This can be explained that shared filters between different sentences are not flexible enough and cannot generalize well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Effects of Underlying Pre-trained Models To test the generality of the benefits of our CDN to other PrLMs, we alter the underlying PrLMs to other variants in different sizes or types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' As shown in Table VII, we see that our CDN is generally effective for the widely-used PrLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Influence of Domain-adaptive Post-Training We are interested in the influence of the post-training ob- jectives (MLM and NSP) on overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Firstly, we compare the three kinds of masking strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', subword, span, and word level masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then, we evaluate the impact of the NSP objective to see if it works in the dialogue scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For the analysis, we use the Ubuntu dataset instead of the MuTual dataset because MuTual is a very small-scale dataset (only 8K context-response pairs) even with high quality, which is not appropriate for evaluating post-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Therefore, we 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='95 1 BiGRU Layer Accuracy(%) R@1 R@2 MRR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Influence of the number of BiGRU layers on MuTual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' TABLE VII PERFORMANCES OF SINGLE PRLM BASELINE AND WITH CDN ON THE MUTUAL DEV SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' PrLMs R@1 R@2 MRR Baseline +CDN Baseline +CDN Baseline +CDN BERTbase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='818 RoBERTabase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='846 ELECTRAbase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='893 BERTlarge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='844 RoBERTalarge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='914 ELECTRAlarge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='958 used Ubuntu which is widely used by existing studies (those reported in Table II), for assessing post-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Table VIII shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We see that the contribution of the masking methods does not vary too much, and the NSP objective boosts the performance remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Although NSP has been shown trivial in RoBERTa [25] during general- purpose pre-training, it yields surprising gains in dialogue sce- narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The most plausible reason is that dialogue emphasizes the relevance between dialogue context and the subsequent response, which shares a similar goal with NSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The findings indicate that the utterance-level language modeling would be effective for dialogue-related tasks, which would be a possible research topic in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' As pre-trained weights from the free-domain corpus are widely-used as the initialization, we are also interested in whether the pre-trained weights are dispensable when us- ing our domain-adaptive training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The comparison between general pre-training and domain-adaptive training without the weights from the general pre-training is shown in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We observe that training with the domain-aware corpus is much more efficient in our dialogue task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' With much less training data and computation cost, our method outperforms all the baselines without large-scale pre-training on the domain-free corpus in Table II, and even achieves better performance than various pre-trained methods such as BERT and TADAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Application Potentials In terms of applications, our model can be used to build the retrieval-based chatbot, and the model latency can be tolerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For our CDN model, the model latency mainly depends on the underlying PrLMs and batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Taking the MuTual dataset 10 TABLE VIII ABLATION STUDY OF DOMAIN-ADAPTIVE POST-TRAINING ON UBUNTU (ELECTRA-BASED CDN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Model R10@1 R10@2 R10@5 Subword 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='987 Span 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='988 WWM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='989 Subword + NSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='992 Span + NSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='992 WWM + NSP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='993 TABLE IX COMPARISON OF OF GENERAL PRE-TRAINING AND DOMAIN-ADAPTIVE TRAINING (WITHOUT THE WEIGHTS FROM THE GENERAL PRE-TRAINING).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Model Data Steps Batch Device R10@1 R10@2 R10@5 Vanilla 160GB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='75M TPU 2048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 Domain 340M 470K 64 GPU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='979 as an example, the inference phase is 2 min/epoch using our settings, where 886 instances are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Such velocity is supposed to be tolerated in real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' For real- world scenarios, our model can be applied to build a chatbot if responses can be first selected coarsely and then ranked by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Besides the retrieval-based chatbots, we have two ways to generalize CDN to generation-based dialogue systems as (1) re-ranking module (2) the encoder in the traditional encoder- decoder framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' A generation-based model can generate some candidate responses, then to choose the better response, our model can be used to rank the candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Another way is using the major parts of CDN (without the final classification layer) as an encoder to learn the utterance-aware and speaker- aware representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Then a decoder of advanced generation- based models can be utilized upon CDN to generate better responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' CONCLUSION In this paper, we propose a novel and simple Channel-aware Decoupling Network (CDN), which decouples the utterance- aware and speaker-aware information, tackling the problem of role transition and noisy distant texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Experiments on four retrieval-based multi-turn dialogue datasets show the superior- ity over existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' The ablation study of different sub- modules explains their effectiveness and relative importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' We find that domain-adaptive post-training also enhances the model performance substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Even on the backbone with post-training, CDN still yields consistent gains, showing that CDN is generally effective no matter how strong the backbone model is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Our work reveals a way to make better use of the semantically rich contextualized representations from pre- trained language models and gives insights on how to combine the traditional RNN models with powerful transformer-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' In the future, we will study dialogue-aware pre- training techniques and in-depth modeling of dialogue struc- tures such as discourse-aware parsing and abstract meaning representation for dialogue texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' REFERENCES [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} 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Proceedings, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Bengio and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' LeCun, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Zhuosheng Zhang received his Bachelor’s degree in internet of things from Wuhan University in 2016, his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' degree in computer science from Shanghai Jiao Tong University in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He is working towards the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' degree in computer science with the Center for Brain-like Computing and Machine Intelligence of Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He was an intern- ship research fellow at NICT from 2019-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' His research interests include natural language process- ing, machine reading comprehension, and dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Hai Zhao received the BEng degree in sensor and instrument engineering, and the MPhil degree in control theory and engineering from Yanshan University in 1999 and 2000, respectively, and the PhD degree in computer science from Shanghai Jiao Tong University, China in 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He is currently a full professor at department of computer science and engineering, Shanghai Jiao Tong University after he joined the university in 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He was a research fellow at the City University of Hong Kong from 2006 to 2009, a visiting scholar in Microsoft Re- search Asia in 2011, a visiting expert in NICT, Japan in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He is an ACM professional member, and served as area co-chair in ACL 2017 on Tagging, Chunking, Syntax and Parsing, (senior) area chairs in ACL 2018, 2019 on Phonology, Morphology and Word Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' His research interests include natural language processing and related machine learning, data mining and artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' Longxiang Liu received Bachelor’s degree in the department of computer science and engineering from Shanghai Jiao Tong University in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He is now a graduate student studying in the University of Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He is working to- wards the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' degree in computer science with the Natural Language Processing Group of Key Labora- tory of Intelligent Information Processing of Institute of Computing Technology in Chinese Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' He has done researches in Center for Brain-like Computing and Machine Intelligence of Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'} +page_content=' His research interests include natural language processing, machine reading comprehension and dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ztE2T4oBgHgl3EQfiAcw/content/2301.03953v1.pdf'}